Symmetry breaking in non conservative systems

Martínez-Pérez, N. E.; Ramı́rez, C.

doi:10.48550/arxiv.1602.05255

Cited by 1 publication

(2 citation statements)

References 15 publications

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“…The canonical momenta ( 16) and ( 17) are p 1 = m q1 + c 2 q 2 and p 2 = m q2 + c 2 q 1 , and the Hamiltonian is (45). From Noether theorem there are four quantities, from (33) and (34) the energies E 1 and E 2 , and from ( 35) and ( 36) the momenta P 1 and P 2 , which are related to the Hamiltonian (28) and to the generator of translations (47).…”

Section: Free Motionmentioning

confidence: 99%

“…Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. Otherwise the symmetries can be studied considering the operator algebra of the system, as done in [7,34,35] In this paper, we start from the nonconservative Lagrangian of Galley.…”

Section: Introductionmentioning

confidence: 99%

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On the Lagrangian description of dissipative systems

Martínez-Pérez

Ramı́rez

2018

Journal of Mathematical Physics

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We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution.PACS numbers: 04.20.Fy,12.60.Jv * Electronic address: nephtalieliceo@hotmail.com † Electronic address: cramirez@fcfm.buap.mx 2 Hamilton variational principle. In this approach the variation of the action is done with boundary conditions only at the initial time, independently for each of both variables, and at the final time these variables must coincide. A similar development for classical and quantum mechanics was given by means of an extension of the Closed Time Path formalism to classical mechanics by Polonyi [40,41], who in [42] considers the issue of breaking of time reversal symmetry.The main interest in the study of phenomenological dissipative systems is on their quantum description. Classically, the doubled variable formalism allows to write the equations of motion and after that the additional variables are somehow discarded. In fact, these variables are considered as an artifice which takes account of the dissipative external influence, the whole system being isolated. However, from its construction, the nonconservative Lagrangian has not the standard form due to the time reversed characteristics of the additional sector, i.e. the kinetic term is not positive definite and the potential appears with an unstable term. Thus, an interpretation of its outcome as a whole is not obvious. On the other side, in a quantum theory every interacting degree of freedom in general contributes to the probabilities, spectra and mean values, as they form part of the operator algebra. Thus, it would be desirable to consider the classical theory taking into account the whole dynamics. Moreover, a general knowledge of the relevant quantities in the theory, as delivered e.g. by Noether theorem, is necessary for the definition of the Hilbert space. Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. ...

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