The adiabatic invariance of the action variable in classical dynamics

Wells, Clive G.; Siklos, S. T. C.

doi:10.1088/0143-0807/28/1/011

Cited by 23 publications

(22 citation statements)

References 1 publication

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“…This is not surprising since under adiabatic variations of the potential it is the time average of the actions that remains constant (see Goldstein et al 2001; Wells & Siklos 2006). In general, for a given orbit the amplitude of the oscillation depends on the time‐scale on which the potential grows.…”

Section: Frequency Space For Time‐dependent Potentialsmentioning

confidence: 95%

On the identification of substructure in phase space using orbital frequencies

Gómez¹,

Helmi²

2010

Monthly Notices of the Royal Astronomical Society

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We study the evolution of satellite debris to establish the most suitable space to identify past merger events. We confirm that the space of orbital frequencies is very promising in this respect. In frequency space individual streams can be easily identified, and their separation provides a direct measurement of the time of accretion. We are able to show for a few idealized gravitational potentials that these features are preserved also in systems that have evolved strongly in time. Furthermore, this time evolution is imprinted in the distribution of streams in frequency space. We have also tested the power of the orbital frequencies in a fully self‐consistent (live) N‐body simulation of the merger between a disc galaxy and a massive satellite. Even in this case, streams can be easily identified and the time of accretion of the satellite can be accurately estimated.

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Section: Frequency Space For Time‐dependent Potentialsmentioning

confidence: 95%

On the identification of substructure in phase space using orbital frequencies

Gómez¹,

Helmi²

2010

Monthly Notices of the Royal Astronomical Society

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“…For epicyclic particles additional adiabatic invariants are identified with the action variables J i of classical dynamics, with i = R, z (see for example Refs. [36][37][38]. These are defined in terms of line integrals over complete periods of the orbit in the (p i , q i ) plane as…”

Section: Adiabatic Invariantsmentioning

confidence: 99%

Kinetic Theory for Gravitating Systems of Collisionless Neutral Matter

Cremaschini

Stuchlík

2013

Int. J. Mod. Phys. D

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The study of the statistical properties of collisionless systems formed by neutral particles subject to gravitational field represents an intriguing theoretical issue. In astrophysics, it directly relates to the description of collisionless gravitating dark matter (DM) halos. Structures of this type are expected to be characterized by intrinsically non-Maxwellian kinetic distribution functions (KDFs) and to exhibit temperature anisotropy, i.e. an anisotropy in the directional particle velocity dispersions. In this paper, a theoretical analysis of the issue is proposed, based on the kinetic theory developed in the framework of the Vlasov-Poisson description for nonrelativistic DM systems at equilibrium. By implementing the method of invariants, explicit solutions for the equilibrium KDFs are constructed and expressed through generalized Gaussian distributions. A perturbative theory is developed which allows them to be cast in terms of Chapman-Enskog series representations and to evaluate analytically the corresponding fluid fields. The conditions for the occurrence of temperature anisotropy are investigated for different physical and geometrical configurations. It is shown that this feature can arise at equilibrium due to specifically-kinetic effects associated with phase-space conservation laws in the presence of a nonuniform gravitational field. PACS Number(s): 05.20.Dd, 95.35.+d, 95.30.Cq, 97.20.Vs, 98.35.Gi, 98.62.Gq, 98.80.Cq 1350077-1 Int. J. Mod. Phys. D 2013.22. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ DAVIS on 02/03/15. For personal use only. C. Cremaschini and Z. StuchlíkThe statistical investigation developed here can have several physical applications. In astrophysical context, it concerns in particular the case of dark matter (DM) halos. According to the widespread theoretical scenarios, DM should be responsible for determining approximately 25% of the total energy density of the universe and to play a crucial role in cosmological large-scale structure and galaxy formation and evolution processes. 1 Although direct detection of DM particles is still missing and consequently its true nature remains to be ascertained, theoretical studies from elementary particle physics support the possibility that DM could consist of neutral weakly interacting massive particles (WIMPs). 2 Several studies have focused on determining the structure of the KDF for the DM in relaxed (i.e. equilibrium) collisionless structures. The majority of these works is based on data analysis of numerical simulations of DM halos, which can provide information regarding the DM mass density and velocity distributions (see for example Refs. 1-5). A number of common conclusions are inferred from these results. First, the fact that the DM equilibrium KDF exhibits non-Maxwellian features with significant departures from an isotropic Maxwell-Boltzmann distribution. 6-9 Second, the occurrence of a temperature anisotropy, namely a phase-space anisotropy characterizing the particle velocity dispersion along different ...

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“…There are in fact, exact invariants, such as the Ermakov-Lewis invariant, which remain unchanged even under more general time-dependent situations [24]. A proper proof of the adiabatic invariance of the classical action was constructed in [23] for a general Harmonic oscillator. Our aim is to check the whether the action variable J proposed here is such an invariant for the corresponding nonlinear oscillator.…”

Section: Remarksmentioning

confidence: 99%

“…Furthermore, as mentioned in the introduction, an important reason for the Action-Angle method is its application to situations where the Hamiltonian of the dynamical system is time-dependent. In this situation, the action-angle method gives rise to adiabatic invariants [23], which do not change and are important in checking the integrability of the model in question. As in the harmonic oscillator case, one can express the frequency of a general system as,…”

Section: )mentioning

confidence: 99%

Action-angle variables for the purely nonlinear oscillator

Ghosh

Bhamidipati

2019

International Journal of Non-Linear Mechanics

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In this letter, we study the purely nonlinear oscillator by the method of action-angle variables of Hamiltonian systems. The frequency of the non-isochronous system is obtained, which agrees well with the previously known result. Exact analytic solutions of the system involving generalized trigonometric functions are presented. We also present arguments to show the adiabatic invariance of the action variable for a time-dependent purely nonlinear oscillator.

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The adiabatic invariance of the action variable in classical dynamics

Cited by 23 publications

References 1 publication

On the identification of substructure in phase space using orbital frequencies

On the identification of substructure in phase space using orbital frequencies

Kinetic Theory for Gravitating Systems of Collisionless Neutral Matter

Action-angle variables for the purely nonlinear oscillator

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