Modification of Dirac's method of Hamiltonian analysis for constrained systems

Abstract

A slight modification of Dirac's method of Hamiltonian analysis for constrained systems is introduced. It leads to a verification of Dirac's conjecture that first-class secondary constraints are always symmetry generators in each of the counterexamples to that conjecture which have appeared in the recent literature. Those counterexamples associated with differentiable Hamiltonians are studied here; the cases involving nondifferentiable Hamiltonians will be considered separately. The relationship between the Lagrangian and Hamiltonian descriptions is studied in some detail and is used to motivate our calling the form of the secondary constraints derived via this modified method the natural form of the secondary constraints. Along the way we distinguish between symmetries of the Lagrangian (or Hamiltonian) and symmetries of the Euler-Lagrange (or Hamilton's) equations; we also distinguish between form and content invariance.