Higher order first integrals of autonomous dynamical systems

Mitsopoulos, Antonios; Tsamparlis, Michael

doi:10.1016/j.geomphys.2021.104383

Cited by 9 publications

(14 citation statements)

References 43 publications

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“…The seminal work by Carter motivated further studies on the subject [19,20]. For more results on higher order or hidden symmetries in various systems, and the geometric conditions of the involved objects in their construction, see also [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Hidden symmetries from distortions of the conformal structure

Dimakis

2022

Preprint

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It is well established that the mass parameter breaks the conformal symmetries in the case of geodesic motion. The proper conformal Killing vectors cease to generate conserved charges when non-null geodesics are considered. We examine how the introduction of the mass is actually related to the appearance of appropriate distortions in the conformal sector, which lead to new conservation laws. As a prominent example we use a general pp-wave metric, which exploits this property to the maximum. We study the necessary geometric conditions, so that such types of distortions are applicable. We show that the relative vectors are generators of disformal transformations and prove their connection to higher order (hidden) symmetries. Except from the pp-wave geometry, we also provide an additional example in the form of the de Sitter metric. Again, the proper conformal Killing vectors can be appropriately distorted to generate conserved quantities for massive geodesics. Subsequently, we proceed by introducing an additional symmetry breaking effect. The latter is realized by considering a Bogoslovsky type of line-element, which involves a Lorentz violating parameter. We utilize once more the pp-wave case as a guide to study how the broken symmetries -this time also related to Killing vectors -are substituted by distortions of the original generators. We further analyze and discuss the necessary geometric conditions that lead to the emergence of these distortions.

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Section: Introductionmentioning

confidence: 99%

Hidden symmetries from distortions of the conformal structure

Dimakis

2022

Preprint

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“…Because the mth-order polynomial in velocities FIs of the geodesic equations in a Riemannian space are known (see e.g. [8,9]), one transfers the dynamical equations to the geodesic equations of another Riemannian space. One approach is to introduce two extra degrees of freedom in the configuration space.…”

Section: The Geodesic Approachmentioning

confidence: 99%

“…Equation (9) implies that K ab is a CKT of order two of the kinetic metric γ ab with associated vector X a . Contracting (9) with γ bc , we find…”

Section: Qfis Of Constrained Autonomous Conservative Dynamical System...mentioning

confidence: 99%

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Quadratic First Integrals of Constrained Autonomous Conservative Dynamical Systems with Fixed Energy

Mitsopoulos

Tsamparlis

2022

Symmetry

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We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For a zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics.

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“…Bertrand 10 in the study of integrable surfaces and, later on, used by Whittaker 11 in the determination of the integrable autonomous conservative Newtonian systems with two degrees of freedom. In the course of time, this method has been used by various authors 3,12,13,14,15,16,17,18,19,20,21,22,23,24 .…”

Section: Introductionmentioning

confidence: 99%

Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach

Mitsopoulos,

Tsamparlis

2023

Preprint

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In a recent paper (A. Mitsopoulos and M. Tsamparlis, J. Geom. Phys. 170, 104383, 2021), a general theorem is given which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of the symmetries of the kinetic metric defined by the dynamical equations of the system. In the present work, we apply this theorem to compute the cubic FIs of autonomous conservative Newtonian dynamical systems with two degrees of freedom. We show that the known results on this topic, which have been obtained by means of various different methods, and additional ones derived in this work can be obtained by the single algorithmic method provided by this theorem. The results are collected in four Tables which can be used as an updated reference of this type of integrable and superintegrable potentials. The results we find are for special values of free parameters; therefore, using the methods developed here, other researchers by different suitable choice of the parameters will be able to find new integrable and superintegrable potentials.

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Higher order first integrals of autonomous dynamical systems

Cited by 9 publications

References 43 publications

Hidden symmetries from distortions of the conformal structure

Hidden symmetries from distortions of the conformal structure

Quadratic First Integrals of Constrained Autonomous Conservative Dynamical Systems with Fixed Energy

Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach

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