<s>[INST] Determine the stability of an equilibrium point for a given Lagrangian 
I have to determine the equilibria and the stability for a Lagrangian given by 
\begin{align}
\frac{\dot{x}^2+\dot{y}^2}{2}-V(x,y)
\end{align}
where $V(x,y)=-(x^2+y^2)^4+(2x^2+y^2)^2-1$.

The origin $(0,0)$ is an equilibria, but since the Hessian of the potential energy is indefinite when evaluated at this point, I need an alternative way to move.
I've tried so far to see if $V(x,y)-V(0,0)>0$ in a neighbourhood of the origin, but I can't prove it. The previous point asked me to see if there's any first integral, and I know that $E(x,y,\dot x, \dot y)$ (the energy) is preserved by the "Jacobi function theorem", so maybe I have to use it but I don't know how.
 [/INST] We have that
$$(2x^2+y^2)^2=x^4+2x^2(x^2+y^2)+(x^2+y^2)^2$$
And if $0<q<1$, we have that $q>q^2$, so we have that (for $0<x^2+y^2<1$)
$$(x^2+y^2)^2 > (x^2+y^2)^4$$
Which means that
$$x^4+2x^2(x^2+y^2)+(x^2+y^2)^2>x^4+2x^2(x^2+y^2)+(x^2+y^2)^2 +(x^2+y^2)^4$$
So
$$(2x^2+y^2)^2-(x^2+y^2)^4 > x^4+2x^2(x^2+y^2)$$
</s>