Branched Hamiltonians and time translation symmetry breaking in equations of the Liénard type

Abstract

Shapere and Wilczek [Phys. Rev. Lett. 109, 160402 and 200402 (2012)] have recently described certain singular Lagrangian systems which display spontaneous breaking of time translation symmetry. We begin by considering the standard Liénard equation for which a Lagrangian is constructed by using the method of Jacobi Last Multiplier. The velocity dependence of the Lagrangian is such that the momentum may exhibit multi-valuedness, thereby leading to the so-called branched Hamiltonian. Next, with a quadratic velocity dependence in the Liénard equation, one can construct a Hamiltonian description involving a position-dependent mass. We compute the Lagrangian and Hamiltonian of this system and show that the canonical Hamiltonian is single valued. However, we find that up to a constant shift, the square of this Hamiltonian describes systems giving rise to spontaneous time translation symmetry breaking provided the potential function is negative.