Lagrange qustion, a partilcle confined to a spherical cone

[mjmontgo]

Homework Statement

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)

Homework Equations

x=rsinθcosø y=rsinθsinø z=rcosθ
the constraint for a circular cone is z=( x^2 + y^2)^1/2

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.

 

[BvU]

d/dt (x^2+y^2+z^2) is not the same as (dx/dt)^2 + etc.

 

[ehild]

You need to know the conversion between Cartesian and spherical polar coordinates.

See http://en.wikipedia.org/wiki/Spherical_coordinate_system (scroll down page).

$$x=r\sin(\theta)\cos(\phi)$$
$$y=r\sin(\theta)\sin(\phi)$$
$$z=r\cos(\theta)$$

Find the derivatives with respect time when θ=constant=α. Substitute for $\dot x$, $\dot y$, $\dot z$ in the formula for the KE. It simplifies to
$$KE = \frac{m}{2} \left(\dot r^2+(r \sin(\theta)\dot {\phi} )^2\right)$$