The Diffiety Institute Preprint Series

Preprint DIPS–1/2009 March 26, 2009

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Partial Differential Hamiltonian Systems

by

L. Vitagliano

DIPS-1/2009

                                                       arXiv:0903.4528

Partial Differential Hamiltonian Systems

L. VITAGLIANO

Abstract. We define partial differential (PD in the following), i.e., field theoretic, analogues of hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD-Hamilton equations, PD-Noether theorem, PD-Poisson bracket, etc.. Unlike in standard multisymplectic approach to hamiltonian field theory, in our formalism the geometric structure (kinematics) and the dynamical information on the “phase space” appear as just different components of one single geometric object.

Introduction

First order lagrangian mechanics can be naturally generalized to higher order lagrangian field theory. Moreover, the latter can be presented in a very elegant and precise algebro-geometric fashion [1]. In particular, it is clear what are all the involved geometric structures (higher order jets, Cartan distribution, 𝒞 -spectral sequence, etc., [1, 2]). On the other hand it seems to be quite hard to understand what is the most “reasonable, unambiguous, higher order, field theoretic generalization” of hamiltonian mechanics on abstract symplectic manifolds. Actually, there exists a universally accepted generalization of the standard mechanical picture

lagrangian mechanics on T Q ⟹ hamiltonian mechanics on T^{∗}Q,

Q being a smooth manifold, to the picture

lagrangian field theory on J^{1}π ⟹ hamiltonian field theory on ℳ π,  (1)

π being a fiber bundle, J^{1}π its first jet space and ℳ π its multimomentum space [3] (see also [4] for a recent review). Picture (1) includes, in particular, a generalization of the Legendre transform. Along this path it has been discovered an analogous structure of the symplectic structure on T^{∗}Q, which has been named the multisymplectic structure of ℳ π (see, for instance, [5]). A whole literature exists about properties of such structure, which are generically referred to as multisymplectic geometry of ℳ π (see references in [4]). In particular, efforts were made to find multisymplectic analogues of all properties of T^{∗}Q (including, for instance, the Poisson bracket [6, 7, 8, 9, 10]). Now, it is natural to wonder if it is possible to reasonably further generalize in two different directions. The first one is towards a picture

lagrangian field theory on J^{∞}π ⟹ higher order hamiltonian field theory, (2)

2000 Mathematics Subject Classification. 53C80, 70S05.

Key words and phrases. Field Theory, Multisymplectic Geometry, Hamiltonian Systems.

J^{∞}π being the ∞th jet space of π, including a higher order generalization of the Legendre transform. There is no universally accepted answer about picture (2) (see, for instance, [11, 12, 13, 14, 15, 16, 17, 18] and references therein). Moreover, it seems that all the proposed answers are ambiguous in some aspects. Most often they involve the choice of some extra structure other than the natural ones on J^{∞}π. The second direction in which to generalize picture (1) can be illustrated as follows. T^{∗}Q is just a very special example of (pre)symplectic manifold. Actually hamiltonian mechanics can be (and should, in some cases [19]) formulated on abstract (pre)symplectic manifolds. Similarly, it is natural to wonder if there exists the concept of abstract multi(pre)symplectic manifolds in such a way that hamiltonian field theory could be reasonably formulated on them. In literature they can be found some proposals of should be abstract multi(pre)symplectic manifolds (see, for instance, [20, 21]). In particular, definitions have been given in such a way to be able to prove multisymplectic analogues of the celebrated Darboux lemma [22, 23]. The recent definitions by Forger and Gomes [23] appear to be the most satisfactory in that they are “minimal” on one side and duly model in an abstract fashion the relevant geometric properties of ℳ π on the other side. In their work Forger and Gomes illustrate, in particular, the role played by fiber bundles in the should be definition of multi(pre)symplectic structure. The next step forward should be to formulate hamiltonian field theory on multisymplectic bundles. In this paper we present our own proposal about what should it be an abstract, first order, hamiltonian field theory. We call such proposal the theory of partial differential (PD in the following) hamiltonian systems so to 1) stress that it is a natural generalization of the theory of hamiltonian systems on abstract symplectic manifolds,

2) distinguish it from the special case of hamiltonian field theory on ℳ π. A PDhamiltonian system encompasses both the kinematics (encoded, in picture (1), by the multisymplectic structure in ℳ π) and the dynamics (encoded, in picture (1), by the so called hamiltonian section [4]) which appear as just different components of one single geometric object. Namely, the main difference between a PD-hamiltonian system and a multi(pre)symplectic structure (whatever the reader understand for this) is the dynamical content of the former (as opposed to the just kinematical one of the latter). The paper is divided into 6 sections. In Section 1 we collect our notations and conventions and recall basic differential geometric facts that will be used in the main part of the paper. In Sections 2 we define what we call affine forms on fiber bundles. The introduction of affine forms can be motivated as follows. Trajectories in hamiltonian mechanics are curves, whose 1st derivative at a point is naturally understood as a tangent vector. In their turn, tangent vectors can be inserted into differential forms and, in particular, a symplectic one, and Hamilton equations can be written in terms of such insertion. Trajectories in field theory are sections of a fiber bundle α : P ⟶ M, whose 1st derivative at a point is naturally understood as a point in J^{1}α. In their turn, points of J^{1}α can be inserted into affine forms and, in particular, a PD-hamiltonian system (see Section 4), and PD-Hamilton equations can be written in terms of such insertion. Recall now that the natural projection J^{1}α ⟶ P is an affine bundle whose sections are naturally interpreted as (Ehresmann) connections in α. Thus, connections and affine geometry play a prominent role in the theory of PD-hamiltonian systems. The affine geometry is hidden in standard hamiltonian mechanics by an a priori choice of the parametrization of the time axis (see [24, 25], and references therein, for the role of affine geometry in theoretical mechanics). Similarly, even if the role of connections in field theory has been often recognized (see, for instance, [26, 27]) their affine geometry is sometimes hidden in hamiltonian field theory on ℳ π by the use of multivectors, or even decomposable ones [28, 29] (which is just a multidimensional analogue of choosing a parameterization of time). Actually, we show in Subsection 2.3 that affine forms can be understood as standard differential forms of a special kind. Nevertheless, we prefer to keep the distinction for foundational reasons. In Section 3 we discuss standard operations with affine forms. Essentially because of the interpretation of affine forms as standard differential forms we mentioned above, some of this operations (for instance, the insertion of a connection into an affine form [30], or the differential of an affine form) were actually already defined in the literature, or can be understood as standard operations with forms. We stress again that we will keep the distinction. Finally, we discuss relevant affine form cohomologies proving an affine form version of the Poincaré lemma. In Section 4 we introduce PD-(pre)hamiltonian systems, discuss their geometry and the geometry of the associated PD-Hamilton equations, with some references to the singular, constrained case (see [31, 32, 33, 34] for an account of the constraint algorithm in first order field theory). For completeness, we also relate PD-hamiltonian systems to multi(pre)symplectic structures à là Forger [23] and the calculus of variations. In Section 5 we introduce PD-Noether symmetries and currents of a PD-hamiltonian system. In view of the dynamical content of the latter we are able to prove a Noether theorem (see also [35]). Moreover, there is a natural Lie bracket (named PD-Poisson bracket) among PD-Noether currents. As already mentioned, multisymplectic analogues of the Poisson bracket have been already discussed in the literature [6, 7, 8, 9, 10]. However, we emphasise here the dynamical nature of PD-Poisson bracket (see also [36, 37]). Namely, such bracket is just part of the Peierls bracket [37] among conservation laws of the underlying lagrangian theory and we don’t try to extend it to non-conserved currents. Indeed, our opinion is that the existence of a Poisson bracket among non-conserved functions in hamiltonian mechanics is essentially due to the existence of a preferred hamiltonian system on any symplectic manifold N, i.e., the one with 0 hamiltonian, for which every function on N is a conservation law. Finally we discuss the (gauge) reduction of a degenerate (but unconstrained) PD-hamiltonian system. In Section 6 we propose few examples of PD-hamiltonian systems, including the computation of their PD-Noether symmetries and currents or, in one case, their reduction.

1. Notations and Conventions

In this section we collect notations and conventions about some general constructions that will be used in the following. Let N be a manifold. We denote by C^{∞}(N) the ℝ-algebra of smooth, ℝ-valued functions on N. A vector field X over N will be always understood as a derivation X : C^{∞}(N) ⟶ C^{∞}(N). We denote by D(N) the C^{∞}(N)-module of vector fields over N, by Λ(M) = ⨁_{k} Λ^{k}(N) the graded ℝ-algebra of differential forms over N, by d_{N} : Λ(N) ⟶ Λ(N) (or, simply, d : Λ(N) ⟶ Λ(N) if this does not lead to confusion) the de Rham differential and by H(N) = ⨁_{k} H^{k}(N) the de Rham cohomology. If F : N_{1} ⟶ N is a smooth map of manifolds, we denote by F^{∗} : Λ(N) ⟶ Λ(N_{1}) its pull-back. We will understand everywhere the wedge product ∧ of differential forms, i.e., for ω, ω_{1} ∈ Λ(N), instead of writing ω ∧ ω_{1}, we will simply write ωω_{1}. We assume the reader to be familiar with Frölicher-Nijenhuis calculus on form valued vector fields (insertion i_{Z}ω of a form valued vector field Z into a differential form ω, Lie derivative L_{Z}ω of a differential form ω along a form valued vector fields Z, Frölicher-Nijenhuis bracket, etc.). Let ϖ : W ⟶ N be an affine bundle (or, possibly, a vector bundle) and F : N_{1} ⟶ N a smooth map of manifolds. The affine space of smooth sections of ϖ will be denoted by Γ(ϖ). For x ∈ N, we put, sometimes, Γ(ϖ)|_{x} := ϖ^{−1}(x) and, for χ ∈ Γ(ϖ), we also put χ_{x} := χ(x). The affine bundle on N_{1} induced by ϖ via F will be denote by ϖ|_{F} : W |_{F} ⟶ N:

\xymatrix{W |_{F} \ar[r] \ar[d]_{ϖ|_{F}} & W \ar[d]^{ϖ} \\ N_{1} \ar[r]^{F} & N} .

We also denote Γ(ϖ)|_{F} := Γ(ϖ|_{F}). For any section s ∈ Γ(ϖ) there exists a unique section, which abusing the notation we denote by s|_{F} ∈ Γ(ϖ|_{F}), such that the diagram

\xymatrix{W |_{F} \ar[r] & W \\ N_{1} \ar[r]^{F} \ar[u]^{s|_{F}} & N \ar[u]_{s}}

commutes. Elements in Γ(ϖ)|_{F} are called sections of ϖ along F. If F is an embedding ϖ|_{F}, Γ(ϖ)|_{F} and s|_{F} will be referred to as the restriction to N_{1} of ϖ, Γ(ϖ) and s, respectively. If ϖ_{1} : W_{1} ⟶ N is an other affine bundle and A : Γ(ϖ) ⟶ Γ(ϖ_{1}) is an affine map then there exists a unique affine map A|_{F} : Γ(ϖ)|_{F} ⟶ Γ(ϖ_{1})|_{F} such that A|_{F} (s|_{F}) = A(s)|_{F} for all s ∈ Γ(ϖ). Let α : P ⟶ M be a fiber bundle. A vector field X ∈ D(P) is called α-projectable iff there exists X̌ ∈ D(P) such that X ∘ α^{∗} = α^{∗} ∘ X̌. X̌ is called the α-projection of X. α-projectable vector fields form a Lie subalgebra in D(P) denoted by D_{V} (P, α) (or simply D_{V} if this does not lead to confusion). An α-projectable vector field projecting onto the 0 vector field is a α-vertical vector field. α-vertical vector fields form an ideal in D_{V} denoted by V D(P, α) (or simply V D). Notice that, if α has connected fiber, then D_{V} is the stabilyzer of V D in D(P), i.e., D_{V} = {X ∈ D(P) | [X, V D] ⊂ V D}. Let α : P ⟶ M be as above, dim M = n, dim P = m + n. Denote by α_{1} : J^{1}α ⟶ M the bundle of 1-jets of local sections of α [38, 2], and by α_{1,0} : J^{1}α ⟶ P the canonical projection. For any local section σ : U ⟶ P of α, U ⊂ M being an open subset, we denote by σ̇ : U ⟶ J^{1}α its 1th jet prolongation. Any system of adapted to α coordinates (…, x^{i}, …, y^{a}, …) on P, …, x^{i}, … being coordinates on M and …, y^{a}, … fiber coordinates on P, gives rise to the system of jet coordinates (…, x^{i}, …, u^{a}, …, y_{i}^{a}, …) on J^{1}α, i = 1, …, n, a = 1, …, m. Recall that α_{1,0} is an affine bundle and a section ∇ : P ⟶ J^{1}α of it is naturally interpreted as a (Ehresmann) connection in α. We assume the reader to be familiar with the geometry of connections (see, for instance, [39, 40]). ∇ is locally represented as

∇ : y_{i}^{a} = ∇_{i}^{a},

with ∇_{i}^{a} = ∇_{i}^{a}(x^{1}, …, x^{n}, y^{1}, …, y^{m}), i = 1, …, n, a = 1, …, m. The space Γ(α_{1,0}) of all such sections will be also denoted by C(P, α) (or simply C). Let α : P ⟶ M be as above, α′ : P′ ⟶ M another fiber bundle and G : P ⟶ P′ a bundle morphism (over the identity id_{M} : M ⟶ M), i.e., a smooth map such that α′ ∘ G = α. First of all, recall that there exists a unique bundle morphism j^{1}G : J^{1}α ⟶ J^{1}α′ such that j^{1}G ∘ σ̇ = (G ∘ σ)˙ for all local sections σ of α. j^{1}G is the first jet prolongation of G and diagram

\xymatrix{J^{1}α \ar[rr]^{j^{1}G} \ar[d]_{α_{1,0}} &  & J^{1}α′ \ar[d]^{α′_{1,0}} \\ P \ar[rr]^{G} \ar[dr]_{α} &  & P′ \ar[dl]^{α′} \\  & M}

commutes. Now, a connection ∇ ∈ C(P, α) and a connection ∇′ ∈ C(P′, α′) are said G-compatible iff ∇′ ∘ G = j^{1}G ∘ ∇. Let

\xymatrix{⋯ \ar[r] & K_{l−1} \ar[r]^{δ_{l−1}} & K_{l} \ar[r]^{δ_{l}} & K_{l+1} \ar[r]^{δ_{l+1}} & ⋯}

be a complex. Put K := ⨁_{l} K_{l} and δ := ⨁_{l} δ_{l}. We denote by H(K, δ) := ⨁_{l} H^{l}(K, δ), the cohomology space of (K, δ), H^{l}(K, δ) := ker δ_{l}/ im δ_{l−1}. Let A be a commutative ℝ-algebra, M, M_{1} be A-modules and A an affine space modelled over M. We denote by Aff_{A}(A, M_{1}) (resp. Hom_{A}(M, M_{1})) the A-module of affine (resp. A-linear) maps A ⟶ M_{1} (resp. M ⟶ M_{1}). If ϕ ∈ Aff_{A}(A, M_{1}), its linear part \underline{ϕ} is an element in Hom_{A}(M, M_{1}).

