Nb. This is a past exam paper question I am attempting to answer
Given the following Lagrangian $$\mathcal{L}=\frac{1}{2}mg_{ij}\dot x^{i}\dot x^{j}-U(x^4)+\lambda(g_{ij}x^ix^j-a^2)$$ $(a)$ Specify any continuous symmetries and use Noether’s theorem to construct the corresponding conserved quantities.
$(b)$ Use the conserved quantities to show that the system is effectively one-dimensional, and derive the effective potential and effective Lagrangian of the resulting onedimensional system.
Answer
Using Noether's Theorem the conserved quantity is $$J_{ij}=p_ix^{j}-p_jx^i$$ Also, the energy is $$E=\frac{1}{2}m\dot x^Tg\dot x+U(x^4)$$ I realise I can reduce the dimensionality by one if I use a parameterisation of $S^3$ however this is very computational, is there a faster way to eliminate dimensions?