Symmetry-preserving perturbations of the Bateman Lagrangian and dissipative systems
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ELEMENTARY PARTICLES AND FIELDS Theory
Symmetry-Preserving Perturbations of the Bateman Lagrangian and Dissipative Systems∗ Rutwig Campoamor-Stursberg** Instituto de Matematica ´ Interdisciplinar and Departamento Geometr ´ıa y Topolog ´ıa, Faculted de Ciencias Matematicas Universidad Complutense, Madrid, Spain Received May 18, 2016
Abstract—Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, R) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2, R), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems. DOI: 10.1134/S1063778817020107
1. INTRODUCTION Within the frame of classical dynamics, dissipative forces are understood as those types of interactions for which energy is lost during motion. Assuming that the consequence of such interactions is the energy transfer from the dissipative part of a system to the heath bath, successful models to describe various types of phenomena have been developed, albeit the quantization and its interpretation, due to the wellknown ambiguities of the Lagrangian (Hamiltonian) formalism, can give rise to inconsistencies between the canonical commutation relations and the equations of motion (see [1–4] and references therein). Dissipative systems of various types have been considered by different authors by means of the Lagrangian formulation, and first integrals have been obtained by using either the classical Noether theorem or some of its generalizations [5–10]. One of the principal difficulties arises from the correct identification of a Lagrangian or Hamiltonian that displays correctly the physical properties of the system. In this context, although mathematical Lagrangians or Hamiltonians have been shown to be useful, they must be handled with care in order to avoid ambiguities and misleading interpretations within the socalled canonical formalism [11, 12]. A standard procedure to circumvent the difficulties arising from such phenomenological approaches consists in coupling the dissipative system to an environment with additional degrees of freedom, in order that the systemplus-reservoir is a Hamiltonian system [13]. One of ∗ **
The text was submitted by the author in English. E-mail: [email protected]
the first examples to be analyzed from this perspective was the damped harmonic oscillator, completed with one additional degree of freedom by Bateman in [14]. The added “dual" equation is a time-reversed version of the oscillator, and the corresponding variable fulfills the absorption of the energy dissipated by the damped oscillator [14–16]. In this work we reconsider the classical Bateman system from the perspective of Noether symmetries. Using that the system is linearizable as a system of second-order ordinary differential equations, we analyze the non-linear perturbations of the Bateman Lagrangian that preserve a certain subalgebra of No
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