Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 75–87

LIOUVILLE AND JACOBI THEOREMS FOR VECTOR DISTRIBUTIONS^{1}

Olga Krupková

Abstract. The paper contains a generalization of the Liouville and Jacobi integration methods, known from the classical calculus of variations, to the case of characteristic distributions of closed 2-forms of constant rank on smooth manifolds. It is shown that the results can be used to find integral manifolds of completely integrable distributions on smooth manifolds. In this way, the theorems of Liouville and Jacobi are generalized to a wide class of generally higher order ordinary and partial differential equations, possibly degenerate, which are not supposed to come from a Lagrangian. Keywords: completely integrable distribution, closed 2-form, characteristic distribution of a closed 2-form, Liouville theorem, Jacobi theorem, symmetries of differential equations, integration of differential equations, degenerate equations, non-variational equations, connections on fibered manifolds. MS classification: 58A30, 34A05, 34A08, 70H10, 70H20.

1. Introduction

The classical calculus of variations provides us with powerful integration methods for regular second order ordinary differential equations which are variational, i.e. are the equations for extremals of a certain regular Lagrangian. It is, in particular, the Liouville theorem, saying that once one knows “a half” of first integrals “in involution”, remaining ones can be computed by “quadratures”. Another one is the Jacobi theorem, according to which one can find first integrals providing that a “complete integral” of the Hamilton-Jacobi equation is known.

A generalized version of the Liouville theorem, which applies to semiregular variational problems of higher-order mechanics has been proved in [4]. A more general version, Liouville theorem for closed 2-forms of constant rank, which applies to any distribution which can be represented as the characteristic distribution of a closed 2-form of constant rank, has been proved in [9]. Some possible generalizations of the Jacobi integration method for the case of closed 2-forms of maximal rank have been studied in [1, 11]. For the case of (possibly degenerate) higher order ordinary variational equations the Jacobi theorem has been generalized in [9].

The purpose of this paper is to propose integration techniques for characteristic distributions of closed 2-forms of constant rank on smooth manifolds, based on a general idea of the Liouville integration method: to use a knowledge of a part of first integrals of the given equations (or equivalently, a knowledge of a part of symmetries of the associated closed 2-form) for finding adapted charts to the distribution in question. Another aim is to understand properly the geometry “behind” these integration techniques.

The plan of the paper is as follows: in Sections 2 and 3 it is shown that the Liouville theorem and the Liouville and Jacobi integration methods can be generalized in such a form which does not need the assumption of variationality (i.e. the existence of a Lagrangian). In Section 2 we study the geometric background of the Liouville integration method. Among others, the geometric nature of such notions, as e.g. that of a “complete integral”, or “first integrals in involution” (commonly used in classical mechanics), becomes transparent. This part of the paper is a more detailed exposition of ideas and results contained in [9, Sect.4.1]. Section 3 contains a geometric look on the Jacobi integration method, and a generalization of the classical Jacobi theorem to an arbitrary distribution which is a characteristic distribution of a closed 2-form. It also provides us with a new understanding of the Jacobi theorem as a special case of the Liouville theorem. In comparison to [9], we present a general theory without any reference to the calculus of variations.

1This paper is in final form and no version of it will be submitted for publication elsewhere.

The integration methods introduced in this paper can be applied to any completely integrable distribution on a manifold, or on a jet-prolongation of a fibered manifold, resp. to any integrable system of ordinary or partial differential equations (possibly of higher order), which arises as a system for integral mappings of a vector distribution. These applications are briefly discussed in Sect. 4. In particular, the family of equations covered by this theory contains (1) semisprays on jet prolongations of fibered manifolds (“higher-order vector fields“), i.e. higher-order ordinary and partial differential equations “explicitly solved with respect to the highest derivatives”, and not necessarily variational, or (2) degenerate equations of variational higher-order mechanics (cf. [9]).

We work in the category of smooth finite dimensional manifolds. We denote by d, i, ∂, and * the exterior derivative, the contraction, the Lie derivative, and the pullback, respectively. The summation convention is used as far as possible.

Throughout the paper, by a distribution of rank r on a manifold M we mean a mapping which to each point x of M assigns an r-dimensional subspace of T_{x}M. If a distribution Δ is spanned by r (smooth, linearly independent) vector fields ξ_{1}, ⋯, ξ_{r}, or equivalently by k = dim M − r (smooth, linearly independent) 1-forms ω_{1}, ⋯, ω_{k}, we write

Δ = span{ξ_{1}, ⋯, ξ_{r}} = span{ω_{1}, ⋯, ω_{k}}.

For a vector field ξ (resp. a 1-form ω) the notation ξ ∈ Δ (resp. ω ∈ Δ) means that ξ (resp. ω) belongs to Δ, i.e. it can be expressed by means of generators of Δ.

2. Liouville theorem for closed 2-forms

In what follows, M is a smooth manifold of dimension n, and α is a closed 2-form of constant rank = 2p on M.

