nLab canonical transformation

This entry is about the notion in Hamiltonian mechanics/symplectic geometry. For a different notion of the same name in category theory see at canonical morphism.

To be merged with generating function in classical mechanics.

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Idea

In physics and specifically in Hamiltonian mechanics what is called a canonical transformation is an equivalence or at least a homomorphism of phase spaces. The term is to be read as short for “transformation of canonical coordinates”.

Since phase spaces are mathematically identified with symplectic manifolds, canonical transformations are symplectomorphisms. In fact, the notion as used in physics is a bit more general, and subsumes also certain Lagrangian correspondences between symplectic manifolds $(X_{1,2}, \omega_{1,2})$. The “generating functions” for canonical transformations as used in the physics literture correspond to functions on the correspondence space $(X_1 \times X_2, p_1^\ast \omega_1 - p_2^\ast \omega_2)$. (Weinstein 83)

References

The standard definition, interpreted in symplectic geometry appears for instance on p. 206 of

• Vladimir Arnold, Mathematical methods of classical mechanics

Discussion of generating functions starts on p. 266 there.

The formalization in terms of Lagrangian correspondences is discussed on p. 5 of

• Alan Weinstein, Lectures on Symplectic Manifolds, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third

  printing.