canonical transformations

madness

Do canonical transformations simply transform the coordinates of a particular system, leaving the physics unchanged? or can they transform between physically different systems? I haven't seen any evidence which shows that they keep the physics the same, but I don't see their usefulness otherwise.

Ben Niehoff

They do both, actually. In other words, they establish a link between different physical systems, such that one can be described in terms of the other.

For example, one canonical transformation can describe a damped harmonic oscillator in terms of variables that behave like an undamped oscillator (obviously, the transformation equations must be time-dependent for this to work). This allows us to talk about one problem in terms of another, perhaps simpler, problem.

dx

Canonical transformations are important for classical mechanics for the same reason linear transformations are important for vector space theory. The important structure on a vector space is its linear structure, and linear transformations are transformations that preserve this. The important structure on phase space is the symplectic structure, and canonical transformations preserve this.

Similarly, diffeomorphims are important in the theory of smooth manifolds, and conformal transformations are important in complex analysis and Riemann surface theory, and Unitary transformations are important in quantum mechanics. All these transformations preserve the relevant structure. The structure is usually defined by an algebra on the set, like multilinear/tensor algebra on vector spaces, or the Poisson bracket algebra on phase space for classical mechanics, or the inner product on quantum mechanical Hilbert spaces. But another point of view of thinking about the structure of sets was pioneered by Felix Klein, i.e. defining the structure of the sets by giving the transformations that preserve that structure. This is the idea behind the Erlangen program.

madness

So canonical transformations are changes of coordinate sytems and the transformed system is the same as the old but viewed in the new coordinates?
The fact that they are canonical is what preserves the symplectic structure and hence ensures we are still looking at the same system. Is this correct?

dx

Symplectic structure is not structure of any particular system, its the structure of phase space itself for all systems.

madness

I still don't see how you know the transformed system is the same as the original. The derivation involves applying the modified Hamilton's principle to both systems, so that they both satisfy Hamilton's equations. Couldn't this be done for any two systems, whether they are the same or not? I can't find anything in the derivation which requires the two systems to be the same physically.