Denote by D the characteristic distribution of α. Recall that this distribution is spanned by the kernel of alpha (consisting of all vector fields ζ on M such that i_{ζ}α = 0), or equivalently, by the associated system of α (consisting of the 1-forms i_{ξ}α where ξ runs over the set of all vector fields on M). It holds rank α = corank D, and by a well-known Cartan theorem, D is completely integrable.

Let Δ be a distribution on M. Recall that a function f defined on an open set U ⊂ M is called a first integral of Δ if the form df belongs to the set of the generators of Δ on U. Suppose Δ be completely integrable and of constant rank on M. A set of functions {f_{i}, 1 ≤ i ≤ corank Δ} on U is called a complete set of independent first integrals of Δ if the 1-forms df_{i} are linearly independent and span Δ on U. The existence of a complete set of independent first integrals of Δ in a neighbourhood of each point of M is ensured by Frobenius theorem (existence of adapted charts to the foliation defined by Δ). Conversely, every given complete set of independent first integrals of Δ can be completed in an obvious way to an adapted chart to Δ.

If, in particular, we take for Δ the characteristic distribution D of a closed 2-form α on M, we get important relations between first integrals of D and symmetries of α (i.e. vector fields ξ such that ∂_{ξ}α = 0), which can be used to simplify the integration procedure. (In this context it should be noted that in classical mechanics, where a particular case of a closed 2-form arises, namely, α is the differential of a Cartan form, or more precisely, a Lepagean 2-form, the symmetries of α are called Noetherian symmetries).

Proposition 1. Let α be a closed 2-form on M.

(1) Let ξ be a symmetry of α. Then di_{ξ}α = 0. Hence, locally, i_{ξ}α = df where f is a first integral of the characteristic distribution D of α.

(2) Let ρ be a 1-form such that α = dρ. Let ξ be a symmetry of ρ. Then ξ is a symmetry of α, and i_{ξ}ρ is a first integral of the characteristic distribution D of α.

(3) Let f be a first integral of the characteristic distribution of α. Then there exists a class {ξ} of symmetries of α such that i_{ξ}α = df for each ξ ∈ {ξ}. This class is of the form ξ_{0} + ζ where ξ_{0} is a symmetry of α such that i_{ξ_{0}}α = df, and ζ runs over the set of the generators of the characteristic distribution of α.

(4) Let ξ be a symmetry of α, let ρ be defined on an open set U ⊂ M, and such that α = dρ on U. Then in a neighbourhood of each point x ∈ U there exists a symmetry \overline{ξ} of ρ (and hence of α) such that i\overline{ξ}ρ = 0. If k ≥ 1 is a fixed integer and ζ_{1},…, ζ_{k} are generators of the characteristic distribution D of α, defined in a neighbourhood of x, and such that i_{ζ_{j}}ρ ≠ 0, 1 ≤ j ≤ k, then

(2.1)  \overline{ξ} =]ξ − \frac{1}{k} ∑_{j=1}^{k} \frac{i_{ξ}ρ}{i_{ζ_{j}}ρ}ζ_{j}

is a required symmetry.

(5) The set of all symmetries of α is a Lie algebra. If f_{1}, f_{2} are first integrals of the characteristic distribution D of α, and ξ_{1}, ξ_{2} are some corresponding symmetries of α, then

(2.2)  g ≡ {f_{1}, f_{2}} = i_{ξ_{1}}i_{ξ_{2}}α = i_{ξ_{1}}df_{2}

is another first integral of D, corresponding to the symmetry [ξ_{1}, ξ_{2}] of α.

Proof is an easy exercise and is left to the reader.

The first integral {f_{1}, f_{2}} defined by (2.2) is called the Poisson bracket of the first integrals f_{1}, f_{2}.

Note that if ξ is a symmetry of α and ζ is a generator of the characteristic distribution D of α then [ξ, ζ] = 0. This means that knowing D and an involutive set of symmetries ξ_{1},…, ξ_{k} of α we get a completely integrable distribution Δ = span{ξ_{K}, ζ_{L}, 1 ≤ K ≤ k, 1 ≤ L ≤ rank D}, whose subdistribution is D. There is a very important particular case when the number k of involutive symmetries of α is equal to p = (1/2) rank α. In this case, as we shall see below, Δ can be obtained directly (without the explicit knowledge of D), and D can be solved easily; this will be subject of the generalized Liouville theorem.

Let Δ be a distribution defined on an open subset U of M, let α be a closed 2-form of constant rank = 2p on M. Δ will be called a complete integral of α on U if (1) Δ is completely integrable and corank Δ = p, and (2) α belongs to the differential ideal generated by Δ [9] (cf. also [2]). Note that the second condition means that α = 0 on the maximal integral manifolds of Δ.

The existence of local complete integrals is ensured by Darboux theorem.

