Lie symmetries, nonlinear equations of motion and new Ermakov systems

Abstract

It is shown that a Lagrangian of the form L=1/1( rho 2- omega 2(t) rho 2)+G(t)F(k(t) rho ), said to be in factored form, yields an equation of motion that is equivalent to the most general equation derivable via Noether's theorem from the unfactored Lagrangian L=1/2 rho 2-P( rho ,t). In view of this equivalence, the theory of extended Lie groups is applied to the factored nonlinear equation of motion p+ omega 2(t) rho =G(t)F(k(t) rho ) to obtain its Lie symmetries. The latter are obtained when G(t) and k(t), initially arbitrary, are determined in terms of a function x(t) which satisfies the auxiliary equation x+ omega 2(t)x=K/x3. It is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations, an Ermakov system, whose first integral is not invariant under the action of the symmetry group, in contrast to previous Ermakov systems.