On Linear Friction in Lagrange's Equation

Denman, Harry H.

doi:10.1119/1.1972535

Cited by 34 publications

(18 citation statements)

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“…The first approach is a phenomenological way, by which the effect of dissipation is taken into account by constructing a suitable Lagrangian or Hamiltonian for the system [2,3]. Following this method the first Hamiltonian was proposed by Caldirola [4] and Kanai [5] and afterward by others [6,7]. There are difficulties about the quantum mechanical solutions of the Caldirola-Kanai Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

A quantum dissipative system in an anisotropic non-linear absorbing environment

Amooshahi

Amooghorban

2010

Annals of Physics

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Section: Introductionmentioning

confidence: 99%

A quantum dissipative system in an anisotropic non-linear absorbing environment

Amooshahi

Amooghorban

2010

Annals of Physics

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“…In the Heisenberg picture, the equations of motion for canonical variables X ω and Q ω are the same equations (26) and (27) with the formal solution (29) and likewise the R field is defined by Eq. (30).…”

Section: Radiation Reactionmentioning

confidence: 99%

“…The second approach is essentially a rigorous one in which the effect of dissipation is introduced by ingeniously construction a suitable Lagrangian or Hamiltonian for the system [24,25]. Historically, the first Hamiltonian was introduced by Caldirola [26] and rederived independently by Kanai [27] and afterward by several others [28,29]. They employed a time dependent mass in such a way that a friction term appears in the corresponding equation of motion.…”

Section: Introductionmentioning

confidence: 99%

Relativistic and Non-Relativistic Quantum Brownian Motion in an Anisotropic Dissipative Medium

Amooghorban

Kheirandish²

2014

Int J Theor Phys

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Using a minimal-coupling-scheme we investigate the quantum Brownian motion of a particle in an anisotropic-dissipative-medium under the influence of an arbitrary potential in both relativistic and non-relativistic regimes. A general quantum Langevin equation is derived and explicit expressions for quantum-noise and dynamical variables of the system are obtained. The equations of motion for the canonical variables are solved explicitly and an expression for the radiation-reaction of a charged particle in the presence of a dissipative-medium is obtained. Some examples are given to elucidate the applicability of this approach.

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“…We give here a simple proof of formula (29). Using the hypothesis and notations of section 4, the equality L = L D(t), i.e.…”

Section: Appendixmentioning

confidence: 98%

Classical Noether theory with application to the linearly damped particle

Leone¹,

Gourieux²

2015

Eur. J. Phys.

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This paper provides a modern presentation of Noether's theory in the realm of classical dynamics, with application to the problem of a particle submitted to both a potential and a linear dissipation. After a review of the close relationships between Noether symmetries and first integrals, we investigate the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. This analysis leads to the determination of all the time-independent potentials allowing such symmetries, in the one-dimensional and the radial cases. Then we develop a symmetry-based transformation of Lagrangians into autonomous others, and apply it to our problem. To be complete, we enlarge the study to Lie point symmetries which we associate logically to Noether ones. Finally, we succinctly address the issue of a 'weakened' Noether's theory, in connection with 'on-flows' symmetries and non-local constant of motions, for it has a direct physical interpretation in our specific problem. Since the Lagrangian we use gives rise to simple calculations, we hope that this work will be of didactic interest to graduate students, and give teaching material as well as food for thought for physicists regarding Noether's theory and the recent developments around the idea of symmetry in classical mechanics. Lagrangian ‡ The BCK Lagrangian can be adapted to a time-dependent dissipation rate; it suffices to replace γt by γ(t) dt in the exponential factor of (1).

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On Linear Friction in Lagrange's Equation

Cited by 34 publications

References 0 publications

A quantum dissipative system in an anisotropic non-linear absorbing environment

A quantum dissipative system in an anisotropic non-linear absorbing environment

Relativistic and Non-Relativistic Quantum Brownian Motion in an Anisotropic Dissipative Medium

Classical Noether theory with application to the linearly damped particle

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