Geometric Aspects of Higher Order Variational Principles on Submanifolds
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Geometric Aspects of Higher Order Variational Principles on Submanifolds Gianni Manno · Raffaele Vitolo
Received: 15 November 2007 / Accepted: 3 December 2007 / Published online: 22 January 2008 © Springer Science+Business Media B.V. 2008
Abstract The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided. Keywords Jets of submanifolds · Calculus of variations · Differential forms Mathematics Subject Classification (2000) 58A12 · 58A20 · 58E99 · 58J10 1 Introduction Jets of submanifolds (also known as manifolds of contact elements) are a natural framework for a geometric study of differential equations and the calculus of variations [1, 6–8, 15, 22, 29]. The space of r-th order jets of submanifolds J r (E, n) is introduced through the notion of contact of order r between n-dimensional submanifolds of a given manifold E. These spaces generalize jets of local sections in the sense that submanifolds which are not transversal to a fibration are also considered. In this paper we devote ourselves to the calculus of variations on J r (E, n). This subject has been started in a modern framework in the pioneering papers [7, 28], where the C -spectral sequence was introduced (see Sect. 3). The main problem with respect to jets of fiberings is the absence of a distinguished space of independent variables. This complicates the computations of the terms of the C -spectral sequence. Let us make the above problem more clear with an example. An r-th order Lagrangian on a bundle π : E → M is a section λ : J r π → n T ∗ M, where J r π is the r-th order jet of π and n = dim M. This section can be regarded as an equivalence class λ = [α] ∈ nr /C 1 nr , where nr is the space of n-forms on J r π and C 1 nr is the subspace of 1-contact n forms, i.e. G. Manno () · R. Vitolo Dept. of Mathematics “E. De Giorgi”, Università del Salento, via per Arnesano, 73100 Lecce, Italy e-mail: [email protected] R. Vitolo e-mail: [email protected]
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n-forms vanishing on (the r-order prolongation of) sections of π . A further property of such forms is that they yield no contribution to action-like functionals. The space nr /C 1 nr is an element of the first term of the C -spectral sequence. The problem is then: how to represent an object in nr /C 1 nr in view of the absence of a distinguished space of independent variables? This problem was first considered in [6], where it was proposed to use a sheaf of local Lagrangians whose difference on the intersection of neighborhoods was a contact form. In this paper we discuss the use of the bundle of “truncated” total derivatives H r+1,r → r+1 J (E, n), already introduced in [20], as a natural analogue of the bundle T M in the nonfibered case. We show that this bundle can be used to represent forms in the first term of the C -spectral sequence. Note that the bundle H 1,0 was i
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