Hamiltonian formulation of a class of constrained fourth-order differential equations in the Ostrogradsky framework

TL;DR

This paper develops a Hamiltonian framework for a class of fourth-order differential equations derived from higher-order Lagrangians, utilizing the Ostrogradsky formalism to handle constraints and enable quantization.

Contribution

It introduces a Hamiltonian formulation for constrained fourth-order equations using Ostrogradsky's method, addressing stability and quantization issues.

Findings

Hamiltonian structure allows natural constraints to eliminate instabilities.

Constraints facilitate canonical quantization via Dirac's method.

Framework extends Hamiltonian analysis to higher-order Lagrangian systems.

Abstract

We consider a class of Lagrangians that depend not only on some configurational variables and their first time derivatives, but also on second time derivatives, thereby leading to fourth-order evolution equations. The proposed higher-order Lagrangians are obtained by expressing the variables of standard Lagrangians in terms of more basic variables and their time derivatives. The Hamiltonian formulation of the proposed class of models is obtained by means of the Ostrogradsky formalism. The structure of the Hamiltonians for this particular class of models is such that constraints can be introduced in a natural way, thus eliminating expected instabilities of the fourth-order evolution equations. Moreover, canonical quantization of the constrained equations can be achieved by means of Dirac's approach to generalized Hamiltonian dynamics.