Let m, r be positive integers and …, A_{a_{1}···a_{r}}, … be elements in a real vector space, a_{1}, …, a_{r} = 1, …, m. We denote by …, A_{[a_{1}···a_{r}]}, … their skew-symmetrization, i.e.,

A_{[a_{1}···a_{r}]} := \frac{1}{s!} ∑_{σ∈S_{r}} ε(σ)A_{a_{σ(1)}···a_{σ(r)}},

S_{r} being the group of permutations of {1, …, r} and ε(σ) the sign of σ ∈ S_{r}. We denote by ≃ (resp. ≈) a canonical (resp. non-canonical) isomorphism between algebraic structures and by ≡ an equivalence of notations. For instance, for α : P ⟶ M as above, V D ≡ V D(P, M). Finally, we understand the sum over upper-lower pairs of repeated indexes.

2. Affine Forms on Fiber Bundles

2.1. Special Forms on Fiber Bundles. Let α : P ⟶ M be a fiber bundle, A := C^{∞}(P), A_{0} := C^{∞}(M), x^{1}, …, x^{n} coordinates on M, dim M = n, and y^{1}, …, y^{m} fiber coordinates on P, dim P = n + m. In the following we will often understand the monomorphism of algebras α^{∗} : A_{0} ⟶ A, whose image is made of functions on P which are constant along the fibers of α. D_{V} (resp. V D) is made of vector fields X locally in the form X = X^{i}∂_{i} + Y^{a}∂_{a} (resp. X = Y^{a}∂_{a}) where X^{i} = X^{i}(x^{1}, …, x^{n}), ∂_{i} := ∂/∂x^{i}, i = 1, …, n, ∂_{a} = ∂/∂y^{a}, a = 1, …, m. Denote by Λ_{1}(P, α) = ⨁_{k} Λ_{1}^{k}(P, α) (or simply Λ_{1} = ⨁_{k} Λ_{1}^{k}) the differential (graded) ideal of differential forms on P vanishing when restricted to fibers of α, i.e., ω ∈ Λ_{1}^{k}, k ≥ 0 iff ω ∈ Λ^{k}(P) and i_{α^{−1}(x)}^{∗}(ω) = 0 for all x ∈ M, i_{α^{−1}(x)} : α^{−1}(x) ⟶ P being the embedding of the fiber α^{−1}(x) of α through x ∈ M. Moreover, denote by Λ_{p}(P, a) = ⨁_{k} Λ_{p}^{k}(P, α) (or simply Λ_{p} = ⨁_{k} Λ_{p}^{k}) the p-th exterior power of Λ_{1}. For all k and p, Λ_{p}^{k} is made of differential k-forms ω such that (i_{Y_{1}} ∘ ⋯ ∘ i_{Y_{k−p+1}})ω = 0 for every Y_{1}, …, Y_{k−p+1} ∈ V D or, which is the same, differential k-forms ω locally in the form

ω = ∑_{l≥0} ω_{i_{1}···i_{p+l}a_{1}···a_{k−p−l}}dx^{i_{1}} ⋯ dx^{i_{p+l}}dy^{a_{1}} ⋯ dy^{a_{k−p−l}},

…, ω_{i_{1}···i_{p+l}a_{1}···a_{k−p−l}}, … being local functions on P, i_{1}, …, i_{p+l} = 1, …, n, a_{1}, …, a_{k−p−l} = 1, …, m. Denote by VΛ(P, α) = ⨁_{k} VΛ^{k}(P, α) (or simply VΛ = ⨁_{k} VΛ^{k}) the quotient differential algebra Λ(P)/Λ_{1}, with d^{V} : VΛ ⟶ VΛ its differential and with p^{V} : Λ(P) ∋ ω ⟼ ω^{V} := ω + Λ_{1} ∈ VΛ the projection onto the quotient. Notice that d^{V} is A_{0}-linear. An element ρ^{V} in VΛ^{k} is locally in the form

ρ^{V} = ρ_{a_{1}···a_{k}}d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k}},

…, ρ_{a_{1}···a_{k}}, … being local functions on P, and d^{V}ρ^{V} is locally given by

d^{V}ρ^{V} = ∂_{a}ρ_{a_{1}···a_{k}}d^{V}y^{a}d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k}} = ∂_{[a}ρ_{a_{1}···a_{k}]}d^{V}y^{a}d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k}}.

Clearly, VΛ^{1} is the dual A-module of V D and VΛ its exterior algebra. In particular, elements in VΛ may be interpreted as multilinear, skew-symmetric forms on V D.

Denote by \overline{Λ}(P, α) = ⨁_{k} \overline{Λ}^{k}(P, α) := ⨁_{k} Λ_{k}^{k} ⊂ Λ(P) (or simply \overline{Λ} = ⨁_{k} \overline{Λ}^{k}) the sub-algebra generated by Λ_{1}^{1}. An element ω ∈ \overline{Λ}^{k} is locally in the form

ω = ω_{i_{1}···i_{k}}dx^{i_{1}} ⋯ dx^{i_{k}}.

Notice that \overline{Λ} is naturally isomorphic to A ⊗_{A_{0}} Λ(M) as an A-algebra. For any p, the quotient (graded) differential module E_{0}^{p,•} ≡ E_{0}^{p,•}(P, α) := Λ_{p}/Λ_{p+1}^{1} is naturally isomorphic to VΛ ⊗_{A} \overline{Λ}^{p} (or, which is the same, VΛ ⊗_{A_{0}} Λ^{p}(M)) via the correspondence

E_{0}^{p,q} ∋ ω + Λ_{p+1}^{p+q} ⟼ ϖ ∈ VΛ^{q} ⊗_{A} \overline{Λ}^{p},  (3)

well defined by putting

ϖ(Y_{1}, …, Y_{q}) := (i_{Y_{q}} ∘ ⋯ ∘ i_{Y_{1}})(ω) ∈ \overline{Λ}^{p},

Y_{1}, …, Y_{q} ∈ V D, E_{0}^{p,q} being the qth homogeneous piece of E_{0}^{p,•}, q ∈ ℤ. In particular, VΛ ⊗_{A} \overline{Λ} (or, which is the same, VΛ ⊗_{A_{0}} Λ(M)) is the graded object associated with the filtration Λ(P) ⊃ Λ_{1} ⊃ ⋯ ⊃ Λ_{p} ⊃ · · ·. As we will see in the next subsection, a connection in α allows one to identify the filtration with its graded object. Let us now focus on the ideals Λ_{n−1} and Λ_{n}. Put d^{n}x := dx^{1} ⋯ dx^{n} and d^{n−1}x_{i} := i_{∂_{i}}d^{n}x, so that dx^{j}d^{n−1}x_{i} = δ_{i}^{j}d^{n}x, i, j = 1, …, n. Then an element ω ∈ Λ_{n−1}^{q+n−1} (resp. ω ∈ Λ_{n}^{q+n−1}) is locally in the form

ω = ω_{a_{1}...a_{q}}^{i}dy^{a_{1}} ⋯ dy^{a_{q}}d^{n−1}x_{i} + ω_{a_{1}...a_{q−1}}dy^{a_{1}} ⋯ dy^{a_{q−1}}d^{n}x

(resp.

ω = ω_{a_{1}...a_{q−1}}dy^{a_{1}} ⋯ dy^{a_{q−1}}d^{n}x),

…, ω_{a_{1}...a_{q}}^{i}, …, ω_{a_{1}...a_{q−1}}, … being local functions on P. In particular Λ_{n−1}^{q+n−1} (resp. Λ_{n}^{q+n−1}) is the module of sections of an [n\binom{q}{m} + \binom{q−1}{m}](resp. \binom{q−1}{m})-dimensional vector bundle over P. In few lines we will provide an alternative description of Λ_{n−1} and Λ_{n} (Theorem 1). In our opinion, such description is more suitable for a better understanding of the role of Λ_{n−1} and Λ_{n} in first order field theories (see, for instance, [5]).

2.2. Affine Forms. Let ∇ ∈ C(P, α). Recall, preliminarily, that C is an affine space modelled over the A-module \overline{Λ}^{1} ⊗_{A} V D, or, which is the same, Λ^{1}(M)⊗_{A_{0}} V D. ∇ allows one to split the tangent bundle T P to P into its vertical part VP and a horizontal part H_{∇}P. denote by H_{∇}D(P, α) ⊂ D(P) (or simply H_{∇}D ⊂ D(P)) the submodule of ∇- horizontal vector fields. An element X ∈ H_{∇}D is locally in the form X = X^{i}∇_{i}, where ∇_{i} := ∂_{i} + ∇_{i}^{a}∂_{a}, i = 1, …, n. Splitting

D(P) = V D ⊕ H_{∇}D  (4)

determines a splitting of the de Rham differential d : Λ(P) ⟶ Λ(P) into a horizontal part d_{∇} : Λ(P) ⟶ Λ(P), and a vertical part d_{∇}^{V} : Λ(P) ⟶ Λ(P), d = d_{∇} + d_{∇}^{V}, where d_{∇} (resp. d_{∇}^{V}) is the Lie derivative along the horizontal-form valued vector field (resp. the form valued vertical vector field) H_{∇} : A ⟶ \overline{Λ}^{1}(P) (resp. V_{∇} : A ⟶ Λ^{1}(P)) determined by ∇. H_{∇} (resp. V_{∇}) is locally given by H_{∇} = dx^{i}∇_{i} (resp. V_{∇} = (dy^{a} − ∇_{i}^{a}dx^{i})∂_{a}). Notice that (Λ(P), d_{∇}^{V}, d_{∇}) is not a bi-complex unless ∇ is flat. Splitting (4) also determines an isomorphism ϕ_{∇} : VΛ ⊗_{A} \overline{Λ} ⟶ Λ(P) locally given by

1This last notation is motivated by the fact that A–modules E_{0}^{p,•} are columns of the first term of the (cohomological) Leray-Serre spectral sequence of the fiber bundle α (see [mvv??]).

ϕ_{∇}(d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{q}} ⊗ dx^{i_{1}} ⋯ dx^{i_{q}}) = d_{∇}^{V}y^{a_{1}} ⋯ d_{∇}^{V}y^{a_{q}}dx^{i_{1}} ⋯ dx^{i_{q}}.

In particular, for any q, p, there is an obvious projection 𝔭_{∇}^{q,p} : Λ(P) ⟶ VΛ^{q} ⊗_{A} \overline{Λ}^{p}. For any k ≥ 0 put ′Ω^{k+1} := Aff_{A}(C, VΛ^{k} ⊗_{A} \overline{Λ}^{n}). An element ′ϑ ∈ ′Ω^{k+1} is locally given by

′ϑ(∇) = (′ϑ_{a,a_{1}···a_{k}}^{i}∇_{i}^{a} +′ϑ_{a_{1}···a_{k}})d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k}} ⊗ d^{n}x, ∇ ∈ C,

…,′ϑ_{a,a_{1}···a_{k}}^{i}, …,′ϑ_{a_{1}···a_{k}}, … local functions on P. The linear part ′\underline{ϑ} of an element ′ϑ ∈ ′Ω^{k+1} is an element in the A-module

Hom_{A}(\overline{Λ}^{1} ⊗_{A} V D, VΛ^{k} ⊗_{A} \overline{Λ}^{n}) ≃ Hom_{A}(V D, VΛ^{k} ⊗_{A} \overline{Λ}^{n−1})
⊃ VΛ^{k+1} ⊗_{A} \overline{Λ}^{n−1}
=: \underline{Ω}^{k+1}(P, α)
≡ \underline{Ω}^{k+1},

where we identified VΛ^{k} ⊗ \overline{Λ}^{n−1} and Hom_{A}(\overline{Λ}^{1}, VΛ^{k} ⊗ \overline{Λ}^{n}) via the isomorphism

VΛ^{k} ⊗ \overline{Λ}^{n−1} ∋ σ ⊗ ρ ⟼ φ_{σ⊗ρ} ∈ Hom_{A}(\overline{Λ}^{1}, VΛ^{k} ⊗ \overline{Λ}^{n}),

σ ∈ VΛ^{k}, ρ ∈ \overline{Λ}^{n−1}, defined by putting

φ_{σ⊗ρ}(η) := (−)^{k}σ ⊗ (η · ρ) ∈ VΛ^{k} ⊗ \overline{Λ}^{n}, η ∈ \overline{Λ}^{1},

and embedded VΛ^{k+1} ⊗_{A} \overline{Λ}^{n−1} into Hom_{A}(V D, VΛ^{k} ⊗_{A} \overline{Λ}^{n−1}) via the correspondence

VΛ^{k+1} ⊗_{A} \overline{Λ}^{n−1} ∋ σ′ ⊗ ρ ⟼ φ′_{σ′⊗ρ} ∈ Hom_{A}(V D, VΛ^{k} ⊗_{A} \overline{Λ}^{n−1}),

σ′ ∈ VΛ^{k}, ρ ∈ \overline{Λ}^{n−1}, defined by putting

φ′_{σ′⊗ρ}(Y) := i_{Y} σ′ ⊗ ρ ∈ VΛ^{k} ⊗_{A} \overline{Λ}^{n−1}, Y ∈ V D.

