Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into
Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.
The analogous argument for the generalized momenta Pm leads to two other sets of equations:
Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:[5]
If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.
and the bilinear invariance condition can be stated as a local conservation of the symplectic product.
[8] The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,
[9] To guarantee a valid transformation between (q, p, H) and (Q, P, K), we may resort to a direct generating function approach.
Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative dG/dt and yield the same equations of motion (as discussed on Wikibooks).
dG/dt is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.
Here G is a generating function of one old canonical coordinate (q or p), one new canonical coordinate (Q or P) and (possibly) the time t. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), depending on the choice of variables.
Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates.
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p).
and K = H. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.
terms represent a Legendre transformation to change the right-hand side of the equation below.
Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates.
terms represent a Legendre transformation to change the left-hand side of the equation below.
Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates.
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p).
Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates.
However, as a feature of canonical transformations, it is always possible to choose 2N such independent functions from sets (q, p) or (Q, P), to form a generating function representation of canonical transformations, including the time variable.
Proof of local invertibility of the set of coordinates is given by proving non singularity of
Similarly the canonical transformation rules are obtained by equating the left hand side as
The canonical transformation relations can now be restated to include time dependance:
, this type of generating function can be used for infinitesimal canonical transformation by restricting
Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.
Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.
Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis.
The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system.
This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867.