Canonical formalism for degenerate Lagrangians

Abstract

A Lagrangian is degenerate when the Hessian matrix whose elements consist of all the second‐order derivatives of the Lagrangian with respect to the generalized velocities has (for simplicity) a constant singular rank everywhere in the space of the arguments of the Lagrangian. This singularity entails a definite number of first‐order Lagrange equations, which act as subsidiary conditions on the coordinates and velocities. Consistency of these subsidiary conditions with the Langrange system requires them to be an invariant system with respect to the second‐order Lagrange equations. An invariant system is analogous to a system of first integrals except that absolute constants appear where arbitrary constants characterize first integrals. The usual definitions of momenta and of Hamiltonian make the Hamiltonian a function of the functionally dependent canonical variables only. Introduction of the momentum variables into the subsidiary conditions on the coordinates and velocities yields under certain circumstances additional subsidiary conditions on the canonical variables only. All the subsidiary conditions on the canonical variables are determined before setting up the multiplier rule for the canonical equations of motion. The multiplier rule is exploited to deduce the invariant system among the subsidiary conditions, the explicit modifications of the canonical equations by the other susidiary conditions, and Dirac's formula for the corresponding modified Poisson brackets. The modifications are caused by the reduction in the number of independent canonically conjugate pairs. A canonical transformation adapted to the subsidiary conditions, which is found with the help of Lie's theory of function groups, transforms the canonical system from the multiplier rule into a canonical system in terms of physical variables. The invariant system is then used to reduce the order of the resulting canonical system by following Levi‐Civita. This reduced canonical system is suitable for integration or quantization.