Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension

Tegmen, A.; Verçin, A.

doi:10.1142/s0217751x04017112

Cited by 23 publications

(21 citation statements)

References 25 publications

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“…Equation (26) is an implicit definition for the stationary probability density because in general, the expression dB/dy depends on P, as well; see Equation (24). However, if the entropy measure does not exhibit an outer function, that is if B(y) = y holds, then Equation (26) becomes an explicit definition for the stationary probability density and reads:…”

Section: Approach To Stationarity and Stationary Solutionsmentioning

confidence: 99%

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

Frank

2016

Entropy

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Abstract:In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker-Planck equation approach is used to describe stochastic aspects of active Nambu systems. Different thermostatistic settings are considered that are characterized by appropriately-defined entropy measures, such as the Boltzmann-Gibbs-Shannon entropy and the Tsallis entropy. In general, the free energy Fokker-Planck equations associated with these generalized entropy measures correspond to nonlinear partial differential equations. Irrespective of the entropy-related nonlinearities occurring in these nonlinear partial differential equations, it is shown that semi-analytical solutions for the stationary probability densities of the active Nambu systems can be obtained provided that the pumping mechanisms of the active systems assume the so-called canonical-dissipative form and depend explicitly only on Nambu invariants. Applications are presented both for purely-dissipative and for active systems illustrating that the proposed framework includes as a special case stochastic equilibrium systems.

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Section: Approach To Stationarity and Stationary Solutionsmentioning

confidence: 99%

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

Frank

2016

Entropy

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“…Certain electrodynamic [5,[11][12][13][14][15] and biochemical [16,17] problems have been re-formulated using the multi-Hamiltonian approach of Nambu mechanics. Multi-Hamiltonian oscillators generalizing the harmonic oscillator [18,19], chiral models [8] and the Calogero-Moser system [20,21] have been studied in the context of Nambu mechanics. While one key benefit of Nambu mechanics is that it provides us with a principled way to construct evolution equations from invariants of motion, this benefit also limits the applicability of Nambu's multi-Hamiltonian approach to systems that features such invariants.…”

Section: Introductionmentioning

confidence: 99%

“…However, problems in interdisciplinary research fields such as biophysics frequently address nonequilibrium systems and are dissipative in nature. Consequently, they often require a generalization of the classical concept of invariants of (Yamaleev) [18,19] Chiral models [8] Calogero-Moser model [20,21] motion. For example, in order to study swarm dynamics and animal mobility we need to consider agents that take up energy from the environment and/or exhibit energy depots.…”

Section: Introductionmentioning

confidence: 99%

Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator

2012

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Publication informationThe European Physical Journal B, 85 (3):Publisher Abstract. We study the emergence of oscillatory self-sustained behavior in a nonequilibrium Nambu system that features an exchange between different kinetical and potential energy forms. To this end, we study the Yamaleev oscillator in a canonical-dissipative framework. The bifurcation diagram of the nonequilibrium Yamaleev oscillator is derived and different bifurcation routes that are leading to limit cycle dynamics and involve pitchfork and Hopf bifurcations are discussed. Finally, an analytical expression for the probability density of the stochastic nonequilibrium oscillator is derived and it is shown that the shape of the density function is consistent with the oscillator properties in the deterministic case.

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“…The explicit general form of C has been derived in detail for superintegrable systems. [3] The aim of this paper is to obtain C for a Hamiltonian system with n degrees of freedom and m constants of motion C 1 , · · · , C m , with m ≥ 2n. …”

Section: Introductionmentioning

confidence: 99%

Nambu Brackets With Constraint Functionals

Tegmen

2006

Int. J. Mod. Phys. A

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If a Hamiltonian dynamical system with n degrees of freedom admits m constants of motion more than 2n − 1, then there exist some functional relations between the constants of motion. Among these relations the number of functionally independent ones are s = m − (2n − 1). It is shown that for such a system in which the constants of motion constitute a polynomial algebra closing in Poisson bracket, the Nambu brackets can be written in terms of these s constraint functionals. The exemplification is very rich and several of them are analyzed in the text. IntroductionThe concept of generalized Hamiltonian dynamics arouse in 1973 with an article by Y. Nambu.[1] In his proposal, Nambu employed an N -ary bracket, generically called Nambu bracket (NB), to describe the time evolution of the dynamical system in N -dimensional (N D) phase space. His bracket includes N − 1 functionally independent constants of motion, the so-called generalized Hamiltonians. As an illustrative example, Nambu considered the Euler equations of free rigid body for a 3D phase space and this was the only example given. Finding examples in higher odd-dimensions is still very tedious matter.In Nambu formalism, dynamical systems produce inevitably a nontrivial normalization factor C at least when N is an even integer grater than three. [2] In words, in order to get the correct Hamiltonian dynamics the NBs must be normalized properly. The explicit general form of C has been derived in detail for superintegrable systems.[3] The aim of this paper is to obtain C for a Hamiltonian system with n degrees of freedom and m constants of motion C 1 , · · · , C m , with m ≥ 2n. defines NB of N -th order satisfying the properties skew-symmetry, Leibniz rule and generalized Jacobi identity (fundamental identity).[4] When one considers this NB structure C ∞ (M) admits another algebra structure.A. Tegmen Nambu brackets with constraint functionals 2

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Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension

Cited by 23 publications

References 25 publications

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator

Nambu Brackets With Constraint Functionals

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