Put also Ω^{0} ≡ Ω^{0}(P, α) := \overline{Λ}^{n−1}, \underline{Ω}^{0} ≡ \underline{Ω}^{0}(P, α) := Ω^{0}, and for k ≥ 0,

Ω^{k+1} ≡ Ω^{k+1}(P, α) := \{ϑ ∈ ′Ω^{k+1} | \underline{ϑ} ∈ \underline{Ω}^{k+1}\},

Ω ≡ Ω(P, α) := ⨁_{q≥0} Ω^{q} and \underline{Ω} ≡ \underline{Ω}(P, α) := ⨁_{q≥0} \underline{Ω}^{q}. Elements in Ω^{k} will be called affine k-forms over α. It is easy to show that an element ϑ ∈ ′Ω^{k+1} is an affine (k + 1)- form iff it is locally given by

ϑ(∇) = (ϑ_{aa_{1}···a_{k}}^{i}∇_{i}^{a} + ϑ_{a_{1}···a_{k}})d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k}} ⊗ d^{n}x, ∇ ∈ C.

…, ϑ_{aa_{1}···a_{k}}^{i}, …, ϑ_{a_{1}···a_{k}}, … being local functions on P such that ϑ_{aa_{1}···a_{k}}^{i} = ϑ_{[aa_{1}···a_{k}]}^{i}, i = 1, …, n, a, a_{1}, …, a_{k} = 1, …, m.

According to the above said, the linear part \underline{ϑ} ∈ \underline{Ω}^{k+1} of ϑ is implicitly defined by the formula

ϑ(∇ + η ⊗ Y)(Y_{1}, …, Y_{k}) − ϑ(∇)(Y_{1}, …, Y_{k}) = (−)^{k}η · \underline{ϑ}(Y, Y_{1}, …, Y_{k}) ∈ \overline{Λ}^{n},

∇ ∈ C, η ∈ \overline{Λ}^{n−1}, Y, Y_{1}, …, Y_{k} ∈ V D, and it is locally given by

\underline{ϑ} = \frac{(−)^{k}}{k+1} ϑ_{a_{1}···a_{k+1}}^{i}d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{k+1}} ⊗ d^{n−1}x_{i}.  (5)

2.3. Affine Forms and Differential Forms. Let Ω_{0}(P, α) = ⨁_{q≥0} Ω_{0}^{q}(P, α) (or, simply, Ω_{0} ≡ ⨁_{q≥0} Ω_{0}^{q}) be the kernel of the projection Ω ∋ ϑ ⟼ \underline{ϑ} ∈ \underline{Ω}. Clearly, Ω_{0}^{q} is canonically isomorphic to V^{q−1}Λ ⊗_{A} \overline{Λ}^{n} for q > 0 (in the following we will understand such isomorphism), while Ω_{0}^{0} = 0. Moreover, Ω^{q} (resp. Ω_{0}^{q}) is the module of sections of an [n\binom{q}{m} + \binom{q−1}{m}] (resp. \binom{q−1}{m})-dimensional vector bundle over P.

Theorem 1. There are canonical isomorphisms of A-modules

ι_{0,q} : Λ_{n}^{q+n−1} ⟶ Ω_{0}^{q},
ι_{q} : Λ_{n−1}^{q+n−1} ⟶ Ω^{q},
\underline{ι}_{q} : E_{0}^{n−1,q} ⟶ \underline{Ω}^{q},

q ≥ 0, such that diagram

\xymatrix{0 \ar[r] & Λ_{n} \ar[d]_{ι_{0}} \ar[r] & Λ_{n−1} \ar[d]_{ι} \ar[r] & E_{0}^{n−1} \ar[d]_{\underline{ι}} \ar[r] & 0 \\ 0 \ar[r] & Ω_{0} \ar[r] & Ω \ar[r] & \underline{Ω} \ar[r] & 0}  (6)

commutes, where ι_{0} := ⨁_{q} ι_{0,q}, ι := ⨁_{q} ι_{q} and \underline{ι} := ⨁_{q} \underline{ι}_{q}.

Proof. Let q > 0. First of all, denote by \underline{ι}_{q} : E_{0}^{n−1,q} ⟶ \underline{Ω}^{q} the already mentioned natural isomorphism (3) and notice that for any ω ∈ Λ_{n}^{q+n−1} and Y_{1}, …, Y_{q−1} ∈ V D, (i_{Y_{1}} ∘ ⋯ ∘ i_{Y_{q−1}})(ω) ∈ \overline{Λ}^{n}. Therefore, it is well defined an element ι_{0,q}(ω) ∈ Ω_{0}^{q} by putting ι_{0,q}(ω)(Y_{1}, …, Y_{q−1}) := (i_{Y_{1}} ∘⋯∘i_{Y_{q−1}})(ω) ∈ \overline{Λ}^{n}, Y_{1}, …, Y_{q−1} ∈ V D. Moreover, the correspondence Λ_{n}^{q+n−1} ∋ ω ⟼ ι_{0,q}(ω) ∈ Ω_{0}^{q} is an isomorphism of A-modules. Indeed, let ω ∈ Λ_{n}^{q+n−1} and (i_{Y_{1}} ∘ ⋯ ∘ i_{Y_{q−1}})(ω) = 0 for all Y_{1}, …, Y_{q−1} ∈ V D, then ω ∈ Λ_{n+1}^{q+n−1} = 0, so that ι_{0,q} is injective. Moreover, Λ_{n}^{q+n−1} and Ω_{0}^{q} are locally free A-modules of the same local dimension. We conclude that

ι_{0} := _{⨁q}ι_{0,q} : Λ_{n} ⟶ Ω_{0}

is a canonical isomorphism of A-modules as well. Moreover, it sends Λ_{n}^{q+n−1} into Ω_{0}^{q}, q ≥ 0. Finally, if ω ∈ Λ_{n}^{q+n−1} is locally given by

ω = ω_{a_{1}...a_{q−1}}dy^{a_{1}} ⋯ dy^{a_{q−1}}d^{n}x,

then ι_{0}(ω) ∈ Ω_{0} is locally given by ι_{0}(ω) = ω_{a_{1}...a_{q−1}}d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{q−1}} ⊗ d^{n}x.

Now, for ω ∈ Λ_{n−1}^{q+n−1} and ∇ ∈ C put

ι_{q}(ω)(∇) := 𝔭_{∇}^{q−1,n}(ω).

If ω is locally given by

ω = ω_{a_{1}...a_{q}}^{i}dy^{a_{1}} ⋯ dy^{a_{q}}d^{n−1}x_{i} + ω_{a_{1}...a_{q−1}}dy^{a_{1}} ⋯ dy^{a_{q−1}}d^{n}x,

then

ω = ω_{a_{1}...a_{q}}^{i}(d_{∇}^{V} + d_{∇})(y^{a_{1}}) ⋯ (d_{∇}^{V} + d_{∇})(y^{a_{q}})d^{n−1}x_{i}

+ ω_{a_{1}...a_{q−1}}d_{∇}^{V}y^{a_{1}} ⋯ d_{∇}^{V}y^{a_{q−1}}d^{n}x

= ω_{a_{1}...a_{q}}^{i}d_{∇}^{V}y^{a_{1}} ⋯ d_{∇}^{V}y^{a_{q}}d^{n−1}x_{i}

+ ∑_{s}(−)^{p−s}ω_{a_{1}...a_{q}}^{i}∇_{j}^{a_{s}}d_{∇}^{V}y^{a_{1}} ⋯ { \atop d_{∇}^{V}y^{a_{s}}} ⋯ d_{∇}^{V}y^{a_{q}}dx^{j}d^{n−1}x_{i}

+ ω_{a_{1}...a_{q−1}}d_{∇}^{V}y^{a_{1}} ⋯ d_{∇}^{V}y^{a_{q−1}}d^{n}x

= ω^{q,n−1} + [q(−)^{q−1}ω_{aa_{1}...a_{q−1}}^{i}∇_{i}^{a} + ω_{a_{1}...a_{q−1}}]d_{∇}^{V}y^{a_{1}} ⋯ d_{∇}^{V}y^{a_{q−1}}d^{n}x

where a cap “ ” denotes omission of the factor below it, and ω^{q,n−1} ∈ Λ(P) is a suitable form such that 𝔭_{∇}^{q−1,n}(ω^{q,n−1}) = 0. Therefore, locally

ι_{q}(ω)(∇) = 𝔭_{∇}^{q−1,n}(ω)
= [q(−)^{q−1}ω_{aa_{1}...a_{q−1}}^{i}∇_{i}^{a} + ω_{a_{1}...a_{q−1}}]d^{V}y^{a_{1}} ⋯ d^{V}y^{a_{q−1}} ⊗ d^{n}x.  (7)

This shows simultaneously that ι_{q}(ω) is affine, that it is in Ω^{q} and that ι_{q} is injective. Since Λ_{n−1}^{p+n} and Ω^{q} are locally free A-modules of the same local dimension, then the correspondence ι_{q} : Λ_{n−1}^{q+n−1} ∋ ω ⟼ ι_{q}(ω) ∈ Ω^{q} is an isomorphism. Commutativity of diagram (6) immediately follows from local formulas (5) and (7). □

Notice that isomorphism ι generalizes considerably the well known isomorphism Λ_{n−1}^{n} ≃ Aff_{A}(C, \overline{Λ}^{n}) [5]. Finally, let π : E ⟶ M be a fiber bundle and …, q^{A}, … fiber coordinates on E. Notice that Ω^{1}(E, π) (resp. \underline{Ω}^{1}(E, π)) is the C^{∞}(E)-module of sections of a vector bundle μ_{0}π : ℳ π ⟶ E (resp. τ_{0}^{†}π : J^{†}π ⟶ E). Recall that there is a distinguished element Θ in Ω^{1}(ℳ π, μπ) (resp. \underline{Θ} ∈ \underline{Ω}^{1}(J^{†}π, τ^{†}π)), the tautological one [5], which in standard coordinates

…, x^{i}, …, q^{A}, …, p_{A}^{i}, …, p

on ℳ π (resp. …, x^{i}, …, q^{A}, …, p_{A}^{i}, … on J^{†}α) is given by

Θ = p_{A}^{i}dq^{A}d^{n−1}x_{i} − pd^{n}x (resp. \underline{Θ} = p_{A}^{i}d^{V}q^{A} ⊗ d^{n−1}x_{i}).

3. Affine Form Calculus

3.1. Natural Operations with Affine Forms. In this section we derive the main formulas of calculus on affine forms. Such formulas will turn useful in generalizing proofs from the context of hamiltonian systems to the context of PD-hamiltonian systems (see Section 5). Let α : P ⟶ M be as in the previous section. Isomorphism ι (resp. ι_{0}, \underline{ι}) can be used to “transfer structures” from Λ_{n−1} (resp. Λ_{n}, E_{0}^{n−1}) to Ω (resp. Ω_{0}, \underline{Ω}) and back. As an instance, notice that Ω has got a natural structure of Λ(P)-module given by

λϑ := ι(λω),

λ ∈ Λ(P), ϑ = ι(ω) ∈ Ω, ω ∈ Λ_{n−1}. Moreover, Ω is generated by Ω^{0} as a Λ(P)-module. Similarly, Λ_{n} (resp. E_{0}^{n−1}) has a structure of VΛ-module given by

λ^{V} ω_{0} := ι_{0}^{−1}(λ^{V} ρ^{V} ⊗ ν) (resp. λ^{V} \underline{ω} := ι_{0}^{−1}(λ^{V} ρ^{V} ⊗ σ)),

λ^{V} ∈ VΛ, ω_{0} = ι_{0}^{−1}(ρ^{V} ⊗ ν) (resp. \underline{ω} = \underline{ι}^{−1}(ρ^{V} ⊗ σ)), ρ^{V} ∈ VΛ, ν ∈ \overline{Λ}^{n} (resp. σ ∈ \overline{Λ}^{n−1}), so that ρ^{V} ⊗ ν ∈ VΛ ⊗_{A} \overline{Λ}^{n} = Ω_{0} (resp. ρ^{V} ⊗ σ ∈ VΛ ⊗_{A} \overline{Λ}^{n−1} = \underline{Ω}). Clearly, Λ_{n} (resp. E_{0}^{n−1}) is generated by \overline{Λ}^{n} (resp. \overline{Λ}^{n−1}) as a VΛ-module. Finally, the presented structures are compatible in the sense that for ω_{0} ∈ Λ_{n}, ω ∈ Λ_{n−1} and λ ∈ Λ(P), we have

λ^{V} ω_{0} = λω_{0} and \underline{λω} = λ^{V} \underline{ω}.

As a last instance of how to use isomorphisms in (6) to transfer a structure from one space to the other we define the insertion of a connection ∇ ∈ C into a differential form ω ∈ Λ_{n} as follows

i_{∇}ω := ι_{0}^{−1}(ϑ(∇)) = (ι_{0}^{−1} ∘ 𝔭_{∇}^{q−1,n})(ω) ∈ Ω_{0},

ϑ = ι(ω). Notice that the just defined insertion of a connection in an element ω ∈ Λ_{n} has been already discussed in [30]. In the following we will always understand isomorphisms ι, ι_{0}, \underline{ι}. Notice that \underline{Ω} inherits many operations from Ω. Indeed, let ∇ ∈ C, Z ∈ \overline{Λ}^{1} ⊗_{A} V D ⊂ Λ(P) ⊗_{A} D(P), Y ∈ V D, X ∈ D_{V}, q ≥ 0. Then

• i_{Z}(Ω) ⊂ Ω_{0} and i_{Z}(Ω_{0}) = 0 so that it is well defined an operator, which, abusing the notation, we again denote by i_{Z} : \underline{Ω} ⟶ Ω_{0}, via the formula

i_{Z} \underline{ω} := i_{Z}ω ∈ Ω_{0},

ω ∈ Ω. Moreover, it is easy to show that

i_{Z} \underline{ω} = i_{∇+Z}ω − i_{∇}ω.

Finally, for Z = η ⊗ Y_{1}, and \underline{ω} = ρ^{V} ⊗ σ, η ∈ \overline{Λ}^{1}, Y_{1} ∈ V D, ρ^{V} ∈ VΛ^{q} and σ ∈ \overline{Λ}^{n−1}, we have

i_{Z} \underline{ω} = (−)^{q−1}i_{Y_{1}}ρ^{V} ⊗ ησ.

