Lie symmetries, nonlinear equations of motion and new Ermakov systems

Publisher Website
Google Scholar
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.

This publication has 16 references indexed in Scilit: