Helmholtz and Liénard-type oscillators from non-standard Lagrangians

This study investigates the nonlinear second-order differential equation \(\ddot{x} + f(x)\dot{x}^n + g(x)w(x) = 0,\; n = 1, 2, 3, \ldots\) and \(\ddot{x} + g(t)\dot{x}^m + k(t)W(x) = 0,\; m = 1, 2, 3, \ldots\) which model a class of dynamical systems characterized by velocity-dependent nonlinearities and nontrivial oscillatory behavior. By employing the method of nonstandard Lagrangians, we derive a systematic variational framework for such equations, enabling the identification of a number of properties that are not accessible through traditional Lagrangian formulations. The resulting first integrals and implicit solutions provide insight into the complex dynamics of the system, including amplitude-dependent oscillations and stability properties. This approach highlights the relevance of nonstandard Lagrangians in capturing the behavior of nonlinear, dissipative, or nonconservative systems, offering both theoretical and practical tools for the analysis of advanced mechanical, physical, and engineering models. Our approach may be used also, after a suitable change of coordinate, to describe complex biological systems such as the Susceptible–Infectious–Recovered model. An analogy with a position-dependent mass system is also addressed.