Small oscillations of a dynamical system near stable equilibrium points

Question

I'm having problems solving this lagrangian dynamical system:
Let $P$ and $Q$ be two points in $\mathbb R^2$ s.t. $P\in\Gamma_1\equiv y=x^2$ and $Q\in\Gamma_2\equiv y=-x^2$.
The two points are connected by a spring of constant $k>0$, with rest lenght zero. I have to find the frequency of small oscillations in a neighbourhood of the stable equilibrium points, but I don't manage to deduce their staility. This is what I've done: $$P=(u,u^2)\text{ and }Q=(v,-v^2)\\V(u,v)=mgu^2-mgv^2+\dfrac k2(d(P,Q))^2=mg(u^2-v^2)+\dfrac k2[(u-v)^2+(u^2-v^2)^2]\\ T(\dot u,\dot v)=\dfrac{m}{2}(\dot u^2+\dot v^2+4(u^2\dot u^2+v^2\dot v^2).$$ In order to find the points of equilibrium I imposed $\nabla V(u,v)=(0,0)$, which (I think) generates the solutions $E=\{(u,v)\in\mathbb R^2:u=v\}$. The Hessian of the potential is given by $$D^2V=\begin{pmatrix}k[(2u^2+2uv+1)+(4u+2v)(u-v)]+2mg &&-k(2u^2+2uv+1)\\-k[(2u^2+2uv+1)+2u(u-v)]&&-k[-(2v^2+2uv-1)+(u-v)(4v+2u)]-2mg\end{pmatrix}$$ and now I should evaluate this matrix in $(u,v)\in E$, but it seems to be really tedious.
I always found stable equilibrium points of the form $(x_0,y_0)\in\mathbb R^2$ so I don't know how to find small oscillations via linearized Lagrange equation $D^2T\cdot\ddot q +D^2V\cdot q=0$ .

Answer 1

Indeed, setting $\ddot u=\ddot v= \dot u=\dot v=0$ for a stationary solution, one finds that the ODE reduces to $∇V(u,v)=(0,0)$. In detail these equations read \begin{align} 0&=2mgu+k(u-v)+2ku(u^2-v^2)\\ 0&=-2mgv+k(v-u)+2kv(v^2-u^2) \end{align} Combining the equations to eliminate the third-degree terms gives $$ 0=2mg(u^2-v^2)+k(u-v)^2 $$ so that either $u=v$ leading to $u=v=0$, or $$ 0=2mg(u+v)+k(u-v)\implies v=-\frac{2mg+k}{2mg-k}u=-au $$ Inserting into the first equation then gives $$ 0=2mgau+2ku^3(1-a^2)\\ u^2=\frac{mga}{k(a^2-1)}=\frac{4m^2g^2-k^2}{8k^2} $$ So as conjectured in the comments, if $k$ is small enough, one gets additional non-zero stationary positions, at $k=2mg$ there is a fork bifurcation.