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CH5_CENTRAL FORCES

# CH5_CENTRAL FORCES - CENTRAL FORCES Definition(1 The...

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CENTRAL FORCES Definition: (1) The potential is only dependent on the distance from source. i.e. (2) From (1): , i.e. the central force is radical in direction and its magnitude is only dependent on the distance from the source. Theorem 1: If a particle is subject to a central force, the particle angular momentum is conserved. Proof: Theorem 2: If a particle is subject to a central force, its motion is on a plane. Proof: Consider at some time t 0 , construct the plane P containing the vectors position vector and the instantaneous momentum vector . The plane P has the normal direction of . Consider the cross product of the position vector and the momentum vectors at some other time t 1 . Notice that because of the conservation of angular momentum. The position vector is always proportional to the constant vector and thus on a plane. The Euler-Lagrange Equation Define a physical quantity called Lagrangian which is defined by: L = T – V where T(x 1 ,x 2 ,…x n ) and V(x 1 ,x 2 ,…x n ) are the kinetic energy and the potential energy of the system. Then the equation of motion is given by the n differential equations:

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i = 1,2,…N This is known as The Euler-Lagrange Equation and we are not going to give a proof on the present course. This equation is indeed logical equivalent to the Newton’s laws of Motion. Example: Consider the spring-mass system: . Applying the Euler-Lagrange Equation yielded: The Effective Potential of a Central Force Consider a mass subject to a central force which has a potential of V(r) . Using the polar coordinates (r, θ ) with the source placed at the origin. Applying the Euler-Lagrange Equation on this case yields: The first equation is the equation of motion for radical direction. The angular momentum has magnitude of , where is the tangential component of the linear momentum. But , therefore Therefore the second equation of the Euler-Lagrange Equation is the conservation of angular momentum. L is determined by the initial condition.
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CH5_CENTRAL FORCES - CENTRAL FORCES Definition(1 The...

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