Canonical transformations and generating functions

New variables Q and P are canonical when their mutual Poisson brackets have the same elementary structure as those of q and p. The transformation must also be locally invertible. For one pair, an orientation-preserving area-preserving map satisfies the local symplectic condition, but for several pairs the preservation of total volume alone is weaker. Mixing one position with the wrong momentum can preserve an ordinary determinant while destroying the specific pairings needed for canonical dynamics.

Scaling illustrates why a reciprocal change matters. If Q equals aq with a constant nonzero scale factor, the matching canonical momentum is P equal to p divided by a. Scaling both variables by a would multiply their bracket by the square of a instead. In physical applications the units assigned to the new coordinates may differ from the old units; the canonical product retains action units even though the axes acquire different individual meanings.

A type-2 generating function uses old positions q, new momenta P, and time as independent arguments. Its q derivative gives old momentum; its P derivative gives new position. The transformed Hamiltonian K equals old H expressed in the new variables plus the explicit time derivative of the generating function. That last derivative is taken before substituting the old-to-new relations, holding q and P fixed. Interchanging the argument types without changing the rules is a common source of incorrect signs.

The construction follows by comparing the old and new phase-space action one-forms up to a total differential. For a type-2 function, the relevant boundary expression is F₂ minus the product QP. Boundary terms change endpoint bookkeeping without changing the interior equations under appropriate variations. A single generating-function type need not cover every canonical map globally: its chosen mixed coordinates must form a valid local chart, and another type may be needed where that chart degenerates.

Take $F_2=(q-vt)P$ for constant v. The new position is the old position measured from a uniformly moving origin, and the canonical momentum remains p. The extra time derivative contributes minus vP to K. For a free particle this term makes the new position rate equal to laboratory velocity minus v, exactly as a moving-origin description requires. Merely substituting q in H without adding the time term would predict the wrong relative velocity.

This transformation has not yet redefined momentum as mass times relative velocity. That would require an additional momentum shift and a different generating function, with an associated constant energy term. Both descriptions can be canonical, but their variables have different meanings. The example is therefore a reminder to distinguish a change of spatial origin from a complete Galilean boost convention. Verify the equations and the variable definitions together instead of deciding correctness by whether K resembles a familiar kinetic energy expression.