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For a particle in circular motion or in a central force field, the radial frequency (\omega_r) and angular frequency (\omega_\theta) are defined as follows: Consider a particle of mass (m) moving under a central potential (V(r)). The Lagrangian in polar coordinates ((r, \theta)) is: [ L = \frac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2\right) - V(r) ] The radial motion equation is obtained from the Euler-Lagrange equation: [ m \ddot{r} = m r \dot{\theta}^2 - \frac{dV}{dr} ] Angular momentum is conserved: [ L_z = m r^2 \dot{\theta} ______ _________ ______ ______ ____ __________ _____ ____.
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