3

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I was only able to find descriptions of canonical transformations, wherever I searched.

Qmechanic
  • 201,751
BB_
  • 81
  • 3
    If you can tell your example explicitly, it would be more helpful !!! – user35952 Jun 19 '16 at 10:30
  • Is there no general method for non canonical transformations? Any paper on the subject? – BB_ Jun 19 '16 at 15:40
  • I am unclear as to what it would mean to say that Lagrangian and Hamiltonian are known in non-canonical coordinates for the following reasons: 1. By definition, there cannot be any quantity $K(\textbf{P}{nc}, \textbf{Q}{nc})$ that follows Hamilton's equations in non-canonical coordinates $\textbf{P}{nc}, \textbf{Q}{nc}$. So, what exactly is meant by the "Hamiltonian" then? 2. Canonical coordinates refer to phase-space coordinates--not configuration-space coordinates. Therefore, the notion of Lagrangian in the paradigm of canonical coordinates seems problematic. [...] –  Sep 17 '18 at 04:35
  • [...] Maybe it can be viewed as the Legendre transform of the Hamiltonian but again, due to my point 1, it is unclear to me what would mean by the Hamiltonian in a non-canonical coordinate set-up. [...] –  Sep 17 '18 at 04:38
  • [...] All in all, what I feel is that there seems to be no way to start with a non-canonical coordinate set-up in the language of either Hamiltonian or Lagrangian. What one could do is simply specify the equations of motion in the coordinates they like and then ask the question how do we go to a canonical coordinate description of the system, i.e. a description in terms of a Hamiltonian that produces equations of motion through Hamilton's equations. –  Sep 17 '18 at 04:38
  • 1
    Possibly helpful: https://physics.stackexchange.com/a/53637/20427 –  Sep 17 '18 at 04:48

0 Answers0