It is clear that the characteristic distribution D of α is a subdistribution of every complete integral of α. Hence, first integrals of Δ are first integrals of D, and by Proposition 1 they are in correspondence with symmetries of α. Namely, if Δ is a complete integral and (x^{J}, a^{K}), 1 ≤ J ≤ rank Δ, 1 ≤ K ≤ p = corankΔ, is an adapted chart to Δ at a point x ∈ M, i.e. Δ = span{da^{K}, 1 ≤ K ≤ p} in a neighbourhood of x, then the functions a^{K}, 1 ≤ K ≤ p, are independent first integrals of D, and the vector fields ∂/∂a^{k}, 1 ≤ K ≤ p, are symmetries of α.

There arises a question about the converse: under what conditions a family of symmetries of α, resp. a family of first integrals of α defines a complete integral of α? This question is solved by the following proposition.

Proposition 2. Let α be a closed 2-form of rank 2p on M. Let a^{K}, 1 ≤ K ≤ p, be independent first integrals of the characteristic distribution D of α. Let ξ_{K}, 1 ≤ K ≤ p, be symmetries of α such that i_{ξ_{K}} = da^{K}, 1 ≤ K ≤ p. The following conditions are equivalent:

(1) The distribution Δ = span{da^{k}, 1 ≤ K ≤ p} = span{ξ_{K}, ζ_{L}, 1 ≤ K ≤ p, 1 ≤ L ≤ rank D}, where ζ_{L} are independent generators of D, is a complete integral of α.

(2) i_{ξ_{K}}da^{L} = i_{ξ_{K}}i_{ξ_{L}}α = 0 for all 1 ≤ K, L ≤ p.

(3) The vector fields ξ_{K}, 1 ≤ K ≤ p, are involutive, i.e. [ξ_{K}, ξ_{L}] = 0, for all 1 ≤ K, L ≤ p.

Proposition 2 can be obviously reformulated in the following way:

Let α be a closed 2-form of rank 2p on M. Let ξ_{K}, 1 ≤ K ≤ p, be symmetries of α, defined on an open subset of M and independent at each point of their domain of definition. The following conditions are equivalent:

(1) The distribution Δ = span{i_{ξ_{K}}α, 1 ≤ K ≤ p} is a complete integral of α,

(2) i_{ξ_{K}}i_{ξ_{L}}α = 0 for all 1 ≤ K, L ≤ p,

(3) [ξ_{K}, ξ_{L}] = 0 for all 1 ≤ K, L ≤ p.

Proof of Proposition 2. The equivalence of (1) and (2) has been proved in [9]. (1)⇒(3) is trivial. (2) follows from (3) by the assertion (4) of Proposition 1, if we use the formula i_{[\overline{ξ}_{K},\overline{ξ}_{L}]}ρ = ∂_{\overline{ξ}_{K}}i_{\overline{ξ}_{L}}ρ−i_{\overline{ξ}_{L}}∂_{\overline{ξ}_{K}}ρ = i_{\overline{ξ}_{K}}i_{\overline{ξ}_{L}}α = i_{ξ_{K}}i_{ξ_{L}}α (in the notation of Proposition 1(4)).

If any of the equivalent conditions of Proposition 2 is satisfied we shall say that the independent first integrals a^{K}, 1 ≤ K ≤ p, are in involution (cf. [9],[4]).

The following theorem, which is a generalization of the classical Liouville theorem from the calculus of variations to closed 2-forms of constant rank, is of fundamental importance for integration of characteristic distributions of closed 2-forms [9].

Theorem 1. Let Δ be a complete integral of a closed 2-form α of rank 2p on M. Then at each point of dom Δ there exists a chart (U, φ), φ = (a^{K}, b_{K}, y^{S}), 1 ≤ K ≤ p, 1 ≤ S ≤ dim M − 2p, such that (1) Δ = span{da^{K}, 1 ≤ K ≤ p},

(2) α = da^{K} ∧ db_{K},

(3) the set of functions {a^{K}, b_{K}, 1 ≤ K ≤ p} is a complete set of independent first integrals of the characteristic distribution of α.

Proof [9]. Let (W, χ), χ = (x^{J}, a^{K}), 1 ≤ J ≤ dim M − p, 1 ≤ K ≤ p, be a chart adapted to Δ at a point of M. Then in this chart

(2.3)  α = (g_{JK}dx^{J} + h_{LK}da^{L}) ∧ da^{K},

where we can set h_{LK} = −h_{KL}. Let x be a point in W. Since α is closed, there exist a neighbourhood of x, and a form ρ defined on this neighbourhood, such that α = dρ. Denote

(2.4)  ρ = ρ_{J}dx^{J} + ρ̂_{K}da^{K}.

