 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/5/solution

Complete the square in the velocity variable:

H = V (x) + p v - \frac{1}{2} m v^{2} = V (x) + \frac{p ^{2}}{2 m} - \frac{m}{2} (v - \frac{p}{m})^{2} .

Since

m > 0

, the final term is nonpositive and vanishes at exactly one velocity. Therefore

v_{*} = \frac{p}{m}, H_{m a x} (x, p) = \frac{p ^{2}}{2 m} + V (x) .

This is the Legendre transform in mechanics that constructs the Hamiltonian mechanics energy from the kinetic-minus-potential Lagrangian.

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