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

In this study we apply partial Noether and λ-symmetry approaches to a second-order nonlinear autonomous equation of the form y″ + f(y)y′ + 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 λ-symmetry-method, we analyze λ-symmetries for the case that λ-function is in the form of λ(x, y, y′) = λ ₁ (x, y)y′ + λ₂ (x, y). Finally, a classification problem for the conservation forms and invariant solutions are considered.