Then it holds

(2.5)  \frac{∂ρ_{I}}{∂x^{J}} − \frac{∂ρ_{J}}{∂x^{I}} = 0, \frac{∂ρ̂_{K}}{∂x^{J}} − \frac{∂ρ_{J}}{∂a^{K}} = g_{JK}, \frac{∂ρ̂_{K}}{∂a^{L}} − \frac{∂ρ̂_{L}}{∂a^{K}} = 2h_{KL},

where 1 ≤ I, J ≤ dim M − p, 1 ≤ K, L ≤ p, and we conclude that

(2.6)  ρ_{J} = \frac{∂f}{∂x^{J}}

for a function f(x^{J}, a^{K}) defined in a neighbourhood of x. Hence, we have shown that to each point of W there exist a neighbourhood U ⊂ W and functions ρ̂_{K}, 1 ≤ K ≤ p, and f on U such that α = dρ, where

(2.7)  ρ = \frac{∂f}{∂x^{J}} dx^{J} + ρ̂_{K}da^{K} = df + (ρ̂_{K} − \frac{∂f}{∂a^{K}} )da^{K}.

Put

(2.8)  b_{K} = \frac{∂f}{∂a^{K}} − ρ̂_{K}.

Since ∂b_{K}/∂x^{J} = −g_{JK} and the matrix (g_{JK}) has maximal rank, there can be found functions y^{S}, 1 ≤ S ≤ dim M −2p, on U such that the mapping (x^{J}, a^{K}) → (a^{K}, b_{K}, y^{S}) is a coordinate transformation. Finally, from the expression of α in the chart (U, φ), φ = (a^{K}, b_{K}, y^{S}) we see that the characteristic distribution of α is spanned by the forms da^{K}, db_{K}, 1 ≤ K ≤ p, which are linearly independent at each point of U. This proves that {a^{K}, b_{K}, 1 ≤ K ≤ p} is a complete set of independent first integrals of the characteristic distribution of α.

The following corollary is a reformulation of Theorem 1 in a way similar to the classical Liouville theorem, and could be referred to as “generalized Liouville integration method”:

Corollary 1. Let α be a closed 2-form of constant rank 2p on M. Let a^{K}, 1 ≤ K ≤ p, be independent first integrals in involution of the characteristic distribution D of α, defined on an open subset of M. Then the set of functions {a^{K}, b_{K}, 1 ≤ K ≤ p}, where b_{K} are defined by (2.8) is a complete set of independent first integrals of D.

Note that the explicit formulas for ρ_{J}, ρ̂_{K}, and f can be easily found with help of the Poincaré lemma. Hence the first integrals b_{K} can be found explicitly by means of differentiation and integration (“by quadratures”). For example, if we have found a form ρ such that α = dρ, we can take f in the form

(2.9)  f = x^{J} ∫_{0}^{1}(ρ_{J} ∘ μ)dυ + h(a^{K}),

where ρ_{J} are defined by ρ = ρ_{J}dx^{J} + ρ̂_{K}da^{K}, the mapping μ : [0, 1] × U → U is defined by

(2.10)  μ(v, (x^{J}, a^{L})) = (vx^{J}, a^{L}),

and h(a^{K}) is an arbitrary (but fixed) function on U (note that ∂h/∂x^{J} = 0, i.e. h is a first integral of the distribution Δ).

Another possible reformulation of Theorem 1 is the following “coordinate-free Liouville theorem”, providing a geometric look on the integration procedure described by Theorem 1.

Corollary 2. Let Δ_{1} be a complete integral of a closed 2-form α of constant rank 2p on M. Then to each point of dom Δ_{1} there exists a neighbourhood U and a complete integral Δ_{2} of α on U such that Δ_{1} ∩ Δ_{2} is the characteristic distribution of α on U.

There is another possibility than that described by Corollary 1 how to use Theorem 1 to integrate the characteristic distribution of α. Consulting the proof of Theorem 1 we get

Corollary 3. Let Δ be a complete integral of a closed 2-form α of constant rank 2p on M. Then to each point x of dom Δ there exists a neighbourhood W and a 1-form ρ̄ on W such that α = dρ̄, and ρ̄ belongs to Δ (i.e. i_{ξ}ρ̄ = 0 for all vector fields spanning Δ on U).

Hence, once we know ρ (defined on an open subset U ⊂ M) such that α = dρ on U, we are able to find a form ρ̄ (on a certain open subset of U) such that α = dρ̄ and ρ̄ ∈ Δ. This is equivalent to the problem of finding a function f satisfying

(2.11)  ρ − df ∈ Δ .

In every chart (x^{J}, a^{K}), 1 ≤ J ≤ rank Δ, 1 ≤ K ≤ corank Δ, adapted to Δ (i.e. such that Δ = span{da^{K}, 1 ≤ K ≤ corank Δ}) the condition (2.11) reads

(2.12)  ρ̄ = ρ − df = b_{K}da^{K},

i.e., expressing ρ in the form ρ = ρ_{J}dx^{J} + ρ̂_{K}da^{K}, we get the equations (2.6) and (2.8). Now, obviously, the functions b_{K} = i_{∂/∂a^{K}} ρ̄, 1 ≤ K ≤ p, are independent first integrals of the characteristic distribution D of α. The above described integration method could be referred to as “generalized Jacobi integration method”.

