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Effective potential in a central field
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[QUOTE="vanhees71, post: 6310484, member: 260864"] The trick is to use the conservation laws, and the conserved quantities are usually energy (Lagrangian doesn't depend explicitly on time) and the canonical momenta of cyclic coordinates (which come also from symmetries, i.e., the choice of the variables takes into account some symmetry like rotational symmetry around an axis or even around a point). If you have enough conserved quantities "first integrals" you can express the conserved energy in terms of only one variable, and then the entire motion is effectively a one-dimensional motion of this variable in the corresponding effective potential. All other variables can be then found by simple integration after the one-dimensional problem is solved. The reason to use the energy rather than the Lagrangian in this context is simply that energy is often a conserved quantity. [/QUOTE]
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Effective potential in a central field
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