Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

Dynamical systems described by equations of motion with the first-order time derivative (dissipative) terms of even and odd powers, and coefficients varying either in time or in space, are considered. Methods to obtain standard and non-standard Lagrangians are presented and used to identify classes of equations of motion that admit a Lagrangian description. It is shown that there are two general classes of equations that have standard Lagrangians and one special class of equations that can only be derived from non-standard Lagrangians. In addition, each general class has a subset of equations with non-standard Lagrangians. Conditions required for the existence of standard and non-standard Lagrangians are derived and a relationship between these two types of Lagrangians is introduced. By obtaining Lagrangians for several dynamical systems and some basic equations of mathematical physics, it is demonstrated that the presented methods can be applied to a broad range of physical problems.