Nonlinear Mechanics - Physics at Oregon State University

Chapter 2
Canonical Transformations
We saw at the end af the last chapter that a problem in which all the
generalized coordinates are cyclic is trivial to solve. We also saw that there
is a great flixibility allowed in the choice of coordinates for any particular
problem. It turns out that there is an important class of problems for which
it is possible to choose the coordinates so that they are in fact all cyclic.
The choice is usually far from obvious, but there is a formal procedure for
finding the “magic” variables. One formulates the problem in terms of the
natural p’s and q’s and then transforms to a new set of variables, usually
called Qk and Pk, that have the right properties.
2.1 Contact Transformations
The most general transformation is called a contact transformation.
Qk = Qk(q, p, t) Pk = Pk(q, p, t) (2.1)
(In this formula and what follows, the symbols p and q when used as arguments
stand for the complete set, q1, q2, q3, · · · , etc.) There is a certain privileged
class of transformations called canonical transformations that preserve
the structure of Hamilton’s equation of motion for all dynamical systems.
This means that there is a new Hamiltonian function called K(Q, P ) for
which the new equations of motion are
˙Qk = ∂K
∂Pk
Pk
˙ = − ∂K
∂Qk
(2.2)
In a footnote in Classical Mechanics, Goldstein suggested that K be called
the Kamiltonian. The idea has caught on with several authors, and I will
use it without further apology. The trick is to find it.
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