• i_{Y} (Ω) ⊂ Ω (resp. L_{X}(Ω) ⊂ Ω) and i_{Y} (Ω_{0}) ⊂ Ω_{0} (resp. L_{X}(Ω_{0}) ⊂ Ω_{0}) so that it is well defined the quotient map, which, abusing the notation, we again denote by i_{Y} : \underline{Ω} ⟶ \underline{Ω} (resp. L_{X} : \underline{Ω} ⟶ \underline{Ω}), via the formula

i_{Y} \underline{ω} := \underline{i_{Y} ω} ∈ \underline{Ω} (resp. L_{X} \underline{ω} = \underline{L_{X}ω} ∈ \underline{Ω}).

Finally, for \underline{ω} = ρ^{V} ⊗ σ, ρ^{V} ∈ VΛ^{q} and σ ∈ \overline{Λ}^{n−1}, we have

i_{Y} \underline{ω} = i_{Y} ρ^{V} ⊗ σ.

• d_{∇}(Ω) ⊂ Ω_{0} and d_{∇}(Ω_{0}) = 0 so that it is well defined an operator, which, abusing the notation, we again denote by d_{∇} : \underline{Ω} ⟶ Ω_{0}, via the formula

d_{∇}\underline{ω} := d_{∇}ω ∈ Ω_{0},

ω ∈ Ω.

Remark 2. Notice that the insertion i_{∇}ω, being affine in ∇, is actually point wise, i.e., if ∇′ ∈ C is such that ∇′_{y} = ∇_{y} ∈ C|_{y} = α_{1,0}^{−1}(y) for some y ∈ P, then (i_{∇′}ω)_{y} = (i_{∇}ω)_{y}. Therefore, it is well defined the insertion i_{c}ω_{y} of an element c ∈ α_{1,0}^{−1}(y), y ∈ P, into ω_{y}. Similar considerations apply to both the above defined insertions i_{Z} and i_{Y}. Finally, for all y ∈ P, the projection Ω ⟶ \underline{Ω} as well determines a well defined linear map Ω|_{y} ∋ ω_{y} ⟼ \underline{ω}_{y} ∈ \underline{Ω}|_{y} whose kernel is Ω_{0}|_{y}.

In the following we will denote by δ : Ω ⟶ Ω (resp. δ_{0} : Ω_{0} ⟶ Ω_{0}) the restricted de Rham differential, i.e., for ω ∈ Ω (resp. ω_{0} ∈ Ω_{0}), δω := dω ∈ Ω (resp. δ_{0}ω_{0} := dω_{0} ∈ Ω_{0}) and with \underline{δ} : \underline{Ω} ⟶ \underline{Ω} the quotient differential. Then, for ω_{0} = ρ^{V} ⊗ α^{∗}(ν_{0}) (resp. ω = ρ^{V} ⊗ α^{∗}(σ_{0})), ρ^{V} ∈ VΛ, ν_{0} ∈ Λ^{n}(M) (resp. σ_{0} ∈ Λ^{n−1}(M)), we have

δ_{0}ω_{0} = d^{V}ρ^{V} ⊗ α^{∗}(ν_{0}) (resp. \underline{δ}\underline{ω} = d^{V}ρ^{V} ⊗ α^{∗}(σ_{0})).

In other words δ_{0} (resp. \underline{δ}) is isomorphic to the differential d^{V} ⊗ id : VΛ ⊗_{A_{0}} Λ^{n}(M) ⟶ VΛ ⊗_{A_{0}} Λ^{n}(M) (resp. d^{V} ⊗ id : VΛ ⊗_{A_{0}} Λ^{n−1}(M) ⟶ VΛ ⊗_{A_{0}} Λ^{n−1}(M)). All the above mentioned formulas can be proved by straightforward computations. Now, let ∇, Y and X be as above. denote by [[·, ·]] the Frölicher-Nijenhuis bracket in Λ(P) ⊗_{A} D(P). It is easy to see that [[H_{∇}, X]] ∈ \overline{Λ}^{1} ⊗_{A} V D ⊂ Λ(P) ⊗_{A} D(P). It holds the following

Theorem 3. Let ω ∈ Ω, then

[i_{∇}, δ]ω := (i_{∇} ∘ δ − δ_{0} ∘ i_{∇})ω = d_{∇}ω ∈ Ω_{0},
[i_{∇}, i_{Y}]ω := (i_{∇} ∘ i_{Y} − i_{Y} ∘ i_{∇})ω = 0 ∈ Ω_{0},  (8)
[i_{∇}, L_{X}]ω := (i_{∇} ∘ L_{X} − L_{X} ∘ i_{∇})ω = i_{[[H_{∇},X]]}ω ∈ Ω_{0}.

Proof. First prove that i_{∇} : Ω ⟶ Ω_{0} satisfies the “Leibnitz rule”

i_{∇}(λω) = λ · i_{∇}ω + i_{H_{∇}}λ · ω,  (9)

λ ∈ Λ(P), ω ∈ Ω. For ρ ∈ Λ(P), denote ρ_{∇}^{•,p} := ∑_{q} 𝔭_{∇}^{q,p}(ρ), so that ρ = ∑_{p} ρ_{∇}^{•,p}. Notice that for ω ∈ Ω and λ ∈ Λ(P), we have ω = ω_{∇}^{•,n} + ω_{∇}^{•,n−1} so that

i_{∇}(λω) = 𝔭_{∇}^{•,n}(λω) = λ_{∇}^{l,0}ω_{∇}^{•,n} + λ_{∇}^{•,1}ω_{∇}^{•,n−1} = λ · i_{∇}ω + λ_{∇}^{•,1} · ω_{∇}^{•,n−1}.

Moreover, i_{H_{∇}}λ = _{∑p}i_{H_{∇}}λ_{∇}^{•,p} = _{∑p}pλ_{∇}^{•,p}, which in turn implies λ_{∇}^{•,p} = i_{H_{∇}}λ − ∑p>1pλ_{∇}^{•,p}. Therefore

i_{∇}(λω) = λ · i_{∇}ω + λ_{∇}^{•,1} · ω_{∇}^{•,n−1}
= λ · i_{∇}ω + i_{H_{∇}}λ · ω_{∇}^{•,n−1} −_{∑p>1}pλ_{∇}^{•,p}ω_{∇}^{•,n−1}
= λ · i_{∇}ω + i_{H_{∇}}λ · ω.

In view of (9), the above defined operators [i_{∇}, δ], [i_{∇}, i_{Y}], [i_{∇}, L_{X}] : Ω ⟶ Ω_{0}, satisfy analogous “Leibnitz rules”

[i_{∇}, δ](λω) = d_{∇}λ · ω + (−)^{l}λ · [i_{∇}, δ](ω),
[i_{∇}, i_{Y}](λω) = λ · [i_{∇}, i_{Y}](ω),  (10)
[i_{∇}, L_{X}](λω) = i_{[[H_{∇},X]]}λ · ω + λ · [i_{∇}, L_{X}](ω).

Since Ω is generated by \overline{Λ}^{n−1} as a Λ(P)-module, in view of (10), it is enough to prove (8) for ω ∈ \overline{Λ}^{n−1}. In this case

[i_{∇}, δ]ω = i_{∇}dω = (dω)_{∇}^{•,n} = d_{∇}ω,
[i_{∇}, i_{Y}]ω = 0,
[i_{∇}, L_{X}]ω = i_{∇}L_{X}ω = (L_{X}ω)_{∇}^{•,n} = 0 = i_{[[H_{∇},X]]}ω.
□

We now discuss the interaction between affine forms and bundle morphisms. Let α′ : P′ ⟶ M be another fiber bundle and G : P ⟶ P′ a bundle morphism. Clearly, G preserves the ideals Λ_{p}, p ≥ 0, i.e., G^{∗}(Λ_{p}(P′, α′)) ⊂ Λ_{p}(P, α). In particular,

G^{∗}(Ω(P′, α′)) ⊂ Ω(P, α) and G^{∗}(Ω_{0}(P′, α′)) ⊂ Ω_{0}(P, α).

We conclude that it is well defined the quotient map which, abusing the notation, we again denote by G^{∗} : \underline{Ω}(P′, α′) ⟶ \underline{Ω}(P, α). Now, consider G-compatible connections ∇ ∈ C(P, α) and ∇′ ∈ C(P′, α′). It is easy to show that

G^{∗} ∘ i_{∇′} = i_{∇} ∘ G^{∗} : Ω(P′, α′) ⟶ Ω_{0}(P, α).  (11)

3.2. Cohomology.

Remark 4. (see [41]) In the following we denote by F the abstract fiber of α. Notice that, for any q ≥ 0, VH^{q} ≡ VH^{q}(P, α) := H^{q}(VΛ, d^{V}) is the A_{0}-module of sections of a (pro-finite) vector bundle α^{q} : P^{q} ⟶ M over M whose abstract fiber is H^{q}(F). Moreover, α^{q} is endowed with a canonical flat connection ∇^{q} (∇^{q} is a smooth analogue of Gauss-Manin connection in algebraic geometry). Correspondingly, there is a de Rham like complex

\xymatrix{⋯ \ar[r] & Λ^{p−1} ⊗_{A_{0}} VH^{q} \ar[r]^{d_{1}^{p−1,q}} & Λ^{p} ⊗_{A_{0}} VH^{q} \ar[r]^{d_{1}^{p,q}} & ⋯} ,

whose cohomology we denote by E_{2}^{•,q} := ⨁_{p} E_{2}^{p,q}, E_{2}^{p,q} := H^{p}(Λ(M) ⊗_{A_{0}} VΛ^{q}, d_{1}^{•,q})^{2}, q ≥ 0. It can be proved that, if α is trivial or M is simply connected, then there is a (generically non-canonical), isomorphism

E_{2}^{p,q} ≈ H^{p}(M) ⊗ H^{q}(F), p, q ≥ 0.

Finally, notice also that, for any q ≥ 0,

H^{q}(Ω_{0}, δ_{0}) ≃ Λ^{n}(M) ⊗_{A_{0}} VH^{q},
H^{q}(\underline{Ω}, \underline{δ}) ≃ Λ^{n−1}(M) ⊗_{A_{0}} VH^{q}.

Proposition 5. Let α : P ⟶ M be a fiber bundle. Then, for any q ≥ 0, there exists a short exact sequence of vector spaces

\xymatrix{0 \ar[r] & coker d_{1}^{n,q−1} \ar[r] & H^{q}(Ω, δ) \ar[r] & ker d_{1}^{n−1,q} \ar[r] & 0}.

In particular, H^{q}(Ω, δ) ≈ coker d_{1}^{n,q−1} ⊕ ker d_{1}^{n−1,q} = E_{2}^{n,q−1} ⊕ ker d_{1}^{n−1,q}.

Proof. Consider the short exact sequence of complexes

\xymatrix{0 \ar[r] & Ω_{0} \ar[r] & Ω \ar[r] & \underline{Ω} \ar[r] & 0},

and the associated long sequence in cohomology

\xymatrix{⋯ \ar[r] & H^{q−1}(\underline{Ω}, \underline{δ}) \ar[r]^{∂} & H^{q}(Ω_{0}, δ_{0}) \ar[r] & H^{q}(Ω, δ) \ar[r] & H^{q}(\underline{Ω}, \underline{δ}) \ar[r]^{∂} & ⋯} . (12)

We already commented, in the above remark, that, for any q, H^{q}(Ω_{0}, δ_{0}) identifies with Λ^{n}(M) ⊗_{A_{0}} VH^{q} and H^{q}(\underline{Ω}, \underline{δ}) identifies with Λ^{n−1}(M) ⊗_{A_{0}} VH^{q}. Similarly, it is easy to show that the connecting operator

∂ : H^{q−1}(\underline{Ω}, \underline{δ}) ⟶ H^{q}(Ω_{0}, δ_{0})

identifies with the de Rham-like differential

d_{1}^{n−1,q} : Λ^{n−1}(M) ⊗_{A_{0}} VH^{q} ⟶ Λ^{n}(M) ⊗_{A_{0}} VH^{q}.

The thesis then follows from exactness of (12). □

Corollary 6. If F is connected, then H^{0}(Ω, δ) ≃ ker d_{M}^{n−1},

d_{M}^{n−1} : Λ^{n−1}(M) ⟶ Λ^{n}(M)

being the last de Rham differential of M.

2Similarly as above, this last notations are motivated by the fact that the differentials d_{1}^{•,q} (resp. the vector spaces E_{2}^{•,q}) are the ones in the first term (resp. are rows of the second term) of the (cohomological) Leray-Serre spectral sequence of the fiber bundle α [41].

Proof. If F is connected VH^{0} ≃ A_{0} and d_{1}^{n−1,0} identifies with d_{M}^{n−1}. □

Corollary 7. Let q ≥ 0 and ω ∈ Ω^{q} be δ-closed, i.e., δω = 0. Then, 1) if q = 0, ω is locally of the form α^{∗}(η) for some η ∈ Λ^{n−1}(M), 2) if q > 0, then ω is locally δ-exact, i.e., ω is locally of the form δθ, θ being a local element in Ω^{q−1}.

Proof. If F is contractible, then VH^{q} = 0, and therefore H^{q}(Ω, δ) = 0, for all q > 0. □

Let ω ∈ Ω and θ ∈ Ω be such that ω = δθ. Then θ will be said a potential of ω.

4. PD-Hamiltonian Systems

4.1. PD-Hamiltonian Systems and PD-Hamilton Equations. In this section we introduce what we think should be understood as the partial differential, i.e., field theoretic, (1st order) analogue of a hamiltonian (mechanical) system on an abstract symplectic manifold. Let α : P ⟶ M be as in the previous section and ω ∈ Ω^{2}(P, α) be such that δω = 0. Put

ker ω := \{Y ∈ V D | i_{Y} ω = 0\}, ker \underline{ω} := \{Y ∈ V D | i_{Y} \underline{ω} = 0\},
Ker ω := \{∇ ∈ C | i_{∇}ω = 0\}, Ker \underline{ω} := \{Z ∈ V D ⊗_{A} \overline{Λ}^{1} | i_{Z} \underline{ω} = 0\}.

Since ω is closed, both ker ω and ker \underline{ω} are modules of smooth sections of involutive α-vertical distributions D^{ω} and \underline{D}^{ω} on P, where, for y ∈ P,

D_{y}^{ω} := \{ξ ∈ V_{y}P | i_{ξ}ω_{y} = 0\}, \underline{D}_{y}^{ω} := \{ξ ∈ V_{y}P | i_{ξ}\underline{ω}_{y} = 0\}.

