Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients

A general method is developed to derive a Lagrangian and Hamiltonian for a nonlinear system with a quadratic first-order time derivative term and coefficients varying in the space coordinates. The method is based on variable transformations that allow removing the quadratic term and writing the equation of motion in standard form. Based on this form, an auxiliary Lagrangian for the transformed variables is derived and used to obtain the Lagrangian and Hamiltonian for the original variables. An interesting result is that the obtained Lagrangian and Hamiltonian can be non-local quantities, which do not diverge as the system evolves in time. Applications of the method to several systems with different coefficients shows that the method may become an important tool in studying nonlinear dynamical systems with a quadratic velocity term