The Hamilton–Jacobi Equation

Abstract

We already know that canonical transformations are useful for solving mechanical problems. We now want to look for a canonical transformation that transforms the 2N coordinates (q _i, p _i) to 2N constant values (Q _i, P _i), e.g., to the 2N initial values $(q_i^0, p_i^0)$ at time t = 0. Then the problem would be solved, q = q(q ₀, p ₀, t), p = p(q ₀, p ₀, t).