Global versus local superintegrability of nonlinear oscillators

Anco, Stephen C.; Ballesteros, Ángel; Gandarias, M. L.

doi:10.1016/j.physleta.2018.12.007

Cited by 5 publications

(1 citation statement)

References 30 publications

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“…The paper is organized as follows. In Section II the general formalism is presented while Section III is devoted to some examples discussed recently in the literature [7,8,9,10,11,12,13,14]. Darboux and Hietarinta approaches are considered in Sec.…”

Section: Introductionmentioning

confidence: 99%

The families of Hamiltonians sharing common symmetry structure

Gonera¹,

Gonera²,

Jasiński³

et al. 2023

Preprint

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We describe a general procedure which allows to construct, starting from a given Hamiltonian, the whole family of new ones sharing the same set of unparameterized trajectories in phase space. The symmetry structure of this family can be completely characterized provided the symmetries of initial Hamiltonian are known. Our approach covers numerous examples considered in literature. It provides a simple proof of Darboux theorem and Hietarinta et al. coupling-constant metamorphosis method.

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Section: Introductionmentioning

confidence: 99%

The families of Hamiltonians sharing common symmetry structure

Gonera¹,

Gonera²,

Jasiński³

et al. 2023

Preprint

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Symmetries and integrals of motion of a superintegrable deformed oscillator

2021

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Analog of a Laplace–Runge–Lenz vector for particle orbits (time-like geodesics) in Schwarzschild spacetime

Anco

Fazio

2023

Journal of Mathematical Physics

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In Schwarzschild spacetime, time-like geodesic equations, which define particle orbits, have a well-known formulation as a dynamical system in coordinates adapted to the time-like hypersurface containing the geodesic. For equatorial geodesics, the resulting dynamical system is shown to possess a conserved angular quantity and two conserved temporal quantities, whose properties and physical meaning are analogs of the conserved Laplace–Runge–Lenz vector, and its variant known as Hamilton’s vector, in Newtonian gravity. When a particle orbit is projected into the spatial equatorial plane, the angular quantity yields the coordinate angle at which the orbit has either a turning point (where the radial velocity is zero) or a centripetal point (where the radial acceleration is zero). This is the same property as the angle of the respective Laplace–Runge–Lenz and Hamilton vectors in the plane of motion in Newtonian gravity. The temporal quantities yield the coordinate time and the proper time at which those points are reached on the orbit. In general, for orbits that have a single turning point, the three quantities are globally constant, and for orbits that possess more than one turning point, the temporal quantities are just locally constant as they jump at every successive turning point, while the angular quantity similarly jumps only if an orbit is precessing. This is analogous to the properties of a generalized Laplace–Runge–Lenz vector and generalized Hamilton vector which are known to exist for precessing orbits in post-Newtonian gravity. The angular conserved quantity is used to define a direct analog of these vectors at spatial infinity.

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Global versus local superintegrability of nonlinear oscillators

Cited by 5 publications

References 30 publications

The families of Hamiltonians sharing common symmetry structure

The families of Hamiltonians sharing common symmetry structure

Symmetries and integrals of motion of a superintegrable deformed oscillator

Analog of a Laplace–Runge–Lenz vector for particle orbits (time-like geodesics) in Schwarzschild spacetime

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