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 polar
coordinates, with a conservative "central" (radial) force acting on it.
The frame is rotating with angular velocity ω
_{0}. The (stationary) Cartesian coordinates are related to the rotating coordinates by:
(1)
The velocities in the Cartesian frame are:
(2)
The kinetic energy of the particle is given by:
(3)
With that, and the Lagrangian from (
lagrange.4), we have:
(4)
The derivatives we need are:
(
5)
Inserting equations (
5) into (
lagrange.3), and rearranging a little, we obtain the equations of motion:
(6a)
(6b) |
The
right hand side of (6a) is the tangential Coriolis force; it
depends on the radial velocity, and the sum of the angular velocity of
the particle in the frame and rotation rate of the frame. The
first term on the right of (6b) is the centrifugal force due to the
particle's own motion
and the rotation of the frame. The second term is the radial Coriolis force; it's proportional to ω
_{0},
so for a non-rotating frame, it vanishes. The last term in (6b)
is the central force we started with, and is the only "real" force
present.
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created on 11/29/06