Similarly, Ker \underline{ω} is a sub-module in V D ⊗_{A} \overline{Λ}^{1}. As a minimal regularity requirement assume that \underline{D}^{ω} has got constant rank \underline{r}. Then, it is easy to check that, as a consequence, Ker \underline{ω} is the module of sections of a smooth vector bundle ϖ : W ⟶ P. For y ∈ P, denote r(y) = dim D_{y}^{ω}. In general, r(y) will change from point to point y ∈ P. However, we are proving in brief that r(y) cannot change that much. First of all, since, obviously, D^{ω} ⊂ \underline{D}^{ω}, then r(y) ≤ \underline{r} for all y ∈ P. Now, for y ∈ P, denote

Ker ω_{y} := {c ∈ α_{1,0}^{−1}(y) | i_{c}ω_{y} = 0}.

Then, Ker ω_{y} is either empty or an affine space modelled over ϖ^{−1}(y). It holds the

Proposition 8. For any y ∈ P, \underline{r} − r(y) ≤ 1 (see also Theorem 4 of [23]).

Proof. Let y ∈ P and suppose r(y) < \underline{r}. If ξ ∈ \underline{D}_{y}^{ω} then (see Remark 2) \underline{i_{ξ}ω_{y}} = i_{ξ}\underline{ω}_{y} = 0 so that i_{ξ}ω_{y} ∈ Ω_{0}^{1}|_{y} = \overline{Λ}^{n}|_{y}. Then consider the map γ_{y} : \underline{D}_{y}^{ω} ∋ ξ ⟼ γ_{y}(ξ) := i_{ξ}ω_{y} ∈ \overline{Λ}^{n}|_{y}. Since r(y) < \underline{r}, γ_{y} is surjective and the sequence of vector spaces 0 ⟶ D_{y}^{ω} ⟶ \underline{D}_{y}^{ω} ⟶^{γ_{y}} \overline{Λ}^{n}|_{y} ⟶ 0 is exact. Since \overline{Λ}^{n}|_{y} is 1-dimensional, it follows that \underline{r}−r(y) = 1. □

The following proposition characterizes the case r(y) = \underline{r}.

Proposition 9. Let ω be as above. Then r(y) = \underline{r} iff Ker ω_{y} ≠ ∅.

Proof. The result is nothing more than an application of the Rouché-Capelli theorem. We here propose a dual proof. Let ξ ∈ V_{y}P be given by ξ = ξ^{a}∂_{a}|_{y}. Then ξ ∈ \underline{D}_{y}^{ω} iff

ω_{ab}^{i}(y)ξ^{a} = 0, a = 1, …, m, i = 1, …, n.  (13)

Similarly, ξ ∈ D_{y}^{ω} iff they are satisfied both (13) and

ω_{a}(y)ξ^{a} = 0.  (14)

Therefore, D_{y}^{ω} = \underline{D}_{y}^{ω} iff Equation (14) linearly depends on Equations (13), i.e., iff there are real numbers h_{i}^{b}, b = 1, …, m, i = 1, …, n, such that

ω_{a}(y) = ω_{ab}^{i}(y)h_{i}^{b},

a = 1, …, m. Now, let c ∈ α_{1,0}^{−1}(y) be given by y_{i}^{a}(c) = −\frac{1}{2}h_{i}^{a}. Then i_{c}ω_{y} is given by

i_{c}ω_{y} = (−2ω_{ba}^{i}(y)y_{i}^{a}(c) + ω_{a}(y))dy^{a}d^{n}x |_{y}
= −(ω_{ab}^{i}(y)h_{i}^{b} − ω_{a}(y))dy^{a}d^{n}x |_{y}
= 0.

□

Definition 10. A PD-prehamiltonian system on the fiber bundle α : P ⟶ M is a δ-closed element ω ∈ Ω^{2}(P, α). A PD-hamiltonian system on α is a PD-prehamiltonian system ω such that ker \underline{ω} = 0 (and, therefore, ker ω = 0 as well).

Let θ ∈ Ω^{1} be locally given by θ = θ_{a}^{i}dy^{a}d^{n−1}x_{i} − Hd^{n}x, …, θ_{a}^{i}, …, H being local functions on P. Then δθ is locally given by

δθ = ∂_{[a}θ_{b]}^{i}dy^{a}dy^{b}d^{n−1}x_{i} − (∂_{i}θ_{a}^{i} + ∂_{a}H)dy^{a}d^{n}x.

Similarly, let ω ∈ Ω^{2} and Y ∈ V D be locally given by ω = ω_{ab}^{i}dy^{a}dy^{b}d^{n−1}x_{i} + ω_{a}dy^{a}d^{n}x and Y = Y^{a}∂_{a}, respectively. Then δω, i_{Y} ω and i_{Y} \underline{ω} are locally given by

δω = ∂_{[a}ω_{bc]}^{i}dy^{a}dy^{b}dy^{c}d^{n−1}x_{i} + (∂_{i}ω_{ab}^{i} + ∂_{[a}ω_{b]})dy^{a}dy^{b}d^{n}x,
i_{Y} ω = 2ω_{ab}^{i}Y^{a}dy^{b}d^{n−1}x_{i} + ω_{a}Y^{a}d^{n}x,
i_{Y} \underline{ω} = 2ω_{ab}^{i}Y^{a}d^{V}y^{b} ⊗ d^{n−1}x_{i},

so that ω is a PD-prehamiltonian system iff

∂_{[a}ω_{bc]}^{i} = 0, ∂_{i}ω_{ab}^{i} + ∂_{[a}ω_{b]} = 0,  (15)

or, which is the same (see Corollary 7),

ω_{ab}^{i} = ∂_{[a}θ_{b]}^{i}, ω_{a} = −∂_{a}H − ∂_{i}θ_{a}^{i},  (16)

for some …, θ_{a}^{i}, …, H local functions on P. Moreover, ω is a PD-hamiltonian system iff

ω_{ab}^{i}Y^{a} = 0 ⟹ Y^{a} = 0.  (17)

In its turn (17) implies ω_{a} = ω_{ab}^{i}f_{i}^{b} for some …, f_{i}^{b}, … local functions on P (see the proof of Proposition 9).

Let ω be a PD-prehamiltonian system on α, σ : U ⟶ P a local section of α, U ⊂ M an open subset. The first jet prolongation σ̇ : U ⟶ J^{1}α of σ may be interpreted as a “connection in α along σ”, i.e., a section of the restricted bundle α_{1,0}|_{σ} : J^{1}α|_{σ} ⟶ P. Moreover, elements in Ω|_{σ} may be interpreted as affine maps from C|_{σ} to Ω_{0}|_{σ} ≃ VΛ|_{σ} ⊗_{A_{0}} Λ^{n}(M) whose linear part is in \underline{Ω}|_{σ} ≃ VΛ|_{σ} ⊗_{A_{0}} Λ^{n−1}(M). Namely, an element ♢ ∈ C|_{σ} can “be inserted” into an element ρ|_{σ} ∈ Ω|_{σ}, ρ ∈ Ω, giving an element i_{♢}ρ|_{σ} ∈ Ω_{0}|_{σ}. Thus, we can search for local sections σ of α such that

iσ̇ω|_{σ} = 0  (18)

Definition 11. Equations (18) are called the PD-Hamilton equations (of the PDprehamiltonian system ω).

If ω is locally given by ω = ω_{ab}^{i}dy^{a}dy^{b}d^{n−1}x_{i} + ω_{a}dy^{a}d^{n}x, then the associated PD-Hamilton equations are locally given by

2ω_{ab}^{i}∂_{i}y^{a} − ω_{b} = 0.  (19)

Conversely, a system of PDEs in the form (19) is a PD-Hamilton equation for some PDprehamiltonian (resp. PD-hamiltonian) system iff coefficients …, ω_{ab}^{i}, …, ω_{b}, … satisfy (15) (or, which is the same, (16)) (resp. (15) and (17)). Notice that, in view of (19) a general PD-prehamiltonian system ω encode “kinematical information”, which can be identified with \underline{ω}, and “dynamical information”, which can be identified with the specific choice of ω in the class of those PD-hamiltonian systems with linear part \underline{ω} (see the comment at the end of Section 2.3, Remark 17 and Example 18). Searching for solutions of PD-Hamilton equations of a PD-prehamiltonian system ω, we could proceed in two steps:

(1) search for a connection ∇ ∈ Ker ω,

(2) search for n-dimensional integral submanifolds of the horizontal distribution H_{∇}P.

However, a solution to the first step of the above mentioned procedure exists iff ker ω = ker \underline{ω} which is not always the case. Therefore, in general, we are led to weaken 1 and search for connections ∇′ in a subbundle P′ ⊂ P such that i_{∇′}ω|_{P′} = 0. As showed in the next proposition, there is always an algorithmic way to find a maximal subbundle ᾰ : P̆ ⟶ M of α such that the affine equation i ˘∇ω|P̆ = 0, ˘∇ ∈ C(P̆, ᾰ) admits at least one solution. We will refer to the above mentioned algorithm as the PD-constraint algorithm (see also [19, 31, 32, 33, 34]).

Proposition 12. Let ω be as above and Ker ω = ∅ (i.e., D_{y}^{ω} ≠ \underline{D}_{y}^{ω} for some y ∈ P). Under suitable regularity conditions on ω (to be specified in the proof), there exists a (maximal) subbundle P̆ ⊂ P such that i ˘∇ω|P̆ = 0 for some ˘∇ ∈ C(P̆, ᾰ).

Proof. For s = 1, 2, … define recursively

P_{(s)} := {y ∈ P_{(s−1)} | Ker ω_{y} ∩ (α_{(s−1)})_{1}^{−1}(y) ≠ ∅} ⊂ P,

α_{(s)} := α|_{P_{(s)}} : P_{s} ⟶ M,

where P_{(0)} := P, α_{(0)} := α (in particular P_{1} = \{y ∈ P | r(y) = \underline{r}\}). We assume that α_{(s)} : P_{(s)} ⟶ M is a smooth (closed) subbundle for all s (regularity conditions). Then, for dimensional reasons, there exists \overline{s} such that P_{(s)} = P_{(\overline{s})} for all s ≥ \overline{s}. Put P̆ := P\overline{(s)}. □

ᾰ := α|P̆ : P̆ ⟶ M will be called the constraint subbundle. Notice that P̆ can be empty (for instance when r(y) = \underline{r} − 1 for all y ∈ P) and, in this case, PD-Hamilton equations do not possess solutions.

Corollary 13. Let ω be a PD-prehamiltonian system on α and σ a solution of PD-Hamilton equations. Then im σ ⊂ P̆.

Proof. By induction on s, im σ ⊂ P_{(s)} for all s = 1, 2, . . .. □

The converse of the above corollary is, a priori, only true for n = 1. Namely, we may wonder if for any y ∈ P̆ there is a solution σ of PD-Hamilton equations such that y ∈ im σ. We know that there is a connection ˘∇ in P̆ which is “a solution of PD-Hamilton equations up to first order”, i.e., i ˘∇ω̆|P̆ = 0. n-dimensional integral manifolds of the horizontal distribution H ˘∇P̆ determined on P̆ by ˘∇ are clearly images of solutions of PD-Hamilton equations. If n = 1, ˘∇ is trivially flat and Fröbenius theorem guarantees that for any y ∈ P̆ there is a solution “through y”. The same is a priori untrue for n > 2. Integrability conditions on H ˘∇P̆ will be discussed in a separate paper. Clearly, standard examples of PD-prehamiltonian systems come from Lagrangian field theory. Let us briefly recall how. Let π : E ⟶ M be a fiber bundle and consider a lagrangian density ℒ ∈ \overline{Λ}^{n}(J^{1}π, π_{1}). ℒ determines two Legendre transforms (see, for instance, [4]) which we denote by F ℒ : J^{1}π ⟶ ℳ π and \underline{F} ℒ : J^{1}π ⟶ J^{†}π. Consider the submanifold C_{0} := im \underline{F} ℒ ⊂ J^{†}π of primary hamiltonian constraints of the lagrangian theory. F ℒ factorizes as F ℒ = ℋ ∘ \underline{F} ℒ, ℋ : C_{0} ⟶ ℳ π being the hamiltonian section. Then ω_{ℒ} := F ℒ^{∗}(δΘ) (resp. ℋ ^{∗}(δΘ)) is a PD-prehamiltonian system in π_{1} (resp. in C_{0} ⟶ M). Moreover, as it is well known [3], if s : M ⟶ E is a solution of the Euler-Lagrange equations, then ṡ : M ⟶ J^{1}π (resp. \underline{F} ℒ ∘ṡ : M ⟶ C_{0}) is a solution of PD-Hamilton equations of ω_{ℒ} (resp. ℋ ^{∗}(δΘ)).

4.2. PD-Hamiltonian Systems and Multisymplectic Geometry à là Forger. Forger and Gomes have recently proposed a definition of multipresymplectic structure on a fiber bundle [23]. Their work aims to define such a structure so that 1) the differential dΘ of the tautological n-form Θ on the affine dual bundle of the first jet bundle is multisymplectic 2) every multipresymplectic structure is locally isomorphic to the pull-back of Θ along a fibration (Darboux lemma). Since, in our opinion, 1)

[23] is the best motivated and established work about fundamentals of multisymplectic geometry, 2) abstract fiber bundles play in [23] a similar role as in this paper, we analyze in this subsection the relationship between PD-prehamiltonian systems and multipresymplectic structures à là Forger, referring to [23] for the main definition. Here we just mention two of the main results of [23] (which can eventually be understood as definitions of polypresymplectic structure and multipresymplectic structure on a fiber bundle, respectively)

Theorem 14 (Forger and Gomes I). Let α : P ⟶ M be a fiber bundle, …, x^{i}, … local coordinates on M, i = 1, …, n = dim M and \underline{ω} ∈ \underline{Ω}^{2}. \underline{ω} is a polypresymplectic structure on α iff, around every point of P, there are local fiber coordinates …, q^{A}, …, p_{A}^{i}, …, z^{1} …, z^{s}, A = 1, …, m, i = 1, …, n (so that dim P = n + m + mn + s) such that \underline{ω} is locally given by

ω = d^{V}p_{A}^{i}d^{V}q^{A} ⊗ d^{n−1}x_{i}.

