Constrained Hamiltonian Systems with Higher-Order Lagrangians

Abstract

The higher-order regular Lagrangian is reduced to first-order singular Lagrangian. Dirac’s method of discrete regular systems with higher-order Lagrangian, are studied as singular systems with first-order Lagrangian, and the equations of motion are obtained. It is shown that the Hamilton-Jacobi approach leads to the same equations of motion as obtained by Dirac’s method. The second-order and third-order Lagrangian are studied as an examples. The Hamilton-Jacobi formulation for first-order constrained systems has been discussed. In such formalism the equations of motion are written as total differential equations in many variables. We generalize the Hamilton-Jacobi formulation for singular systems with second-order Lagrangians and apply this new formulation to Podolsky electrodynamics, comparing the results with the results obtained through Dirac’s method. The equations of motion for the associated Lagrangian to a nonholonomic Lagrangian of second-order are computed in both methods Dirac and Hamilton-Jacobi. Besides, the canonical path integral quantization was obtained to quantize singular systems. All the results obtained using Hamilton-Jacobi method, are in exact agreement with those results obtained using Dirac’s method.