Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / b

Table of contents
- Solution b

Solution

The Arnold-Liouville theorem states that if

F_{1}, \dots, F_{n}

are independent commuting first integrals and a connected component

L

of a regular common level

F_{1} = c_{1}, \dots, F_{n} = c_{n}

is compact, then

L

is diffeomorphic to the

n

-torus. A neighbourhood of

L

admits action-angle variables

(I_{1}, \dots, I_{n}, θ_{1}, \dots, θ_{n})

in which

ω = \sum_{j} d I_{j} \land d θ_{j}, H = H (I),

and the motion is linear on each invariant torus:

\dot{I}_{j} = 0, \dot{θ}_{j} = \frac{\partial H}{\partial I _{j}} .

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