Vtyjkyuk

I'm a bit rusty in this stuff.
I'm trying to design an energy function that would allow me to derive a motion equation.

I'm considering the vector field

$$
F(x,y,z) = kleft( fracxsqrtx^2+y^2,fracysqrtx^2+y^2,0 right)^T
$$

($k$ positive constant). I've checked the condition

$$
curl(F) = 0
$$
for all $x,y,z$, and indeed I get 0. However the components aren't differentiable in $(x,y) = 0$ which basically confuses me a little bit.

Is there a slightly modification I could do maybe to get something more sensible? But I'm not sure if this matters though.

The other question is I have the potential function

$$
U(x,y,z) = k sqrtx^2 + y^2
$$

and I clearly have

$$
nabla U = F
$$

How do I derive the motion equation? My intuition is that since

$$
F = ma = m fracd^2xdt^2
$$

I have to relate both $U$ and the motion equation, is there some Euler lagrange equation (in general not for this specific case) I should use?

You should use the cylindric coordinates $phi$, $r$, $z$, where
$$[x,y,z]= [rcos(phi),rsin(phi),z].$$
We then have for the kinetic energy
$$
T = fracm2left(r^2dotphi^2 + dotr^2 + dotz^2right)
$$
and for the potential simply
$$
U = k r.
$$
Now apply Euler-Lagrange:
$$
L = T - V
$$
and
$$
d_tpartial_dotqL = partial_q L$$
where $qin lbracephi,r,zrbrace$. This will give you your equations of motion:
$$
ddotz = 0,
$$
conservation of momentum in $z$-direction,
$$
rddotphi + 2dotrdotphi = 0 = dotJ,
$$
conservation of angluar momentum $J = mr^2dotphi$, and finally
$$
mddotr = mrdotphi^2-k = fracJ^2mr^3 - k.
$$
$J$ in the last equation is constant because of the conservation of angluar momentum. So now you are left with a decoupled equation for $r$.