Remarks. (1) The problem of finding a one-form ρ such that α = dρ̄ and ρ̄ vanishes on the maximal integral submanifolds of Δ has been pointed out in [1] where the condition (2.11) has been introduced as a modification of the Hamilton-Jacobi equation.

(2) The Jacobi integration method for characteristic distributions of closed 2-forms can be explained in terms of a pair of complementary distributions. To this purpose notice that if Δ is a complete integral of α and Δ^{c} is a distribution complementary to Δ, then at each point x of the common domain of definition of Δ and Δ^{c} there exists a chart (x^{J}, a^{K}) adapted to both Δ and Δ^{c}, i.e. such that

(2.13)  Δ = span\{da^{K}\} = span\{\frac{∂}{∂x^{J}} \}, Δ^{c} = span\{dx^{J}\} = span\{\frac{∂}{∂a^{K}} \},

where 1 ≤ K ≤ corankΔ, 1 ≤ J ≤ rankΔ. Conversely, if Δ is a complete integral of α and (U, φ), φ = (x^{J}, a^{K}) is a chart adapted to Δ such that Δ = span{da^{K}}, then by Δ^{c} = span{∂/∂a^{K}} = span{dx^{J}} a complementary distribution Δ^{c} to Δ is defined on U. Now, suppose we are given two complementary distributions Δ and Δ^{c}, defined on an open set U ⊂ M, and such that Δ is a complete integral of α. Since any 1-form ω on U can be uniquely decomposed into a sum ω = P ω + P^{c}ω where P ω belongs to Δ and P^{c}ω belongs to Δ^{c}, we get Corollary 3 in the following form:

Let Δ be a complete integral of α, and Δ^{c} be a distribution complementary to Δ, both Δ and Δ^{c} defined on an open set U ⊂ M. Then in a neighbourhood of each point of U there exists a 1-form ρ̄ such that α = dρ̄ and P^{c}ρ̄ = 0.

Hence, if ρ is a 1-form such that α = dρ, then “locally” there exists a function f such that

(2.14)  P^{c}ρ = P^{c}df;

ρ̄ is then given by ρ̄ = P ρ̄ = P ρ−P df. Equation (2.14) covers two special cases discussed for the case rank α = maximal in [11]; namely (1) Δ and Δ^{c} are such that to a given 1-form ρ there exists f satisfying P^{c}ρ = df (cf. the “Noether splitting” in [11]), or (2) Δ and Δ^{c} are such that to a given 1-form ρ there exists f satisfying ρ = P^{c}df (cf. the “Jacobi splitting” in [11]).

We shall use a slightly different description of geometric structures connected with complete integrals than that mentioned in the above remark; this will be subject of the following section.

3. Jacobi theorem for closed 2-forms

In this section we shall show that Corollary 3 to Theorem 1 can be proved directly if one makes use of fibered-manifolds techniques. This approach will also provide us with a possibility to simplify (in a certain situation) the Jacobi integration method.

To this purpose we shall need some basic concepts from the theory of jets and connections on fibered manifolds (see e.g. [12] and [10]). For the reader’s convenience we recall briefly some definitions and assertions from the calculus of horizontal and contact forms on fibered manifolds. For more details we refer to [5], [6].

Let π : Y → X be a fibered manifold, dim X = n, dim Y = n + m. Denote by π_{1} : J^{1}Y → X the first jet prolongation of π (π_{1}, as well as the canonical projection π_{1,0} : J^{1}Y → Y, is a fibered manifold). Points of J^{1}Y (i.e. 1-jets of local sections γ of π) will be denoted by J_{x}^{1}γ.

A vector field ξ on Y is called π-vertical if T π.ξ = 0.

A q-form ρ on Y is called π-horizontal if i_{ξ}ρ = 0 for each π-vertical vector field ξ on Y. ρ is called π-projectable if there exists a form ρ_{0} on X such that π^{∗}ρ_{0} = ρ; (if ρ_{0} exists it is unique, and is called the π-projection of ρ).

Let ρ be a q-form on Y, q ≥ 1. For every J_{x}^{1}γ ∈ J^{1}Y and any vectors ξ_{1}, ..., ξ_{q} ∈ T J^{1}Y at the point J_{x}^{1}γ we set

(3.1)  hρ(J_{x}^{1}γ)(ξ_{1}, ..., ξ_{q}) = ρ(γ(x))(T_{x}γ.T π_{1}.ξ_{1}, ..., T_{x}γ.T π_{1}.ξ_{p}).

Evidently, hρ is a π_{1}-horizontal q-form on J^{1}Y. The mapping h : ρ → hρ is called the π-horizontalization. It has the following properties:

(1) for any q-forms ρ, ω and any p-form η

h(ρ + ω) = hρ + hω, h(ρ ∧ η) = hρ ∧ hη;

(2) if q > n then hρ = 0;

(3) if ρ is π-horizontal then hρ is π_{1,0}-projectable (if there is no danger of confusion we usually write ρ = hρ in this case);

(4) hhρ = hρ .

