I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem:
According to Hamilton's principle a particle moving in a potential $V(x,y,z)$ in space (three dimensions) in such a way that the functional
$\quad\quad S\equiv \int_{t_0}^{t_1}(\frac{m}{2}[\dot{x}(t)^2+\dot{y}(t)^2+\dot{z}(t)^2]-V(x(t),y(t),z(t)))dt$
reaches a extreme ($m>0$ is the mass of the particle, the dots denote time derivatives and $t_0$ and $t_1$ are arbitrary times). Use Hamilton's principle to derive Newton's equations of motion for the coordinates $(u,v,w)$ defined as:
$\quad\quad x=uv\cos(w),\quad\quad y=uv\sin(w),\quad\quad z=(u^2-v^2)/2$
Now, I believe those coordinates are called parabolic, and they should be orthogonal to each other (if that helps us). We have three positional coordinates, which gives us three Euler-Lagrange equations:
$\quad\quad \frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}=0,\quad\quad i=1,2,3\quad and\quad \textbf{q}=(x,y,z)$.
The function $L$ (Lagrangian) is the integrand of $S$ in this case.
I really don't know where to go from here; I just get confused about the last part where I am supposed to write them in parabolic coordinates. When do I transform from Cartesian to parabolic, the Euler-Lagrange equations?
Best regards//
Best Answer
Hint: $$ \left.\begin{align} \dot x&=(u\dot v+\dot u v)\cos w-uv\dot w \sin w\\ \dot y&=(u\dot v+\dot u v)\sin w+uv\dot w \cos w\\ \dot z&=u\dot u-v\dot v \end{align}\right\} \implies \dot x^2+\dot y^2+\dot z^2=(u^2+v^2)(\dot u^2+\dot v^2)+u^2v^2\dot w^2. $$ Now plug in the Euler-Lagrange equation the Lagrangian: $$ {\cal L}=\frac{m}{2}\left[(u^2+v^2)(\dot u^2+\dot v^2)+u^2v^2\dot w^2\right]-V(u,v,w). $$