The Dirac Conjecture and the Non-uniqueness of Lagrangian

TL;DR

This paper completes the proof of the Dirac conjecture by systematically adding total time derivatives of constraints to the Lagrangian, revealing hidden constraints and clarifying the relationship between extended and total Hamiltonians.

Contribution

It provides a complete proof of the Dirac conjecture and demonstrates how adding total derivatives to the Lagrangian uncovers hidden constraints and improves Hamiltonian formulations.

Findings

The Dirac conjecture is now fully proved.

Adding total derivatives reveals hidden constraints.

Extended Hamiltonian can be more effective than total Hamiltonian.

Abstract

By adding the total time derivatives of all the constraints to the Lagrangian step by step, we achieve the further work of the Dirac conjecture left by Dirac. Hitherto, the Dirac conjecture is proved completely. It is worth noticing that the addition of the total time derivatives to the Lagrangian can turn up some constraints hiding in the original Lagrangian. For a constrained system, the extended Hamiltonian $H_E$ considers more constraints, and shows symmetries more obviously than the total Hamiltonian $H_T$. In the Lagrangian formalism, we reconsider the Cawley counterexample, and offer an example in which in accordance with its original Lagrangian its extended Hamiltonian is better than its total Hamiltonian.