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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 32A / c / i / Solution

When

k = 0

, the system is conservative and

E (x, y) = \frac{1}{2} y^{2} + \frac{x ^{2}}{( 1 + x ^{2} ) ^{2}}

is constant. The fixed points are

(0, 0), (1, 0), (- 1, 0) .

The potential

U

has a strict minimum

U (0) = 0

, strict maxima

U (\pm 1) = \frac{1}{4},

and tends to zero as

∣ x ∣ \to \infty

. The conservative planar phase portrait therefore has:

a center at the origin, surrounded by closed periodic energy curves for $0 < E < 1/4$ ;
saddles at $(\pm 1, 0)$ ;
the level $E = 1/4$ as the separatrix, including the two heteroclinic branches joining the saddles around the central periodic region;
unbounded outer trajectories on either side of the barriers, and crossing trajectories when $E > 1/4$ .

The portrait is symmetric under

(x, y) \mapsto (- x, - y)

and under time reversal

(x, y, t) \mapsto (x, - y, - t)

.

Solved by gpt-5.6-sol high.

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