figure 1  Rotating coordinate system:

As another example of a simple use of the
Lagrangian formulation of Newtonian mechanics,
we find the equations of motion of a particle in rotating rectangular
coordinates, with an arbitrary conservative force acting on it.
The particle has mass
m, and the frame is rotating with angular velocity ω. We'll use coordinates (
x,
y) to refer to the
rotating
frame (figure 1). The stationary frame will be referred to
using coordinates (ζ, ξ). The (stationary) Cartesian coordinates
are related to the rotating coordinates by:
(1)
The ζ and ξ velocities of a particle (in the stationary frame) convert to:
(2)
The sum of the squared velocities in the stationary frame is:
(3)
Finally, we can write the Lagrangian in the form of equation (
lagrange.4):
(4)
We write out the partial derivatives we'll need:
(5)
Plugging equations (5) into (
lagrange.3), we obtain:
(6a)
(6b) 
On
the right hand side of both (6a) and (6b), the first term is the
centrifugal force, the second term is the Coriolis force, and the third
term is the arbitrary conservative force which we started with  and
is also the only
real force present, the other two being "fictitious" forces.
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created on 11/29/06