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Homework Statement:

I will not use summation sign: repeated pair of (upper and lower) indices are summed over: [itex]\sum_{a} A_{a} B^{a} \equiv A_{a}B^{a} = A_{c}B^{c}[/itex] (summed over indices are dummy indices so you can rename them as you like).
Consider a Hamiltonian of ##3## degrees of freedom ##q_i, i = 1,...3##. Momentum is given by ##p_i##. Suppose that the Hamiltonian has the following form:
$$H = \alpha^i (p_i)^2 + V(q)$$
The only possible combination of ##q_i## given by the potential function ##V(q)## is ##q_1^2 +q_2^2 +q_3^2##. We can apply the following change of variables* ##X = q_1^2 +q_2^2 +q_3^2## and obtain the potential in function of the new variable: ##V(X)##
If all ##\alpha^i## are the same, then there is extra symmetry and corresponding constants of motion.
a) Find all constants of motion in case all ##\alpha^i## are the same. HINT: Find the Lagrangian first.
EXTRA (which means I added it; we can deal with this after solving a)):
b) Why all ##\alpha^i## need to be the same?
*I learned this technique in PF! :) More: https://www.physicsforums.com/threads/hamiltonsequationandeulerlagrangesequationcomparison.982951/
Relevant Equations:
 $$H = \alpha^i (p_i)^2 + V(q)$$
I do not understand the following sentence (particularly, the concept of extra symmetry): 'If all ##\alpha^i## are the same, then there is extra symmetry and corresponding constants of motion'.
OK so let's find the Lagrangian; we know it has to have the form:
$$L(q, \dot q) = T(q, \dot q)  V(q)$$
The idea is to apply a change of variables to the given ##H(q, p)## to get ##L(q, \dot q)##
This should be straightforward; we pick the change of variables ##p = \dot q (p, q)## and then apply it to the Hamiltonian to get the Lagrangian. But to do so we need an equation for ##\dot q## in function of ##p## and ##q##
The issue is that I do not see how to get such an equation...
Any hint is appreciated.
Thanks.
OK so let's find the Lagrangian; we know it has to have the form:
$$L(q, \dot q) = T(q, \dot q)  V(q)$$
The idea is to apply a change of variables to the given ##H(q, p)## to get ##L(q, \dot q)##
This should be straightforward; we pick the change of variables ##p = \dot q (p, q)## and then apply it to the Hamiltonian to get the Lagrangian. But to do so we need an equation for ##\dot q## in function of ##p## and ##q##
The issue is that I do not see how to get such an equation...
Any hint is appreciated.
Thanks.