TY - JOUR

T1 - Branched Hamiltonians and time translation symmetry breaking in equations of the Liénard type

AU - Choudhury, A. Ghose

AU - Guha, Partha

N1 - Publisher Copyright:
© 2019 World Scientific Publishing Company.

PY - 2019/10/20

Y1 - 2019/10/20

N2 - 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.

AB - 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.

KW - Jacobi last multiplier

KW - multi-valued Hamiltonians

KW - position-dependent mass

KW - time translation symmetry breaking

UR - http://www.scopus.com/inward/record.url?scp=85069941813&partnerID=8YFLogxK

U2 - 10.1142/S0217732319502638

DO - 10.1142/S0217732319502638

M3 - Article

AN - SCOPUS:85069941813

SN - 0217-7323

VL - 34

JO - Modern Physics Letters A

JF - Modern Physics Letters A

IS - 32

M1 - 1950263

ER -