Quadratic first integrals of time-dependent dynamical systems of the form ${\ddot{q}}^a=-{\Gamma }_{bc}^a{\dot{q}}^b{\dot{q}}^c-\omega (t)Q^a(q)$

Abstract

We consider the time-dependent dynamical system ${\ddot{q}}^a=-{\Gamma }_{bc}^a{\dot{q}}^b{\dot{q}}^c-\omega (t)Q^a(q)$ where $\omega (t)$ is a non-zero arbitrary function and the connection coefficients ${\Gamma }_{bc}^a$ are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) $I$ we assume that $I=K_{ab}{\dot{q}}^a{\dot{q}}^b+K_a{\dot{q}}^a+K$ where the unknown coefficients $K_{ab},K_a,K$ are tensors depending on $t,q^a$ and impose the condition $\frac{dI}{dt}=0$. This condition leads to a system of partial differential equations (PDEs) involving the quantities $K_{ab},K_a,K,$ $\omega (t)$ and $Q^a(q)$. From these PDEs, it follows that $K_{ab}$ is a Killing tensor (KT) of the kinetic metric. We use the KT $K_{ab}$ in two ways: a. We assume a general polynomial form in $t$ both for $K_{ab}$ and $K_a$; b. We express $K_{ab}$ 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 $\omega (t)$ or $Q^a(q)$. We consider first that $\omega (t)$ 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 $V=-\frac{\omega (t)}{r^{\nu }}$ and determine the functions $\omega (t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $\ddot{x}=-\omega (t)x^{\mu }+\varphi (t)\dot{x}$ $(\mu \ne -1)$ and compute the relation between the coefficients $\omega (t),\varphi (t)$ so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane-Emden equation.

References in corpus (7)

• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Lie point symmetries of a general class of PDEs: The heat equation
• Generalizing the autonomous Kepler Ermakov system in a Riemannian space
• Quadratic first integrals of autonomous conservative dynamical systems
• First integrals of holonomic systems without Noether symmetries
• Lie and Noether point Symmetries for a Class of Nonautonomous Dynamical Systems
• Symmetries of second-order PDEs and conformal Killing vectors

Cited by in corpus (6)

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• The generalized Ermakov conservative system: A discussion
• Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
• Higher order first integrals of autonomous non-Riemannian dynamical systems
• Integrable time-dependent central potentials