Section 5: Classical Mechanics

One of the many applications of the theory of differential forms in the context of classical mechanics is providing a criterion for checking whether a transformation is canonical. Specifically, for a Hamiltonian system with n degrees of freedom with coordinates (pj,qj), we may change the coordinates to (Pj,Qj) such that the Hamiltonian structure is preserved. A transformation that preserves the exact form of the Hamiltonian: H(pj,qj) = H(pj(Pk,Qk),qj(Pk,Qk)) is called a canonical transformation.It can be proven that a transformation is canonical if and only if (dp1^dq1 + ... + dpn^dqn) = (dP1^dQ1 + ... + dPn^dQn).

However, at this moment this statement is rather mysterious since we haven't dealt with the issue of changing coordinates in the theory of forms. So let's not go into the details. Interested readers can refer to Arnold or Rand.

(I may supply the details later when I have more time. Please bear in mind that this is an assignment for a course I am taking. There is a deadline.)

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).