Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 16B / b

Solution

\overset{x}{¨} + α x^{3} + β x y^{2} = 0, \overset{y}{¨} + α y^{3} + β x^{2} y = 0.

Hence

\dot{J} = (β - α) x y (x^{2} - y^{2}),

which is not generally zero. Conservation requires $β / α = 1$ . Then the potential is

α (x^{2} + y^{2})^{2} /4

J

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