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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. Hence, it can be proven that every finite canonical transformation can be given as a closed but ...
The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above, =, and substitute the expression for P from equation , expressed in terms of p and Q:
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and ...
Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function (not Hamilton's principal function ).Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian (,) is merely the old Hamiltonian (,) expressed in terms of the new canonical coordinates, which we denote as (the action angles, which are the ...
A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the zeta series transformation and its generalizations defined as a derivative-based transformation of generating functions, or alternately termwise by and performing an integral transformation on the sequence ...
To derive the HJE, a generating function (,,) is chosen in such a way that, it will make the new Hamiltonian =. Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial P ˙ = Q ˙ = 0 {\displaystyle {\dot {\mathbf {P} }}={\dot {\mathbf {Q} }}=0} so the new generalized coordinates and momenta are ...
Thus, the time evolution of a function on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian.
Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie group on a coadjoint orbit.