On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

In this study we apply partial Noether andλ-symmetry approaches to a second-order nonlinear autonomous equation of the formy′′+fyy′+g(y)=0, called Liénard equation corresponding to some important problems in classical mechanics field with respect tof(y)andg(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 theλ-symmetry method, we analyzeλ-symmetries for the case thatλ-function is in the form ofλ(x,y,y′)=λ1(x,y)y′+λ2(x,y). Finally, a classification problem for the conservation forms and invariant solutions are considered.