Solution

Let

q = (q_{1}, q_{2}, q_{3})

and

n = (1, 1, 1)

. Since

q_{1} q_{2} + q_{2} q_{3} + q_{3} q_{1} = \frac{1}{2} {(q \cdot n)^{2} - ∣ q ∣^{2}},

the Lagrangian depends on

q

only through

∣ q ∣^{2}

and

(q \cdot n)^{2}

, and on the velocity through

∣ \overset{q}{˙} ∣^{2}

. Rotations about the

n

-axis preserve all three quantities, proving the invariance.

J_{n} = n \cdot (q \times \overset{q}{˙}) = (q_{2} \overset{q}{˙}_{3} - q_{3} \overset{q}{˙}_{2}) + (q_{3} \overset{q}{˙}_{1} - q_{1} \overset{q}{˙}_{3}) + (q_{1} \overset{q}{˙}_{2} - q_{2} \overset{q}{˙}_{1}),

up to an irrelevant normalization of

n

. Since

L

has no explicit time dependence, conservation of energy from time-translation invariance gives the second conserved quantity

E = \frac{1}{2} ∣ \overset{q}{˙} ∣^{2} + \frac{1}{2} ∣ q ∣^{2} + α (q_{1} q_{2} + q_{2} q_{3} + q_{3} q_{1}) .

Solved by gpt-5.6-sol high.

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