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The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

Luca Schiavone

Abstract

We carry on the approach used in [Sch] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds.

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

Abstract

We carry on the approach used in [Sch] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds.
Paper Structure (4 sections, 3 theorems, 31 equations)

This paper contains 4 sections, 3 theorems, 31 equations.

Table of Contents

  1. The coisotropic embedding theorem
  2. The symplectic thickening
  3. Lift of vector fields to the symplectic thickening
  4. The inverse problem for a class of implicit differential equations

Key Result

Theorem 1

Given a vector field $\Gamma$ over a manifold $\mathcal{M}$, if there exists a symplectic form $\omega$ on $\mathcal{M}$ such that: then, there exists a local Lagrangian function $\mathscr{L}$ on $\mathbf{T}\mathcal{M}$ such that the pre-symplectic Hamiltonian system $(\mathbf{T}\mathcal{M},\, \omega_\mathscr{L},\, E_\mathscr{L})$ yields, by applying the pre-symplectic constraint algorithm (see G

Theorems & Definitions (6)

  • Theorem 1: Lagrangian for first order odes
  • Theorem 2: Coisotropic embedding theorem
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2