Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 32C / d

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- Solution d

Solution

Write the Hamiltonian as the sum of separated one-dimensional energies

H = \sum_{i = 1}^{n} H_{i}, H_{i} = \frac{1}{2} (p_{i}^{2} + ω_{i}^{2} q_{i}^{2}) .

Each

(q_{i}, p_{i})

system is a simple harmonic oscillator and is integrable away from

(q_{i}, p_{i}) = (0, 0)

by part (b). The separated energies satisfy

{H_{i}, H_{j}} = 0,

and their differentials are independent whenever every oscillator pair is nonzero. The Poisson-commuting separated energies therefore make the

n

-dimensional oscillator integrable. The exceptional coordinate strata have measure zero, so this proves Liouville integrability in its usual almost-everywhere sense; the origin is the only point where

d H

itself vanishes.

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