A q-form ρ on Y (or, possibly on some r-jet prolongation J^{r}Y of π) is called π-contact if hρ = 0. For any q-form ρ on Y the form π_{1,0}^{∗}ρ on J^{1}Y has the unique decomposition

(3.2)  π_{1,0}^{∗}ρ = hρ + pρ.

Clearly, pρ is a π-contact form. The mapping p assigning to a q-form ρ on Y the πcontact q-form pρ on J^{1}Y is called the contactization (with respect to π). Note that p does not preserve the wedge product.

Let ρ be a π-contact form on Y (i.e. let π_{1,0}^{∗}ρ = pρ). We say that ρ is 1-contact if for each π_{1}-vertical vector field ξ on J^{1}Y the form i_{ξ}(π_{1,0}^{∗}ρ) is π_{1}-horizontal. We say that ρ is k-contact (1 < k ≤ q) if i_{ξ}(π_{1,0}^{∗}ρ) is (k − 1)-contact.

We shall recall two assertions which are of fundamental importance in the calculus on fibered manifolds:

Decomposition theorem [5], [6]. Let ρ be a q-form on Y. Then π_{1,0}^{∗}ρ has the unique decomposition

(3.3)  π_{1,0}^{∗}ρ = hρ + p_{1}ρ + … + p_{q}ρ,

where p_{i}ρ, 1 ≤ i ≤ q, is a i-contact q-form on J^{1}Y.

For each i, 1 ≤ i ≤ q, the mapping p_{i} : ρ → p_{i}ρ defined by (3.3) is called the i-contactization (with respect to the projection π).

Let ρ be a π-contact q-form on Y. Let k be a fixed integer, 1 ≤ k ≤ q. We say that the order of contactness of ρ is equal to k if p_{i}ρ = 0 for all i > k.

Poincaré lemma for contact forms [5], [6]. Let ρ be a q-form on Y, suppose that dρ = 0 and the order of contactness of ρ is equal to k ≥ 1. Then to each point of Y there exists its neighbourhood W and a (q − 1)-form η on W such that

(1) the order of contactness of η is equal to k − 1,

(2) dη = 0 on W.

Now, let us return to closed 2-forms.

Let α be a closed 2-form of rank 2p on M, let Δ be a complete integral of α, defined on an open set U ⊂ M. Denote by F the foliation of U defined by Δ. Lowering U if necessary, we have a (global) submersion π^{F} : U → U/F of U onto the quotientmanifold U/F of U by the foliation F (recall that the equivalence relation in U is the following: x_{1}, x_{2} ∈ U are equivalent iff both they belong to the same leaf of F).

In correspondence with the above notations let us denote by π_{1}^{F} the first jet prolongation of π^{F}, and by h^{F}, p_{1}^{F} and p_{2}^{F} the horizontalization, 1-contactization and 2contactization with respect to π^{F}, respectively. Just from the definition of π^{F} we recognize that the order of contactness of α is equal to 1, i.e. it holds p_{2}^{F}α = 0. Since α is closed, we can apply the Poincaré lemma for contact forms to get

Proposition 3. To each point x ∈ U there exist a neighbourhood W and π^{F}-horizontal 1-form ρ̄ on W such that α = dρ̄ on W.

Note that the condition that ρ̄ is π^{F}-horizontal means h^{F} ρ̄ = ρ̄ and p_{1}^{F} ρ̄ = 0, i.e. Proposition 3 is nothing but a “coordinate free” reformulation of Corollary 3 to Theorem 1 in terms of the theory of fibered manifolds. Expressing the condition

(3.4)  ρ̄ = ρ − df = h^{F} ρ̄

in a fiber chart on π^{F} we get (2.6) and (2.8).

The generalized Jacobi integration method is then expressed in the following way:

Corollary. Let α and ρ̄ be as above, let D be the characteristic distribution of α, corank D = 2p. Then

(1) ρ̄ is invariant with respect to D, i.e. for each vector field ζ belonging to D, ∂_{ζ} ρ̄ = 0.

(2) In each fiber chart (V, ψ), ψ = (a^{K}, x^{J}) on π^{F} it holds ρ̄ = b_{K}da^{K} where b_{K} are first integrals of D. The rank of the matrix (∂b_{K}/∂x^{J}) is constant on V and equal to p (i.e. it is maximal).

Proof. If ζ belongs to D, it is π^{F}-vertical (since D ⊂ Δ), and i_{ζ}α = 0. But ρ̄ is π^{F}-horizontal, i.e. i_{ζ} ρ̄ = 0. Hence, ∂_{ζ} ρ̄ = i_{ζ}dρ̄ + di_{ζ} ρ̄ = 0, proving (1).

