Spherical Pendulum

It consists of a Mass moving without Friction on a Sphere . The only Force s acting on the mass are the Reaction from the sphere and Gravity .

It is convenient to use Spherical Coordinates and describe the position of the mass in terms of

(r,	heta,\phi)

, where ''r'' is fixed.

The Lagrangian is

:

L=rac{1}{2}
m\left(
  r^2\dot{	heta}^2+r^2\sin^2	heta\ \dot{\phi}^2

ight)
+ mgr\cos	heta.

The Euler-Lagrange Equations give

:

rac{d}{dt}
\left(mr^2\dot{	heta}

ight)
-mr^2\sin	heta\cos	heta\dot{\phi}^2+
mgr\sin	heta =0

and
:

rac{d}{dt}
\left(
  mr^2\sin	heta
  \,
  \dot{\phi}

ight)
=0

showing that Angular Momentum is conserved.

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Spherical Pendulum