Theorem 15 (Forger and Gomes II). Let α : P ⟶ M be a fiber bundle, …, x^{i}, … local coordinates on M, i = 1, …, n = dim M, and ω ∈ Ω^{2}. ω is a multipresymplectic structure on α iff, around every point of P, there are local fiber coordinates …, q^{A}, …, p_{A}^{i}, …, p, z^{1} …, z^{r}, A = 1, …, m, i = 1, …, n (so that dim P = (n + 1)(m + 1) + r) such that ω is locally given by

ω = dp_{A}^{i}dq^{A}d^{n−1}x_{i} − dpd^{n}x.

Proposition 16. Let ω be a PD-prehamiltonian system on α. The following two conditions are equivalent.

(1) \underline{ω} is a polypresymplectic structure and r(y) = \underline{r} − 1 for all y ∈ P.

(2) ω is a multipresymplectic structure.

Proof. Recall that, in view of Proposition 5, ω is locally δ-exact. Suppose \underline{ω} is polypresymplectic and r(y) = \underline{r} − 1 for all y ∈ P. Then \underline{r} > 0 and, in view of Theorem 14, (around every point in P) there are α-adapted local coordinates

…, x^{i}, …, q^{A}, …, p_{A}^{i}, z^{0}, z^{1}, …, z^{\underline{r}−1},

such that, locally, \underline{ω} = d^{V}p_{A}^{i}d^{V}q^{A} ⊗ d^{n−1}x_{i} (in particular, ker \underline{ω} is locally spanned by …, ∂/∂z^{α}, …). Therefore, \underline{ω} = \underline{δ}\underline{θ}_{0}, where \underline{θ}_{0} := p_{A}^{i}d^{V}q^{A} ⊗ d^{n−1}x_{i} is a local element of \underline{Ω}^{1}. A general (local) potential of ω is then θ′ ∈ Ω^{1} such that \underline{θ}′ = \underline{θ}_{0} + d^{V}ν, ν being a local element in \underline{Ω}^{0} = \overline{Λ}^{n−1}. The (local) potential θ := θ′ − δν is locally in the form θ = p_{A}^{i}dq^{A}d^{n−1}x_{i} − pd^{n}x, where p is a local function on P. Therefore ω is locally given by

ω = dp_{A}^{i}dq^{A}d^{n−1}x_{i} − dpd^{n}x.

ker ω is locally spanned by those local elements Y^{α} \frac{∂}{∂z^{α}} in ker \underline{ω} such that Y^{α} \frac{∂h}{∂z^{α}} = 0. Since ker ω ≠ ker \underline{ω}, then \frac{∂p}{∂z^{α}} dz^{α} ≠ 0. Let, for instance, be \frac{∂p}{∂z^{0}} ≠ 0. Then …, x^{i}, …, q^{A}, …, p_{A}^{i}, p, z^{1}, …, z^{\underline{r}−1} is a new local coordinate system on P. In view of Theorem 15, ω is then multipresymplectic.

On the other hand, let ω be multipresymplectic. Then \underline{ω} is polypresymplectic. Moreover (around every point in P) there are α-adapted local coordinates

…, x^{i}, …, q^{A}, …, p_{A}^{i}, …, p, z^{1}, …, z^{r}  (20)

such that, locally, ω = dp_{A}^{i}dq^{A}d^{n−1}x_{i} − dpd^{n}x and \underline{ω} = d^{V}p_{A}^{i}d^{V}q^{A} ⊗ d^{n−1}x_{i}. This shows that for all y ∈ P,

D_{y}^{ω} = ⟨…, \frac{∂}{∂z^{α}} |_{y} , …⟩ ≠ \underline{D}_{y}^{ω} = ⟨…, \frac{∂}{∂z^{α}} |_{y} , …, \frac{∂}{∂p} |_{y}⟩.
□

Remark 17. Let ω be a PD-prehamiltonian system ω. First of all, notice that, if ω is a multipresymplectic structure then, in view of Proposition 16, PD-Hamilton equations of ω do not possess solutions. In this sense, multipresymplectic structures do not contain any dynamical information. Now, the proof of Proposition 16 also shows that if \underline{ω} is a polypresymplectic structure and ker ω = ker \underline{ω} then ω is locally in the form

ω = dp_{A}^{i}dq^{A}d^{n−1}x_{i} − dHd^{n}x

where \frac{∂H}{∂z^{α}} = 0, α = 1, 2, …, i.e., H is constant along the leaves of the distribution D^{ω} = \underline{D}^{ω}.

Example 18. Let ω ∈ Ω^{2} be a multisymplectic structure on α. In this case ker ω = 0, while \underline{D}^{ω} is a 1-dimensional (involutive) distribution. Leaves of \underline{D}^{ω} are submanifolds in the fibers of α. denote by \underline{P} the set of leaves of \underline{D}^{ω}. There is an obvious projection \underline{α} : \underline{P} ⟶ M. Suppose that \underline{α} : \underline{P} ⟶ M is a smooth fiber bundle and 𝔭 : P ⟶ \underline{P} a smooth submersion (which is always true locally). There is a distinguished class of (local) PD-hamiltonian systems on \underline{α}. Indeed, let U ⊂ \underline{P} be an open subbundle and ℋ : U ⟶ P a local section of 𝔭. Then ω′ := ℋ ^{∗}(ω) ∈ Ω^{2}(U, \underline{α}) is a PD-hamiltonian system. In particular, if we choose coordinates on P as in (20) (here r = 0), then …, x^{i}, …, q^{A}, …, p_{A}^{i}, … are coordinates on \underline{P}, ℋ is given by

ℋ ^{∗}(p) = H,

H being a local function on \underline{P}, and ω′ is locally given by

ω′ = dp_{A}^{i}dq^{A}d^{n−1}x_{i} − dHd^{n}x,

in particular θ′ := p_{A}^{i}dq^{A}d^{n−1}x_{i} − Hd^{n}x is a local potential of ω′. Finally, PD-Hamilton equations of ω′ read

q_{i}^{A} = \frac{∂H}{∂p_{A}^{i}} ,
p_{A}^{i}_{,i} = −\frac{∂H}{∂q^{A}} ,

which are de Donder-Weyl equations (see, for instance, [3]).

4.3. PD-Hamiltonian Systems and Variational Calculus. We show in this subsection that PD-Hamilton equations are locally variational. First of all, an element θ ∈ Ω^{1} may be understood as a (fiber-wise affine) horizontal n-form over J^{1}α, i.e., as an element ℒ^{θ} ∈ \overline{Λ}^{n}(J^{1}α, α_{1}) via

ℒ_{c}^{θ} := i_{c}θ_{y}, c ∈ J^{1}α, y = α_{1,0}(c) ∈ P.

In its turn ℒ^{θ} is a 1st order lagrangian density in the fiber bundle α determining an action functional which we denote by S^{θ} = ∫ ℒ^{θ}. If θ is locally given by θ = θ_{a}^{i}dy^{a}d^{n−1}x_{i} − Hd^{n}x, then ℒ^{θ} is locally given by ℒ^{θ} = L^{θ}d^{n}x, where L^{θ} is the local function on J^{1}α given by

L^{θ} = (θ_{b}^{i}y_{i}^{b} − H).

In particular, if θ = δν for some ν ∈ Ω^{0} = \overline{Λ}^{n−1} locally given by ν = ν^{i}d^{n−1}x_{i}, then

L^{θ} = (∂_{i} + y_{i}^{a}∂_{a})ν^{i},  (21)

i.e., L^{θ} is a total divergence.

Proposition 19. Let ω ∈ Ω^{2} be a δ-exact PD-prehamiltonian system. Then PD-Hamilton equations of ω coincide with Euler-Lagrange equations associated with the action S^{θ} := ∫ ℒ^{θ}, where θ ∈ Ω^{1} is the opposite of any potential of ω, i.e., −δθ = ω. Moreover, if H^{1}(Ω, δ) = 0 then S^{θ} is independent of the choice of θ and does only depend on ω.

Proof. The first part of the proposition can be proved in local coordinates. Indeed, compute variational derivatives of L^{θ},

\frac{δ}{δy^{b}} L^{θ} := ∂_{b}L^{θ} − (∂_{i} + y_{i}^{a}∂_{a})\frac{∂}{∂y_{i}^{b}} L^{θ}
= y_{i}^{a}(∂_{b}θ_{a}^{i} − ∂_{a}θ_{b}^{i}) − ∂_{a}H − ∂_{i}θ_{a}^{i}
= −2ω_{ab}^{i}y_{i}^{a} + ω_{b}.

where we used (16). To prove the second part of the proposition, use (21) to conclude that, for ν ∈ Ω^{0}, δL^{δν}/δy^{a} = 0. □

Remark 20. Condition H^{1}(Ω, δ) = 0 depends on the topology of the fiber bundle α. It is satisfied, for instance, if H^{n}(M) = 0 and H^{1}(F) = 0, F being, as above, the abstract fiber of α. Indeed, if H^{1}(F) = 0 then H^{1}(\underline{Ω}, \underline{δ}) = 0 so that, the first part of the exact sequence (12) reads

\xymatrix{0 \ar[r] & H^{0}(Ω, δ) \ar[r] & Λ^{n−1}(M) \ar[r]^{d_{M}^{n−1}} & Λ^{n}(M) \ar[r] & H^{1}(Ω, δ) \ar[r] & 0}

and H^{1}(Ω, δ) ≃ H^{n}(M) = 0.

5. PD-Noether Symmetries and Currents

5.1. PD-Noether Theorem and PD-Poisson Bracket. The multisymplectic analogues of 1) hamiltonian vector fields and 2) Poisson bracket in symplectic geometry have been longly investigated [6, 7, 8, 9, 10, 36]. We here propose the natural definitions for general PD-hamiltonian systems. Notice that, even if they look formally identical to (or possibly less general than) the ones proposed in [6, 8, 9, 10], our definitions have actually got a dynamical content, not only a kinematical one (see Remark 17), so that, for instance, we can prove a PD-version of (hamiltonian) Noether theorem. That’s why, e.g., we will better speak about PD-Noether symmetries rather than hamiltonian (multi)vector fields [42]. Let ω be a PD-prehamiltonian system on the bundle α : P ⟶ M. In the following we assume α to have connected fiber.

Definition 21. Let Y ∈ V D and f ∈ Ω^{0}. If i_{Y} ω = δf, then Y and f are said to be a PD-Noether symmetry and a PD-Noether current of ω (relative to each other), respectively.

Denote by 𝒮 (ω) and 𝒞 (ω) the sets of PD-Noether symmetries and PD-Noether currents of ω, respectively. A PD-Noether symmetry Y (relative to a PD-Noether current f) is a symmetry of ω in the sense that

L_{Y} ω = i_{Y} δω + δi_{Y} ω = δδf = 0.

The next proposition clarifies in what sense a PD-Noether current is a conserved current for ω.

Proposition 22 (PD–Noether theorem). Let Y ∈ 𝒮 (ω) and f ∈ 𝒞 (ω) be a PD-Noether symmetry and a PD-Noether current of ω relative to each other. Then σ^{∗}(f) ∈ Λ^{n−1}(M) is a closed form for every solution σ of PD-Hamilton equations.

Proof. First of all, let ϱ ∈ Ω^{1} and τ be a (local) section of α. It is easy to show (for instance, using local coordinates) that τ^{∗}(ϱ) = iτ̇ϱ|_{τ} ∈ Λ^{n}(M). Then

dσ^{∗}(f) = σ^{∗}(df)
= σ^{∗}(δf)
= iσ̇δf|_{σ}
= iσ̇i_{Y} ω|_{σ}
= i_{Y |_{σ}}iσ̇ω|_{σ}
= 0.
□

We are now in the position to introduce a Lie bracket among PD-Noether currents.

Proposition 23. Let Y_{1}, Y_{2} ∈ 𝒮 (ω) be PD-Noether symmetries relative to the PD-Noether currents f_{1}, f_{2} ∈ 𝒞 (ω), respectively. Then [Y_{1}, Y_{2}] ∈ 𝒮 (ω) and f := L_{Y_{1}}f_{2} ∈ 𝒞 (ω) and they are relative to each other. Moreover, f is independent of the choice of Y_{1} among the PD-Noether symmetries relative to the PD-Noether current f_{1}.

Proof. Compute

δL_{Y_{1}}f_{2} = L_{Y_{1}}δf_{2}
= L_{Y_{1}}i_{Y_{2}}ω
= i_{[Y_{1},Y_{2}]}ω + i_{Y_{2}}L_{Y_{1}}ω
= i_{[Y_{1},Y_{2}]}ω.

Now, let V ∈ ker ω. Then L_{V} f_{2} = i_{V} δf_{2} = i_{V} i_{Y_{2}}ω = 0. This proves the second part of the proposition. □

Let Y_{1}, Y_{2}, f_{1}, f_{2} be as in the above proposition.

Proposition 24. The ℝ-bilinear map

𝒞 (ω) × 𝒞 (ω) ∋ (f_{1}, f_{2}) ⟼ {f_{1}, f_{2}} := L_{Y_{1}}f_{2} ∈ H(ω),

Y_{1} being a PD-Noether symmetry relative to f_{1}, is a Lie bracket.

Proof. Let Y_{2} ∈ 𝒮 (ω) be a PD-Noether symmetry relative to f_{2} ∈ 𝒞 (ω). Skewsymmetry of {·, ·} immediately follows from the remark:

{f_{1}, f_{2}} = L_{Y_{1}}f_{2}
= i_{Y_{1}}δf_{2} + δi_{Y_{1}}f_{2}
= i_{Y_{1}}i_{Y_{2}}ω.

Now, check Leibnitz rule. Let Y_{3} ∈ 𝒮 (ω) and f_{3} ∈ 𝒞 (ω be another pair of PD-Noether symmetry, PD-Noether current relative to each other. Then

{f_{1}, {f_{2}, f_{3}}} = L_{Y_{1}}{f_{2}, f_{3}}
= L_{Y_{1}}L_{Y_{2}}f_{3}
= L_{[Y_{1},Y_{2}]}f_{3} + L_{Y_{2}}L_{Y_{1}}f_{3}
= {{f_{1}, f_{2}}, f_{3}} + {f_{2}, {f_{1}, f_{3}}}.
□

PD-Noether symmetries and PD-Noether currents of a PD-hamiltonian system constitute very small Lie subalgebras of the Lie algebras of higher symmetries and conservation laws of PD-Hamilton equations, for which they have been given fully satisfactory definitions and have been developed many infinite jet based computational techniques [2]. Nevertheless, it is worth a definition and a careful analysis, independent on infinite jets, in view of the possibility of developing a “(multi)symplectic theory” of higher symmetries and conservation laws (see, for instance, [37]). In Section 6 we propose some specific examples.

