Phys 325 Spring 2011 Lecture 7 - Physics 325 Lecture 7...

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Physics 325 Lecture 7 Motion in a Central Field The total energy expression, Equation 6.8, can simplified a bit further by the introduction of a new variable, the angular momentum , L, defined as L r p  (7.1) It’s time derivative / dL dt is given by   dL d rp dt dt r p r p v mv r F rF           (7.2) and is called the torque . In an isotropic central field ˆ () F f r r , so the force produces no torque since ˆ 0 rr  . Therefore, angular momentum is conserved for a central force. This also a direct consequence of the spherical symmetry of the potential: a rotation around any fixed axis cannot affect the equations of motion. We will see later that there is a deep connection between the symmetry properties of the system and the invariance of physical quantities . It follows from the conservation of angular momentum that the path of motion of a particle in a central field remains in a single plane since L is a constant vector that is perpendicular to both r and v . In terms of polar coordinates,   2 2 ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ L r mv mr r r r r mrr r r mr r mr k    (7.3) Using this expression in Equation (6.8), we can write the conserved energy as   2 2 2 2 2 2 mK E r r r m L K r mr r       (7.4)
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Upon rearrangement, we can make this look like a kinetic energy term plus a potential energy term. 2 2 2 1 22 LK E mr mr r    (7.5) Since L is constant for a central force, the last term amounts to a potential, which we call the effective potential , () eff U U r . We have   2 1 2 eff E mv U r  (7.6)
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys 325 Spring 2011 Lecture 7 - Physics 325 Lecture 7...

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