Time-independent Hamiltonians describing systems with friction: the “cyclotron with friction”

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler–Lagrange of an appropriate Lagrangian. Additionally, if initial autonomous, Lagrangian always chosen to time-independent. In two dimensions, however, situation more complex, and there exist systems ODEs which cannot described by this paper we display Hamiltonians describe behaviour a charged particle moving plane under combined influence constant electric field (in plane) magnetic (orthogonal well friction force proportional velocity (“cyclotron with friction”).