Gauge invariance and the Helmholtz conditions

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Abstract
In a previous work I have outlined a new formalism of theoretical mechanics, extending the traditional subject in a new way to elementary systems whose second‐order dynamical equations do not satisfy the classical Helmholtz conditions for the existence of a Lagrange function. In the present paper I determine the role of a new gauge invariance (’’dynamical’’ gauge invariance), satisfied by the equations of motion of systems covered by the formalism, and the relationship this bears to the Helmholtz conditions. A certain subgroup, of ’’kinematic’’ gauge transformations, is singled out: the kinematic gauge transformations correspond for the Lorentz force law to the usual gauge transformation of electromagnetism, general dynamical gauge transformations correspond to minimal substitutions. An analogue for the nonalbelian gauge force law is discussed briefly. An implication of the present work is the result, in a sense made explicit within, that (the usual) gauge invariance is an intrinsic property of classical Hamiltonian systems.

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