In this chapter, we return to some structural features of classical mechanics, carrying on from where we left off in Chap.  3 . Recall that canonical variables are a set of position-type and momentum-type coordinates \(q^i\) , \(p_i\) on phase space such that the equations of motion take the standard Hamiltonian form \(\dot{q}^i = \frac{\partial H}{\partial p_i}\) and \(\dot{p}_i = - \frac{\partial H}{\partial q^i}\) ( 3.78 ) for a suitable Hamiltonian H(q, p) and \(i = 1, \cdots , n\) , where n is the number of degrees of freedom. Roughly, we will define a canonical transformation as a change of phase space variables \((q^i, p_j) \mapsto (Q^i(q,p),P_j(q,p))\) such that the equations of motion, when transformed to the new variables, continue to take a Hamiltonian form. Canonical transformations bear a resemblance to some other transformations we encounter in mathematical physics. Recall from Appendix A.10 that an isomorphism L between vector spaces is one that preserves linear combinations: \(L(a u + b v) = a L(u) + b L(v)\) for any vectors u, v and scalars a, b. In other words, it preserves the linear structure of the space. An automorphismGroupautomorphism of a group (see Appendix B.13* ) is a bijective map from a group to itself that preserves the group composition law. An isometry (see Footnote 28 of Appendix B.6* ) is a map between spaces that preserves the distances between points.