Finally, notice that, in general, nor a PD-Noether current is uniquely determined by the relative PD-Noether symmetry nor vice versa (unless ker ω = 0). However, “nontrivial PD-Noether symmetries” are in one to one correspondence with “non-trivial PD-Noether currents” in the following sense. Clearly, ker ω ⊂ 𝒮 (ω) and H^{0}(Ω, δ) ⊂ 𝒞 (ω). We will call elements in ker ω gauge PD-Noether symmetries (see below) and elements in H^{0}(Ω, δ) (i.e., closed (n − 1) -forms on M, see Corollary 6) trivial PD-Noether currents.

Remark 25. It is easy to see that ker ω and H^{0}(Ω, δ) are ideals in the Lie algebras 𝒮 (ω) and 𝒞 (ω), respectively. Let \overline{𝒮} (ω) := 𝒮 (ω)/ ker ω and \overline{𝒞} (ω) := 𝒞 (ω)/H^{0}(Ω, δ) be the quotient Lie algebras. Then the map

\overline{𝒮} (ω) ∋ Y + ker ω ⟼ f + H^{0}(Ω, δ) ∈ \overline{𝒞} (ω),

where Y ∈ 𝒮 (ω) and f ∈ 𝒞 (ω) are relative to each other, is a well defined isomorphism of Lie algebras. It is natural to call elements in \overline{𝒮} (ω) and \overline{𝒞} (ω) non-trivial PD-Noether symmetries and non-trivial PD-Noether currents, respectively. Indeed, elements in ker ω are trivial symmetries in that they are infinitesimal gauge transformations (see next subsection), and elements in H^{0}(Ω, δ) are trivial conserved currents in that they are conserved currents for every PD-prehamiltonian system ω, independently of ω.

5.2. Gauge Reduction of PD-Hamiltonian Systems. From a physical point of view, elements in ker ω are infinitesimal gauge transformations and therefore should be quotiented out via a reduction of the system. In this section we assume ker ω = ker \underline{ω} or, which is the same, Ker ω ≠ ∅. As a further regularity condition we assume that the leaves of D^{ω} = \underline{D}^{ω} form a smooth fiber bundle P̃ over M, whose projection we denote by α̃ : P̃ ⟶ M, in such a way that the canonical projection 𝔭 : P ⟶ P̃ is a smooth bundle. The last condition is always fulfilled at least locally. Notice, also, that, by construction, 𝔭 has connected fiber.

Theorem 26. There exists a unique PD-hamiltonian system ω̃ in α̃ such that 1) ω = 𝔭^{∗}(ω̃), 2) ker ω̃ = ker \underline{ω̃} = 0 and 3) a local section σ of α is a solution of the PD-Hamilton equation of ω iff 𝔭 ∘ σ (which is a local section of α̃) is a solution of PD-Hamilton equations of ω̃.

Proof. Let ˜∇ ∈ C(P̃, α̃). There exists a (non-unique) connection ∇ ∈ C(P, α) such that ∇ and ˜∇ are 𝔭-compatible. To prove this, choose a connection □ in 𝔭 and lift the planes of ˜∇ to P by means of □. It is easy to show that the so obtained distribution on P defines a connection ∇ in α with the required property. Similarly, every vector field X̃ ∈ V D(P̃, α̃) can be lifted to a (non-unique) 𝔭-projectable vector field X ∈ V D(P, α) such that X̃ is its projection. Then X ∈ D_{V} (P, 𝔭). Consider η := ω(∇)(X) ∈ Ω^{0}(P, α) and prove that L_{Y} η = 0 for any Y ∈ V D(P, 𝔭). Indeed, let Y ∈ V D(P, 𝔭). Then [Y, X] ∈ V D(P, 𝔭). Similarly [[Y, H_{∇}]] ∈ \overline{Λ}^{1}(P, α) ⊗ V D(P, 𝔭) ⊂ \overline{Λ}^{1}(P, α) ⊗ V D(P, α).

Now, V D(P, 𝔭) = ker ω by construction, and therefore

L_{Y} η = L_{Y} i_{X}i_{∇}ω
= [L_{Y} , i_{X}]i_{∇}ω + i_{X}L_{Y} i_{∇}ω
= i_{[Y,X]}i_{∇}ω + i_{X}i_{∇}L_{Y} ω + i_{X}[L_{Y} , i_{∇}]ω
= i_{∇}i_{[Y,X]}ω + i_{X}i_{[[Y,H_{∇}]]}ω
= 0.

Since fibers of 𝔭 are connected we conclude that η = 𝔭^{∗}(η̃) for a unique η̃ ∈ Ω^{0}(P̃, α̃). Put

ω̃( ˜∇)(X̃) := η̃.

ω̃ is a well defined element in Ω^{2}(P̃, α̃). Indeed, let ∇′ ∈ C(P, α) be also 𝔭-compatible with ˜∇ and X′ ∈ V D(P, α) be another 𝔭-projectable vector field projecting onto X̃. Then ∇′ − ∇ ∈ \overline{Λ}^{1}(P, α) ⊗ V D(P, 𝔭) and X′ − X ∈ V D(P, 𝔭). Therefore,

ω(∇′)(X′) = i_{X′}i_{∇′}ω
= i_{X′}i_{∇}ω + i_{X′}i_{∇′−∇}\underline{ω}
= i_{X}i_{∇}ω + i_{X′−X}i_{∇}ω
= i_{X}i_{∇}ω + i_{∇}i_{X′−X}ω
= ω(∇)(X).

Moreover, ω = 𝔭^{∗}(ω̃) by construction. Let us compute ker \underline{ω̃}. Thus, let X̃ ∈ V D(P̃, α̃) be such that iX̃ \underline{ω̃} = 0 and X ∈ V D(P, α) be as above. Then i_{X} \underline{ω} = 𝔭^{∗}(iX̃ \underline{ω̃}) = 0. This shows that X ∈ V D(P, 𝔭) and then X̃ = 0. Finally, let σ be a local section of α, σ̃ := 𝔭 ∘ σ, X̃ ∈ V D(P̃, α̃) and X as above. Compute

(iσ̃^{·} ω̃|σ̃)(X̃|σ̃) = iσ̃^{·}(iX̃ ω̃)|σ̃
= σ̃^{∗}(iX̃ ω̃)
= (σ^{∗} ∘ 𝔭^{∗})(iX̃ ω̃)
= σ^{∗}(i_{X}ω)
= iσ̇(i_{X}ω)|_{σ}
= (iσ̇ω|_{σ})(X|_{σ}).

This shows that iσ̇ω|_{σ} = 0 iff iσ̃^{·} ω̃|σ̃ = 0. □

Proposition 27. There are natural isomorphisms of Lie algebras

\overline{𝒮} (ω) ≃ 𝒮 (ω̃),
𝒞 (ω) ≃ 𝒞 (ω̃).

Proof. First of all let f ∈ 𝒞 (ω) and X ∈ 𝒮 (ω) be relative to each other. Then f = 𝔭^{∗}(f̃) for some f̃ ∈ Ω^{0}(P̃, α̃) and X is 𝔭-projectable. Indeed, for all Y ∈ ker ω,

L_{Y} f = i_{Y} δf + δi_{Y} f = i_{Y} i_{X}ω = i_{[Y,X]}ω = 0.

Moreover,

𝔭^{∗}(δf̃) = δ𝔭^{∗}(f̃) = δf = i_{X}ω = 𝔭^{∗}(iX̃ ω̃),

where X̃ denotes the 𝔭-projection of X, and, therefore, δf̃ = iX̃ ω̃, i.e., f̃ ∈ 𝒞 (ω̃) and X̃ ∈ 𝒮 (ω̃) is a PD-Noether symmetry relative to it. Thus, they are well defined maps

\overline{𝒮} (ω) ∋ X + ker ω ⟼ X̃ ∈ 𝒮 (ω̃),  (22)
𝒞 (ω) ∋ f ⟼ f̃ ∈ 𝒞 (ω̃).  (23)

Conversely, let X̃_{1} ∈ 𝒮 (ω̃), f̃_{1} ∈ 𝒞 (ω̃) be relative to each other, X_{1} ∈ V D(P, α) be any 𝔭-projectable vector field, X̃_{1} ∈ V D(P̃, α̃) be its projection, and f_{1} := 𝔭^{∗}(f̃_{1}) ∈ Ω^{0}(P, α). Then X_{1} ∈ 𝒮 (ω) and f_{1} ∈ 𝒞 (ω) is a PD-Noether current relative to it. Indeed,

i_{X_{1}}ω = 𝔭^{∗}(iX̃_{1} ω̃) = 𝔭^{∗}(δf̃_{1}) = δ𝔭^{∗}(f̃_{1}) = δf_{1}.

We conclude that (22) and (23) are inverted by

𝒮 (ω̃) ∋ X̃_{1} ⟼ X_{1} + ker ω ∈ \overline{𝒮} (ω),
𝒞 (ω̃) ∋ f̃_{1} ⟼ f_{1} ∈ 𝒞 (ω),

respectively. □

6. Examples

6.1. Non-Degenerate Examples. Let α : ℝ^{2n+1} ∋ (x^{1}, …, x^{n}, u, u_{1}, …, u_{n}) ⟼ (x^{1}, …, x^{n}) ∈ ℝ^{n}, n > 1. Consider T, V ∈ C^{∞}(ℝ^{2n+1}) of the form T = T (u_{1}, …, u_{n}) and V = V (u), respectively. The form

ω := \frac{∂^{2}T}{∂u_{i}∂u_{j}} du_{i}(dud^{n−1}x_{j} − u_{j}d^{n}x) − V′dud^{n}x,

is a PD-prehamiltonian system in α (here and in what follows a prime “ ′ ” denotes differentiation with respect to u). The associated PD-Hamilton equations read

\begin{matrix}\frac{∂^{2}T}{∂u_{i}∂u_{j}} ∂_{j}u_{i} + V′ = 0, \\ ∂_{i}u = u_{i}, \end{matrix}

which are in turn equivalent to

\frac{∂^{2}T}{∂u_{i}∂u_{j}} ∂_{ij}^{2}u + V′ = 0,  (24)
∂_{i}u = u_{i},

∂_{ij}^{2} := ∂_{i}∂_{j}, i, j = 1, …, n. Moreover, for

det (\frac{∂^{2}T}{∂u_{i}∂u_{j}} ) ≠ 0,

ω is a PD-hamiltonian system. We will only consider this case in the following. Thus, put T^{ij} := \frac{∂^{2}T}{∂u_{i}∂u_{j}}, i, j = 1, …, n, and let (T_{ij}) be the inverse matrix of (T^{ij}). As examples notice that

(1) For T = \frac{1}{2}g^{ij}u_{i}u_{j},

(g^{ij}) = ( \begin{matrix}−1 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \end{matrix} ) ,

(resp., g^{ij} = δ^{ij}, i, j = 1, …, n), (24) reduces to the wave equation (resp., the Poisson equation) with a u-dependent potential V (including the f-Gordon equation as a particular example, if n = 2 and f = −V′).

(2) For n = 2, T = \sqrt{1 + g^{ij}u_{i}u_{j}}, g^{ij} = δ^{ij}, i, j = 1, 2, and V = 0, (24) reduces to the equation for minimal surfaces in ℝ^{3} transversal to the projection ℝ^{3} ∋ (x_{1}, x_{2}, u) ⟼ (x_{1}, x_{2}) ∈ ℝ^{2}.

Let us search for PD-Noether symmetries and currents of ω. Let Y = U \frac{∂}{∂u} + U_{i} \frac{∂}{∂u^{i}} ∈ VD and f = f^{i}d^{n−1}x_{i} ∈ Ω^{0}. Then

i_{Y} ω = T^{ij}(U_{i}du − Udu_{i})d^{n−1}x_{j} − (T^{ij}u_{i}U_{j} + V′U) d^{n}x,
δf = ∂_{i}f^{i}d^{n}x + \frac{∂}{∂u}f^{i}dud^{n−1}x_{i} + \frac{∂}{∂u_{k}} f^{i}du_{k}d^{n−1}x_{i}.

Recall that Y and f are a PD-Noether symmetry and a PD-Noether current relative to each other, respectively, iff i_{Y} ω = δf, i.e.,

∂_{i}f^{i} + T^{ij}u_{i}U_{j} + V′U = 0,  (25)
\frac{∂}{∂u}f^{i} − T^{ij}U_{j} = 0,  (26)
\frac{∂}{∂u_{j}} f^{i} + T^{ij}U = 0.  (27)

It follows from (27) that \frac{∂}{∂u_{j}} f^{i} = \frac{∂}{∂u_{i}} f^{j}, i, j = 1, …, n, and then

\frac{∂^{2}}{∂u_{k}∂u_{j}} f^{i} = \frac{∂^{2}}{∂u_{k}∂u_{i}} f^{j}, i, j, k = 1, …, n.

Now,

\frac{∂^{2}}{∂u_{k}∂u_{j}} f^{i} = \frac{∂}{∂u_{k}} \frac{∂}{∂u_{j}} f^{i}
= −\frac{∂}{∂u_{k}} (T^{ij}U)
= −\frac{∂^{3}T}{∂u_{k}∂u_{i}∂u_{j}} U − T^{ij} \frac{∂}{∂u_{k}} U

Similarly,

\frac{∂^{2}}{∂u_{k}∂u_{i}} f^{j} = \frac{∂}{∂u_{i}} \frac{∂}{∂u_{k}} f^{j} = −\frac{∂^{3}T}{∂u_{i}∂u_{j}∂u_{k}} U − T^{jk} \frac{∂}{∂u_{i}} U.

Therefore,

T^{ij} \frac{∂}{∂u_{k}} U − T^{jk} \frac{∂}{∂u_{i}} U = 0

Contracting with T_{ij} we find (n − 1)\frac{∂}{∂u_{i}} U = 0 and, therefore,

U = U(x^{1}, …, x^{n}, u),

so that (27) can be rewritten as

\frac{∂}{∂u_{j}} (f^{i} + \frac{∂T}{∂u_{i}} U) = 0.

