On conservation forms and invariant solutions for classical mechanics problems of Liénard type (Q471610)

Summary: We apply partial Noether and \(\lambda\)-symmetry approaches to a second-order nonlinear autonomous equation of the form \(y^{\prime\prime} + f (y) y^{\prime} + g(y)= 0\), called Liénard equation corresponding to some important problems in classical mechanics field with respect to \(f(y)\) and \(g(y)\) functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on the \(\lambda\)-symmetry method, we analyze \(\lambda\)-symmetries for the case that \(\lambda\)-function is in the form of \(\lambda(x,y,y^{\prime} + \lambda_2 (x,y))\). Finally, a classification problem for the conservation forms and invariant solutions are considered.