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In a certain problem regarding motion of a particle in a gravitational field with axial symmetry, I have an expression of an effective potential $\Phi_{eff}(r,\theta)$. Now, I am interested to study the motion of the particle in the $r-\theta$ plane. Spherical coordinates $(r,\theta,\phi)$ is used in the problem and the axial symmetry corresponds to the $\phi$ coordinate.

How to obtain the equations of motion from $\Phi_{eff}(r,\theta)$ that I can numerically integrate to obtain the trajectory of the particle?

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  • $\begingroup$ You can use EL to obtain the Equations of motion $\endgroup$
    – Eli
    Commented Apr 11, 2023 at 7:27

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You can use the Lagrange's equations. Taking the Lagrangian $L$ as

$L = T - \Phi_{eff}$

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  • $\begingroup$ Your answer is short enough that it can just be a comment. $\endgroup$
    – naturallyInconsistent
    Commented Apr 23, 2023 at 4:35