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

We consider the time-dependent dynamical system q ¨ a = − Γ b c a q ˙ b q ˙ c − ω ( t ) Q a ( q ) where ω ( t ) is a non-zero arbitrary function and the connection coefficients Γ 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 , ω ( 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 ω ( t ) or Q a ( q ) . We consider first that ω ( 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 = − ω ( t ) r ν and determine the functions ω ( t ) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x ¨ = − ω ( t ) x μ + ϕ ( t ) x ˙ ( μ ≠ − 1 ) and compute the relation between the coefficients ω ( t ) , ϕ ( t ) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.