We conclude that

f^{i} = −\frac{∂T}{∂u_{i}} U + A^{i}  (28)

for some A^{i} = A^{i}(x^{1}, …, x^{n}, u), i = 1, …, n. Notice that (26) can be used to determine the U_{j}’s from the f^{i}’s via

U_{j} = T_{ji} \frac{∂}{∂u}f^{i}.

It remains to solve (25) which, in view of (28), reduces to

(∂_{i} + u_{i} \frac{∂}{∂u}) A^{i} − \frac{∂T}{∂u_{i}} (∂_{i} + u_{i} \frac{∂}{∂u}) U + V′U = 0.  (29)

We cannot go further on in solving (29) without better specifying T. In the following we will only consider two special cases.

(1) T = \frac{1}{2}g^{ij}u_{i}u_{j}, (g^{ij}) being a constant, non degenerate, symmetric matrix with inverse (g_{ij}). In this case (29) reads

∂_{i}A^{i} + V′U + (\frac{∂}{∂u}A^{i} − g^{ij}∂_{j}U) u_{i} − (g^{ij} \frac{∂}{∂u}U) u_{i}u_{j} = 0.  (30)

The left hand side of (30) is polynomial in u_{1}, …, u_{n}. Thus, all the corresponding coefficients must vanish, i.e.,

\frac{∂}{∂u}U = 0,  (31)
\frac{∂}{∂u}A^{i} − g^{ij}∂_{j}U = 0,  (32)
∂_{i}A^{i} + V′U = 0.  (33)

From (31), U = U(x^{1}, …, x^{n}) and, then, from (32), \frac{∂^{2}}{∂u^{2}} A^{i} = 0, i = 1, …, n, which in turn implies, using (32) again,

A^{i} = (g^{ij}∂_{j}U) u + B^{i}

for some B^{i} = B^{i}(x^{1}, …, x^{n}). Finally, (33) implies

(g^{ij}∂_{ij}^{2}U) u + ∂_{i}B^{i} + V′U = 0

and differentiating once more with respect to u

g^{ij}∂_{ij}^{2}U + V′′U = 0.

Since U doesn’t depend on u, if (a) V′′′ ≠ 0. Then U = 0 so that

f^{i} = \frac{1}{2}∂_{j}B^{ji}, U_{j} = 0

for some B^{ij} = −B^{ji} = B^{ij}(x^{1}, …, x^{n}), i.e.,

Y = 0 and f = dβ,

β = B^{ji}d^{n−2}x_{ji}, where d^{n−2}x_{ji} := i_{∂_{j}}d^{n−1}x_{i}, i, j = 1, …, n. Therefore, ω doesn’t posses PD-Noether symmetries nor non-trivial PD-Noether currents.

(b) V′′′ = 0. Then V = \frac{1}{2}μu^{2} for some constant μ and

g^{ij}∂_{ij}^{2}U + μU = 0, f^{i} = g^{ij}(u ∂_{j}U − u_{j}U) + \frac{1}{2}∂_{j}B^{ji}, U_{j} = ∂_{j}U,

for some B^{ij} = −B^{ji} = B^{ij}(x^{1}, …, x^{n}). Thus,

Y = U \frac{∂}{∂u} + ∂_{j}U \frac{∂}{∂u_{j}} and f = g^{ij}(u ∂_{j}U − u_{j}U)d^{n−1}x_{i} + dβ,

β = B^{ji}d^{n−2}x_{ji}, where U is any solution of the PD-Hamilton equation

g^{ij}∂_{ij}^{2}u + μu = 0.  (34)

Let us compute the PD-Poisson bracket. Consider two solutions of (34), say U_{1}, U_{2}, the corresponding PD-Noether symmetries Y_{1}, Y_{2} and associated PD-Noether currents f_{1}, f_{2}. Then

{f_{1}, f_{2}} = L_{Y_{1}}f_{2} = g^{ij}(U_{1}∂_{j}U_{2} − U_{2}∂_{j}U_{1})d^{n−1}x_{i},

which, as can be easily checked, is a trivial conservation law.

(2) n = 2, T = \sqrt{1 + δ^{ij}u_{i}u_{j}} and V = 0. In this case (29) reads

τ^{1/2} (∂_{i} + u_{i} \frac{∂}{∂u}) A^{i} = δ^{ij}u_{j} (∂_{i} + u_{i} \frac{∂}{∂u}) U,  (35)

where τ = 1 + δ^{ij}u_{i}u_{j}. Squaring both sides of (35) we get

τ [(∂_{i} + u_{i} \frac{∂}{∂u}) A^{i}]^{2} − [δ^{ij}u_{j} (∂_{i} + u_{i} \frac{∂}{∂u}) U]^{2} = 0,

whose left hand side is polynomial in u_{1}, u_{2}. Collecting homogeneous terms we get

[(\frac{∂}{∂u}U)^{2} δ^{ij} − (\frac{∂}{∂u}A^{i}) (\frac{∂}{∂u}A^{j})] δ^{kl}u_{i}u_{j}u_{k}u_{l}
+2δ^{ij} [δ^{kl} (\frac{∂}{∂u}U) (∂_{l}U) − (\frac{∂}{∂u}A^{k}) (∂_{l}A^{l})] u_{i}u_{j}u_{k}
− [δ^{ij} (∂_{k}A^{k})^{2} + (\frac{∂}{∂u}A^{i}) (\frac{∂}{∂u}A^{j}) − δ^{ik}δ^{jl} (∂_{k}U) (∂_{l}U)] u_{i}u_{j}
+2 (∂_{j}A^{j})^{2} (\frac{∂}{∂u}A^{i}) u_{i} + (∂_{i}A^{i})^{2} = 0.  (36)

All coefficient of the left hand side of (36) must vanish. It follows that

\frac{∂}{∂u}U = ∂_{1}U = ∂_{2}U = 0, \frac{∂}{∂u}A^{1} = \frac{∂}{∂u}A^{2} = 0, ∂_{1}A^{1} + ∂_{2}A^{2} = 0, i.e., U is a constant while A^{1} = ∂_{2}B, A^{2} = −∂_{1}B for some B = B(x^{1}, x^{2}). Thus,

Y = U \frac{∂}{∂u}, f = U τ^{−1/2} (u_{2}dx^{1} − u_{1}dx^{2}) + dB.

It is obvious that the PD-Poisson bracket is also trivial in this case.

6.2. A Degenerate, Constrained Example. The example in this sunsection is taken from [43]. Let α : ℝ^{3m+2}×ℝ_{+} ∋ (q^{1}, …, q^{m}, s_{1}, …, s_{m}, t_{1}, …, t_{m}, s, t; e) ⟼ (s, t) ∈ ℝ^{2}. The form

ω := −dt_{α}dq^{α}ds + ds_{α}dq^{α}dt − δ^{αβ}(et_{α}dt_{β} − s_{α}ds_{β})dsdt − εdedsdt,

where ε := \frac{1}{2}(δ^{αβ}t_{α}t_{β} − 1), is a PD-prehamiltonian system in α. The associated PD-Hamilton equation reads

\frac{∂}{∂t}t_{α} + \frac{∂}{∂s}s_{α} = 0,
\frac{∂}{∂t}q^{α} = eδ^{αβ}t_{β},
\frac{∂}{∂s}q^{α} = −δ^{αβ}s_{β},
ε = 0,

α = 1, …, m, which is in turn equivalent to

\begin{matrix}e^{−1} \frac{∂^{2}}{∂t^{2}} q^{α} − \frac{∂^{2}}{∂s^{2}} q^{α} = e^{−2} (\frac{∂}{∂t}q^{α}) (\frac{∂}{∂t}e) , \\ e^{2} = δ_{αβ} (\frac{∂}{∂t}q^{α}) (\frac{∂}{∂t}q^{β}) , \\ t_{α} = e^{−1}δ_{αβ} \frac{∂}{∂t}q^{β}, \\ s_{α} = δ_{αβ} \frac{∂}{∂s}q^{β}, \end{matrix} ,

Notice that \underline{D}^{ω} is generated by \frac{∂}{∂e}, while

D_{y}^{ω} = \{ \begin{matrix}0 & for ε(y) ≠ 0 \\ ⟨ \frac{∂}{∂e} |_{y}⟩ & for ε(y) = 0 \end{matrix} , y ∈ P

we conclude that P_{(1)} is the hypersurface defined by δ^{αβ}t_{α}t_{β} = 1. It is easy to see that, actually, P̆ = P_{(1)}. Let us search for PD-Noether symmetries and currents of ω. Let Y = Q^{α} \frac{∂}{∂q^{α}} +S_{α} \frac{∂}{∂s_{α}} + T_{α} \frac{∂}{∂t_{α}} + E \frac{∂}{∂e} ∈ V D and f = αds + βdt ∈ Ω^{0}. Then i_{Y} ω = δf iff

\begin{matrix}\frac{∂}{∂s}β − \frac{∂}{∂t}α = δ^{βγ}(s_{β}S_{γ} − et_{β}T_{γ}) − εE, \\ \frac{∂}{∂q^{α}} α = −T_{α}, \frac{∂}{∂q^{α}} β = S_{α}, \frac{∂}{∂t_{α}} α = \frac{∂}{∂s_{α}} β = Q^{α} \\ \frac{∂}{∂s_{α}} α = \frac{∂}{∂t_{α}} β = 0, \frac{∂}{∂e}α = \frac{∂}{∂e}β = 0, \end{matrix}  (37)

α = 1, …, m. Equations (37) can be easily solved and give quite large 𝒮 (ω) and 𝒞 (ω). Namely,

α = C^{α}t_{α} + A, β = C^{α}s_{α} + B,

and

\begin{matrix}Q^{α} = C^{α}, T_{α} = −\frac{∂C^{β}}{∂q^{α}} t_{β} − \frac{∂A}{∂q^{α}} , S_{α} = \frac{∂C^{β}}{∂q^{α}} s_{β} + \frac{∂B}{∂q^{α}} , \\ εE = \frac{∂C^{α}}{∂s} s_{α} − \frac{∂C^{α}}{∂t} t_{α} + \frac{∂B}{∂s} − \frac{∂A}{∂t} − δ^{αβ} [s_{α} (\frac{∂C^{γ}}{∂q^{β}} s_{γ} + \frac{∂B}{∂q^{β}} ) + et_{α} (\frac{∂C^{γ}}{∂q^{β}} t_{γ} + \frac{∂A}{∂q^{β}} )] \end{matrix}

where A, B, …, C^{α}, …, D^{αβ}, …, E^{α}, … are arbitrary functions of the only s, t, …, q^{β}, . . .. Compute the PD-Poisson bracket. Let f_{1},f_{2} be PD-Noether currents determined by functions A_{1}, B_{1}, …, C_{1}^{α}, … and A_{2}, B_{2}, …, C_{2}^{α}, … respectively. A straightforward computation shows that

{f_{1}, f_{2}} = (C^{α}t_{α} + A)ds + (C^{α}s_{α} + B)dt

with

A = C_{1}^{β} \frac{∂}{∂q^{β}} A_{2} − C_{2}^{β} \frac{∂}{∂q^{β}} A_{1},
B = C_{1}^{β} \frac{∂}{∂q^{β}} B_{2} − C_{2}^{β} \frac{∂}{∂q^{β}} B_{1},
C^{α} = C_{1}^{β} \frac{∂}{∂q^{β}} C_{2}^{α} − C_{2}^{β} \frac{∂}{∂q^{β}} C_{1}^{α},

α = 1, …, m.

6.3. A Degenerate, Unconstrained Example. Finally, we propose an example of reduction. Consider the cotangent bundle π : T^{∗}𝕄 ∋ A_{i}dx^{i}|_{(x^{1},...,x^{n})} ⟼ (x^{1}, …, x^{n}) ∈ 𝕄 and let α := π_{1} : (x^{1}, …, x^{n}, …, A_{i}, …, A_{i,j}, …) ∋ J^{1}π ⟼ (x^{1}, …, x^{n}) ∈ 𝕄, 𝕄 being the n-dimensional Minkowski space. As such 𝕄 is endowed with the metric g := g_{ij}dx^{i} · dx^{j} where

(g_{ij}) = ( \begin{matrix}−1 & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \end{matrix} ) .

In the following we will raise and lower indexes using g. Let

ω := 2dA^{[j,i]} (\frac{1}{2}A_{i,j}d^{n}x − dA_{i}d^{n−1}x_{j})

Then ω is a PD-prehamiltonian system on π whose PD-Hamilton equation reads

∂_{k}A^{[i,k]} = 0,
∂_{[j}A_{i]} = A_{[i,j]},

i, j = 1, …, n, which are equivalent to Maxwell equations for the vector potential

(∂_{k}∂^{k})A_{i} − ∂_{i}∂_{k}A^{k} = 0,
A_{[i,j]} = ∂_{[j}A_{i]}.

Notice that

ker ω = ker \underline{ω} = ⟨…, \frac{∂}{∂A_{i,j}} + \frac{∂}{∂A_{j,i}} , …⟩ .

Therefore ω is “degenerate and unconstrained”. Moreover, leaves of D^{ω} = \underline{D}^{ω} are given by A_{[i,j]} = const. We conclude that J^{1}π “reduces” via

𝔭 : J^{1}π ⟶ T^{∗}𝕄×_{M} ∧^{2} T^{∗}𝕄 ≃ ℝ^{n(n+3)/2}
(x^{1}, …, x^{n}, …, A_{i}, …, A_{i,j}, …) ⟼ (x^{1}, …, x^{n}, …, A_{i}, …, F_{ij}, …)

where F_{ij} = F_{[ij]}, 𝔭^{∗}(F_{ij}) := 2A_{[j,i]} and ω = 𝔭^{∗}(ω̃), with

ω̃ := dF^{ij} (\frac{1}{4}F_{ji}d^{n}x − dA_{i}d^{n−1}x_{j})

is a PD-hamiltonian system on

α̃ : ℝ^{n(n+3)/2} ∋ (x^{1}, …, x^{n}, …, A_{i}, …, F_{ij}, …) ⟼ (x^{1}, …, x^{n}) ∈ ℝ^{n},

whose PD-Hamilton equations read

∂_{k}F^{ik} = 0,
∂_{[j}A_{i]} = 2F_{ji},

which are Maxwell equations for the field strength.

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Luca Vitagliano, DMI, Università degli Studi di Salerno, and Istituto Nazionale di Fisica Nucleare, GC Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy

  E-mail address: lvitagliano@unisa.it