Does Canonical Variable have any physical meaning in Classical Mechanics?

Question

Consider a canonical transformation from variable $(q,p) rightarrow (Q,P)$ generated by the generating function $F(q,Q)=qQ$ so in this
case
$$p=frac{partial F}{partial q}=QRightarrow Q=p$$
and $$P=-frac{partial F}{partial Q}=-qRightarrow P=-q$$
Which says in this space the old coordinate and momenta are changed to new momenta and coordinate. There is nothing wrong here. Every space is equally valid. Now if interpret the position in an ordinary way then Is it possible that if I change the space from one to other with this generating function the meaning also get changed? So what's the point in defining the physical meaning? Is there any physical meaning to these canonical variable at all or not?

spiridon_the_sun_rotator · Answer

I think, it is a philosophical question, which set of coordinates should be regarded as more fundamental. It is like the choice of a certain coordinates system on a given manifold.
The canonical transformations https://en.wikipedia.org/wiki/Canonical_transformation is the change of coordinates, that preserves the form of Hamilton equations.
In more general setup, when the phase space is not a cotangent bundle $T^{*} M$ of a manifold, this transformation is called Symplectomorphism https://en.wikipedia.org/wiki/Symplectomorphism. There is a non-degenerate ($omega^n neq 0$), closed ($d omega =0$) 2-form $omega$, and the symplectic transformations $f$ preserves it:
$$
f^{*}
 omega = omega$$
The distinction between generalized coordinates and momenta is on level on our intuition, coordinates determine location in the space, momenta are some characterstic of the motion. But in Hamiltonian dynamics the difference between the two dissappears, and becomes merely the choice of a convetion.

Does Canonical Variable have any physical meaning in Classical Mechanics?

One Answer

Add your own answers!

Ask a Question