# Revisiting the Hamilton theory for second order Lagrangian

## TL;DR

This paper proposes a novel Hamiltonian formulation for second order Lagrangian systems that eliminates Ostrogradsky's instabilities by redefining canonical momenta and introducing new constraints, leading to more stable dynamics.

## Contribution

It introduces a new definition of second order canonical momentum and a set of Hamilton equations that remove Ostrogradsky's instabilities in higher order Lagrangian theories.

## Key findings

- New second order Hamilton equations derived.
- Constraints depend only on velocities, removing instabilities.
- Canonical variables identified as poles of constraints.

## Abstract

The Hamilton theories for higher orders classical Lagrange functions result on a well known Ostrogradski's instabilities. In this work, we propose a different definition for the second order canonical momentum and obtain a new set of second order's Hamilton equations. The identity transformation introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability. The evolution of the system identifies a new set of canonical variables as the poles of the constraints. The second order momentum shows to be the generator for the negative displacement of poles of such constraints. The momentum first order momentum remains as the generator for the displacement of the coordinate.