Lagrangian mechanics is a powerful reformulation of classical mechanics based on energy rather than force vectors. Instead of analyzing free-body diagrams, it uses the ( ), defined as the difference between kinetic energy ( ) and potential energy ( L=T−Vcap L equals cap T minus cap V

If (\omega^2 < g/R): only (\theta=0,\pi) (top and bottom). If (\omega^2 > g/R): also (\theta = \pm \cos^-1(g/(R\omega^2))).

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Use (\dot\phi = \ell/(mr^2)) in energy: (E = \frac12 m \dotr^2 + \frac\ell^22mr^2 - \frackr). Effective potential: (U_\texteff(r) = \frac\ell^22mr^2 - \frackr).

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