Let (V, ψ), ψ = (x^{J}, a^{K}) be a fiber chart on U, associated with a chart (W, φ), φ = (a^{K}) on U/F. Since ρ̄ = h^{F} ρ̄, we have ρ̄ = b_{K}da^{K} where b_{K} are functions on V. Let ζ belong to D, and let ζ = ζ^{J}∂/∂x^{J} be the chart expression of ζ (note that ζ is π^{F}-vertical). Since i_{ζ}α = 0, we get i_{ζ}(db_{K} ∧ da^{K}) = (∂_{ζ}b_{K})da^{K} = 0, proving that b_{K}, 1 ≤ K ≤ dim U/F, are first integrals of D. Since rank α = 2p and α = db_{K} ∧ da^{K} on V, we get rank (∂b_{K}/∂x^{J}) = const. = p on V.

Let Γ be a connection on π^{F}, i.e. a section of the fibered manifold π_{1,0}^{F} : J^{1}π^{F} → U. Suppose Γ be flat (which means that the associated π^{F}-horizontal distribution on U, denoted also by Γ, is completely integrable). Denote by G the foliation defined by Γ. Lowering U if necessary, we can consider the quotient manifold U/G and the fibered manifold π^{G} : U → U/G. (Notice that, on π^{G}, G is the vertical bundle while F is a connection). Evidently, in general, the order of contactness of α with respect to the projection π^{G} is equal to 2.

Denote by h^{G}, p_{1}^{G} and p_{2}^{G} the horizontalization, 1-contactization and 2-contactization with respect to π^{G}, respectively.

A complete integral Δ of α on U will be called canonically associated with a (flat) connection Γ on π^{F} if p_{2}^{G}α = 0. Again, as a direct consequence of the Poincaré lemma for contact forms we obtain:

Proposition 4. Let α, Δ and π^{F} be as above. If Δ is canonically associated with a flat connection Γ on π^{F}, then to each point x ∈ U there exist a neighbourhood W and a π^{G}-horizontal form ρ̃ on W such that α = dρ̃ on W.

Note that if Δ is a complete integral of α on U, and Γ is a completely integrable distribution on U, complementary to Δ, then Γ is a flat connection on π^{F}. Further note that if (V, ψ), ψ = (x^{J}, a^{K}) is a chart at a point x ∈ U, adapted to the complementary distributions Γ and Δ (i.e. such that Δ = span{da^{K}} and Γ = span{dx^{J}}, where 1 ≤ K ≤ corank Δ, 1 ≤ J ≤ rank Δ), then (V, ψ) can be viewed as a fiber chart on π^{F} associated with a chart (W_{1}, φ_{1}), φ_{1} = (a^{K}) on U/F, or as a fiber chart on π^{G} associated with a chart (W_{2}, φ_{2}), φ_{2} = (x^{J}) on U/G, respectively.

Now, using Proposition 3 and 4 we get immediately a general version of the classical Jacobi theorem.

Theorem 4. Let α be a closed 2-form of rank 2p on M. Let Δ be a complete integral of α, defined on an open set U ⊂ M, and denote by F the foliation of U by Δ. Let Γ be a distribution on U, completely integrable, and complementary to Δ; denote by G the foliation defined by Γ. Suppose that π^{F} : U → U/F and π^{G} : U → U/G are fibered manifolds, and that Δ is canonically associated with Γ. Then to each point x ∈ U there exists a neighbourhood W and, on W

(1) a π^{G}-horizontal 1-form ρ̃ such that α = dρ̃,

(2) a function f such that the 1-form ρ̄ = h^{G}ρ̃ − df is π^{F}-horizontal.

If (V, ψ), ψ = (x^{J}, a^{K}) is a chart at x adapted to Δ and Γ (i.e. Δ = span{da^{K}, 1 ≤ K ≤ corankΔ}, Γ = span{dx^{J}, 1 ≤ J ≤ rankΔ}), then it holds ρ̃ = ρ̃_{J}dx^{J} where

(3.5),  ρ̃_{J} = \frac{∂f}{∂x^{J}} , 1 ≤ J ≤ rank Δ

and the functions

(3.6)  b_{K} = \frac{∂f}{∂a^{K}} , 1 ≤ K ≤ corank Δ

are first integrals of D. The rank of the matrix

(3.7)  (\frac{∂b_{K}}{∂x^{J}} ) = (\frac{∂^{2}f}{∂x^{J}∂a^{K}} )

is constant on V and equal to p.

Theorem 4 can be used to find local solutions of the characteristic distribution of a closed 2-form of constant rank, especially in case we know ρ̃. Then a function f satisfying (3.5) can be easily constructed and the “remaining” first integrals b_{K} can be found according to (3.6).

4. Applications to integration of differential equations

The Liouville and Jacobi integration methods can be used to find local solutions of characteristic distributions of closed 2-forms. The purpose of this section is to give a few examples of systems of ordinary and partial differential equations of order r ≥ 1 which can be interpreted as equations for integral mappings of characteristic distributions of closed 2-forms, and hence integrated by means of the Liouville or Jacobi theorem.

4.1. Integration of vector distributions

Let M be a manifold, let D be a completely integrable distribution on M.

