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Rotating Polar Coordinates

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 ω0.  The (stationary) Cartesian coordinates are related to the rotating coordinates by:

(1)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_BMkD2R.png

The velocities in the Cartesian frame are:

(2)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_YVQcN9.png

The kinetic energy of the particle is given by:

(3)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_uu75lb.png

With that, and the Lagrangian from (lagrange.4), we have:

(4)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_SC9Yi8.png

The derivatives we need are:

(5)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_nUnGwV.png

Inserting equations (5) into (lagrange.3),  and rearranging a little, we obtain the equations of motion:

(6a)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_bwxsrb.png

(6b)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_W5susC.png

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 ω0, 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.

Page created on 11/29/06