A particle is conﬁned to move on the surface of a circular cone with its axis
on the vertical z axis, vertex at origin (pointing down), and half-angle α(alpha)
a) write down the lagrangian in terms of spherical coordinates r and ø (phi)
2. Relevant equations
x=rsinθcosø y=rsinθsinø z=rcosθ
the constraint for a circular cone is z=( x^2 + y^2)^1/2
3. The attempt at a solution
So using this constraint and some definitions of cartesian–> spherical coordinates one can show
that θ is constant, i.e θ=α (alpha)
My problem here is setting up the Kinetic Energy, as the Lagrangian (L) is
L= T (kinetic) – U(potential) energies.
In cartesian T= 1/2m(d/dt(x)^2+d/dt(y)^2+d/dt(z)^2)
My problem is now converting this to spherical polar coordinates, keeping in mind all time derivatives of θ=zero because theta is constant (θ=α)
I’ve found a solution online and it gives the kinetic Energy as
T=1/2m(d/dt(r)^2+(rsinαø^(dot))^2) …so the 1/2m( rdot^2 + (rsinαø(dot)^2)
where ø(dot) is time derivate w.r.t phi…If anyone could help me get to this conclusion it would be appreciated. I’ve tried substituting directly for d/dt (x^2+y^2+z^2) but i do not get this answer,
i think it is just perhaps my math (algrebra) screwing me up.
Thanks in advance.