Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 1 / c

Table of contents
- Solution c

Solution

Let

V = ab c

and write

T = K V^{- (γ - 1)}

for a constant

K

. The axis equations are Newton equations in the effective potential

U (a, b, c) = \frac{1}{2} Ω^{2} (a^{2} + b^{2} + c^{2}) + \frac{2 T}{γ - 1},

because

- \frac{\partial}{\partial a} (\frac{2 T}{γ - 1}) = \frac{2 T}{a},

and similarly for

b, c

. Therefore

E = \frac{1}{2} (\overset{a}{˙}^{2} + \dot{b}^{2} + \overset{c}{˙}^{2}) + \frac{1}{2} Ω^{2} (a^{2} + b^{2} + c^{2}) + \frac{2 T}{γ - 1}

is conserved. Multiplying by the fixed profile-dependent mass moment converts

E

into the physical kinetic, trapping-potential, and internal energy of the star, so it is proportional to total energy.

Solved by gpt-5.6-sol high.

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