Proposition 5. (1) Let corank D be even and equal to 2p. Then to each point x ∈ M there exists a neighbourhood W and a closed 2-form α of rank 2p on W such that D is the characteristic distribution of α.

(2) Let corank D be odd and equal to 2p+1. Then to each point x ∈ M there exists a neighbourhood W and a closed 2-form α of rank 2p on W such that D is a subdistribution of the characteristic distribution of α.

Proof. (1) Frobenius theorem ensures the existence of an adapted chart (W, φ), φ = (y^{J}, f_{K}), 1 ≤ J ≤ rankD, 1 ≤ K ≤ 2p, at x, such that f_{K} are first integrals of D. Putting α = df_{1} ∧ df_{2} + … + df_{2p−1} ∧ df_{2p} we get a desired closed 2-form on W.

(2) Let f_{1},…, f_{2p+1} be independent first integrals of D, defined in a neighbourhood W of x. Putting α = df_{1} ∧ df_{2} + … + df_{2p−1} ∧ df_{2p} the proof is finished.

By the above proposition every involutive vector distribution D of constant rank on a manifold M can be associated with a (generally non-unique, locally defined) closed 2-form, i.e. it can be solved by means of either the Liouville or Jacobi integration method. If corank D is even then these methods provide us directly with a complete set of independent first integrals of D. In case that corank D is odd we get in this way a foliation of dimension equal to rank D + 1, whose subfoliation is D; hence the integration problem for D is reduced to the problem of finding one additional first integral. In concrete practical situations, the problem of finding an appropriate 2-form α can lead to rather unpleasant computations. However, one has several possibilities how to proceed. For example, (1) one can solve the equations i_{ζ_{K}}α = 0, 1 ≤ K ≤ rank D, where ζ_{K} are independent generators of D, with respect to a closed 2-form α of rank 2p, or (2) one can solve the conditions ∂_{ζ_{K}}ρ = 0, di_{ζ_{K}}ρ = 0, 1 ≤ K ≤ rank D, with respect to a 1-form ρ such that rank dρ = 2p, and put α = dρ, or (3) if Δ is spanned by the 1-forms ω_{1}, ⋯, ω_{2p} (resp. ω_{1}, ⋯, ω_{2p+1}), to find functions a^{JK} such that the form α = a^{JK}ω_{J} ∧ω_{K} is closed and of rank 2p; in this case it is convenient to use the fact that the forms ω_{K} satisfy the Frobenius condition dω_{K} = η_{K}^{L} ∧ ω_{L}, and moreover, that they can be completed to a basis of linear forms on M, and to perform the computations in this basis.

4.2. Distributions on fibered manifolds

All above applies to the case when D is a distribution defined on an r-jet prolongation J^{r}Y, r ≥ 0, of a fibered manifold π : Y → X, dim X = n, dim Y = m + n. In this case obviously M = J^{r}Y. In most applications the distributions in question are connections [10], semispray connections [3], [13], or generalized connections [9] on fibered manifolds, and the problem is to search for integral sections of these distributions. Consequently, the Liouville and Jacobi integration methods can be used to find solutions of a wide class of systems of higher-order ordinary and partial differential equations.

4.3. Variational ordinary differential equations

It has been shown in [8] (see also [9], [7]) that a system of generally higher-order variational ordinary differential equations (i.e. arising from a possibly higher-order Lagrangian as its Euler-Lagrange equations) can be interpreted as equations for integral sections of a distribution D on a certain jet-prolongation of a fibered manifold π : Y → X (with the base X of dimension 1); in general, rank D is nonconstant and can be greater than one. The most important cases are those when D is an integrable distribution of constant rank, complementary to a subbundle Γ of the vertical bundle (i.e. when it is a generalized connection) [9]. Then obviously the integration problem for D can be solved by means of the Liouville or Jacobi integration method. However, it should be stressed that due to the existence of Lagrangians the situation simplifies essentially:

(1) the problem of finding an appropriate closed 2-form to D is solved canonically by the existence (and easy construction) of the so called Lepagean 2-form [9],

(2) for even-order equations, D is canonically associated with a subbundle of the vertical bundle, hence the Jacobi theorem can be applied easily to find the extremals.

For details on the integration methods arising from the Liouville and Jacobi theorems for (generally degenerate) higher-order variational equations we refer to [9].

Acknowledgment

I am grateful to Prof. Z. Oziewicz for valuable discussions during the conference meeting.

References

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[12] D. J. Saunders, “The Geometry of Jet Bundles”, London Math. Soc. Lecture Notes Series 142, Cambridge Univ. Press, Cambridge, 1989.

[13] A. Vondra, Semisprays, connections and regular equations in higher order mechanics, in: Differential Geometry and Its Applications, Proc. Conf., Brno, Czechoslovakia, 1989, J. Janyška and D. Krupka eds. (World Scientific, Singapore, 1990), 276–287.

Author’s address: O. Krupková, Department of Mathematics, Silesian University, Bezručovo nám. 13, 746 01 Opava, Czech Republic

Received 15 October 1992