# A Hamilton-Jacobi formalism for higher order implicit systems

## TL;DR

This paper extends Hamilton-Jacobi theory to higher order implicit differential equations using Ostrogradsky and Schmidt approaches, providing new methods to handle implicit systems in geometric mechanics.

## Contribution

It introduces a Hamilton-Jacobi formalism for higher order implicit systems using two different geometric approaches, expanding the theoretical framework for such equations.

## Key findings

- Developed Hamilton-Jacobi equations for implicit systems
- Applied Ostrogradsky and Schmidt transforms to higher order Lagrangians
- Provided examples demonstrating the applicability of the methods

## Abstract

In this paper, we present a generalization of a Hamilton--Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradsky approach involves the addition of new independent variables to account for higher order derivatives, whilst the Schmidt transform adds gauge invariant terms to the Lagrangian function. In these two settings, the implicit character of the resulting equations will be treated in two different ways in order to provide a Hamilton--Jacobi equation. On one hand, the implicit differential equation will be a Lagrangian submanifold of a higher order tangent bundle and it is generated by a Morse family. On the other hand, we will rely on the existence of an auxiliary section of a certain bundle that allows the construction of local vector fields, even if the differential equations are implicit. We will illustrate some examples of our proposed schemes, and discuss the applicability of the proposal.