The Ostrogradsky Method for Local Symmetries. Constrained Theories with Higher Derivatives

TL;DR

This paper extends the Hamiltonian formalism to systems with higher derivatives, clarifying the role of constraints in local symmetries and demonstrating the canonical nature of transformations in Ostrogradsky's extended phase space.

Contribution

It develops a method to construct local symmetry generators for higher-derivative theories within the Ostrogradsky framework, highlighting the role of first-class constraints and the canonical structure.

Findings

Second-class constraints do not affect the symmetry transformations.

Symmetry transformations are canonical in the extended phase space.

Application demonstrated on the spinor Christ-Lee model.

Abstract

In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them). A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. It is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. An application of the method in theories with higher derivatives is demonstrated with an example of the spinor Christ -- Lee model.