Editing Canonical transformation (section)

{{Short description|Coordinate transformation that preserves the form of Hamilton's equations}}
In [[Hamiltonian mechanics]], a '''canonical transformation''' is a change of [[canonical coordinates]] {{math|('''q''', '''p''') → ('''Q''', '''P''')}} that preserves the form of [[Hamilton's equations]]. This is sometimes known as ''form invariance''. Although [[Hamilton's equations]] are preserved, it need not preserve the explicit form of the [[Hamiltonian mechanics|Hamiltonian]] itself. Canonical transformations are useful in their own right, and also form the basis for the [[Hamilton–Jacobi equation]]s (a useful method for calculating [[constant of motion|conserved quantities]]) and [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] (itself the basis for classical [[statistical mechanics]]).

Since [[Lagrangian mechanics]] is based on [[generalized coordinates]], transformations of the coordinates {{math|'''q''' → '''Q'''}} do not affect the form of [[Lagrangian mechanics|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
<math> P_i = \frac{ \partial L }{ \partial \dot{Q}_i }\ ,</math>
where <math>\left\{\ (P_1 , Q_1),\ (P_2, Q_2),\ (P_3, Q_3),\ \ldots\ \right\} </math> are the new co‑ordinates, grouped in canonical conjugate pairs of momenta <math>P_i </math> and corresponding positions <math>Q_i,</math> for <math>i = 1, 2, \ldots\ N,</math> with <math>N </math> being the number of [[degrees of freedom (mechanics)|degrees of freedom]] in both co‑ordinate systems.

Therefore, coordinate transformations (also called ''point transformations'') are a ''type'' of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called ''restricted canonical transformations'' (many textbooks consider only this type).

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 bundle|cotangent bundles]], [[Exterior derivative|exterior derivatives]] and [[Symplectic manifold|symplectic manifolds]].