CentralMotion - Fyta12:1 Motion in a Central Force Field A...

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Fyta12:1 – Motion in a Central Force Field A central force F = F ( r ) e r results from a spherically symmetric potential function V ( r ), as F ( r ) = - V ( r ). The motion is restricted to a plane, and in plane-polar coordinates ( r,ϕ ) the Lagrangian is L = m 2 ˙ r 2 + m 2 r 2 ˙ ϕ 2 - V ( r ) (1) which gives the equations of motion (note the cyclicity of ϕ ) m ¨ r = mr ˙ ϕ 2 - V ( r ) (2) d d t ( mr 2 ˙ ϕ ) = 0 (3) ˙ ϕ = L mr 2 (4) where L is the value of the conserved angular momentum . Inserting the expression (4) for ˙ ϕ into the r equation yields m ¨ r = L 2 mr 3 - V ( r ) (5) Note that the radial force as described by (5) can be seen as formally resulting from an effective potential , V eff ( r ), where a repulsive centrifugal term, L 2 / (2 mr 2 ) mr 2 ˙ ϕ 2 / 2 representing the angular part of the kinetic energy, is added to V ( r ), yielding V eff ( r ) = V ( r ) + L 2 2 mr 2 (6) A first integration of eq. (5) is easily done, yielding m 2 ˙ r 2 + L 2 2 mr 2 + V ( r ) = const. = E (7) which shows the conservation of energy. This implies ˙ r = ± radicalbigg 2 m ( E - V eff ( r ))
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