The Hamilton-Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes

TL;DR

This paper applies the Hamilton-Jacobi framework to analyze topological invariants like Euler and Pontryagin classes, revealing their equations of motion and symmetry differences within a unified phase space approach.

Contribution

It constructs a fundamental Hamilton-Jacobi differential for topological theories and compares the symmetries of Euler and Pontryagin classes using this framework.

Findings

Euler and Pontryagin classes share equations of motion

Their symmetries differ despite similar dynamics

Provides a unified Hamilton-Jacobi formulation for topological invariants

Abstract

By using the Hamilton-Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental $HJ$ differential where the characteristic equations and the symmetries of the theory are identified. Moreover, we work in both theories with the same phase space variables and we show that in spite of Pontryagin and Euler classes share the same equations of motion their symmetries are different. In addition, we report all HJ Hamiltonians and we compare our results with other formulations reported in the literature.