Quadratic First Integrals of Time-Dependent Dynamical Systems of the Form q ¨ a = ? ? b c a q ? b q ? c ? ? ( t ) Q a ( q )

Antonios Mitsopoulos and Michael Tsamparlis
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Antonios Mitsopoulos: Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, 15783 Athens, Greece
Michael Tsamparlis: Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, 15783 Athens, Greece

Mathematics, 2021, vol. 9, issue 13, 1-38

Abstract: We consider the time-dependent dynamical system q ¨ a = ? ? b c a q ? b q ? c ? ? ( t ) Q a ( q ) where ? ( t ) is a non-zero arbitrary function and the connection coefficients ? b c a are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I = K a b q ? a q ? b + K a q ? a + K where the unknown coefficients K a b , K a , K are tensors depending on t , q a and impose the condition d I d t = 0 . This condition leads to a system of partial differential equations (PDEs) involving the quantities K a b , K a , K , ? ( t ) and Q a ( q ) . From these PDEs, it follows that K a b is a Killing tensor (KT) of the kinetic metric. We use the KT K a b in two ways: a. We assume a general polynomial form in t both for K a b and K a ; b. We express K a b in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t . In both cases, this leads to a new system of PDEs whose solution requires that we specify either ? ( t ) or Q a ( q ) . We consider first that ? ( t ) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Q a ( q ) to be the generalized time-dependent Kepler potential V = ? ? ( t ) r ? and determine the functions ? ( t ) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x ¨ = ? ? ( t ) x ? + ? ( t ) x ? ( ? ? ? 1 ) and compute the relation between the coefficients ? ( t ) , ? ( t ) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Keywords: time-dependent dynamical systems; quadratic first integrals; Killing tensors; kinetic metric; Kepler potential; oscillator; Lane-Emden equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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