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

Hans Christian Öttinger

doi:10.1088/2399-6528/aaf6f2

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 for 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.

Highlights

The vector potentials of the Yang-Mills theory are no longer considered as primary fields, but rather as functions of a decomposition of the metric tensor including time derivatives
As P2 = −∂H/∂Q2 reproduces the representation (9) of P1, the evolution equation must be contained in P1 = −∂H/∂Q1
We have introduced a class of Lagrangians L(q, q, q) associated with fourth-order evolution equations by substituting a transformation q(q, q), as well as the consistent transformation q(q, q, q), into a standard Lagrangian

Summary

LAGRANGIAN FORMULATION

In a monumental article written in 1848 (in French), Mikhail Vasilyevich Ostrogradsky laid the foundations for the Lagrangian and Hamiltonian formulation of higher-order differential equations and pointed out their disposition to instability [1]. It is natural to consider theories given by standard Lagrangians L(q, q) where the variables q(q, q ̇) and q(q, q ̇, q ̄) are given in terms of more fundamental variables q Based on this idea, we introduce a class of higher-order models in the Lagrangian (Section II) and Hamiltonian (Section III) settings, where the reformulation is achieved by means of the Ostrogradsky framework. Note that Eq (7) contains third-order time derivatives of q, implying a set of fourth-order differential equations for q. As a consequence of the chain rule, they have the factorized structure of Eq (7) because they result from second-order differential equations by considering the unknowns as functions of potentially fewer, more basic variables and their time derivatives. Note that Eq (5) is a system of third-order differential equation for q(t) consisting of I equations for K ≤ I functions

HAMILTONIAN FORMULATION

OSTROGRADSKY INSTABILITY

SUMMARY AND CONCLUSIONS