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Motion in a Central Potential
Consider a particle of mass moving in two dimensions in the central potential . This is clearly a two degree of freedom dynamical system.
As described in Section 5.5, the particle's instantaneous position
is most conveniently specified in terms of the plane polar
coordinates and . These are our two generalized coordinates.
According to Equation (223), the square of the particle's velocity
can be written

(614) 
Hence, the Lagrangian of the system takes the form

(615) 
Note that
Now, Lagrange's equation (613) yields the equations of motion,
Hence, we obtain
or
where , and is a constant. We recognize Equations (622) and (623) as the equations
we derived in Chapter 5 for motion in a central potential.
The advantage of the Lagrangian method of deriving these equations is
that we avoid having to express the acceleration in terms of the generalized
coordinates and .
Next: Atwood Machines
Up: Lagrangian Dynamics
Previous: Lagrange's Equation
Richard Fitzpatrick
20110331