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Rotating Rectangular Coordinates

figure 1 -- Rotating coordinate system:
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)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_BxAKSV.png

The ζ and ξ velocities of a particle (in the stationary frame) convert to:

(2)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_dk9tzM.png

The sum of the squared velocities in the stationary frame is:

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

Finally, we can write the Lagrangian in the form of equation (lagrange.4):

(4)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_3dFIyy.png

We write out the partial derivatives we'll need:

(5)    file:///nfs/dodo/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_kr8sDG.png

Plugging equations (5) into (lagrange.3), we obtain:

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

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

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.

Page created on 11/29/06