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GPS 8.12_G 8.02

# GPS 8.12_G 8.02 - GPS 8.12(2nd ed 8-2 Start with the...

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GPS 8.12 (2 nd ed. 8-2 ) Start with the kinetic energy written as the sum of the center of mass con- tribution and the contribution from motion relative to the COM: T = M 2 · R 2 + m 1 2 μ · r 0 1 2 + m 2 2 μ · r 0 2 2 , (1) where M = m 1 + m 2 , m i is the mass of particle i , R is the COM position vector, and r 0 i is the position vector of particle i relative to the COM. With the use of the relative position vector r r = r 0 2 r 0 1 , (2) the expression for T simpli f es (cf. Chap. 3) to T = M 2 · R 2 + μ 2 · r 2 , (3) where μ is the reduced mass of the two-body system μ = m 1 m 2 M . (4) The two particles interact with a central force so the potential energy of the system is simply V ( r ) . Hence, the Lagrangian is L = M 2 · R 2 + μ 2 · r 2 V ( r ) . (5) Because the COM coordinates are cyclic variables, the components of the linear COM momentum are conserved quantities, and as a result the COM kinetic energy is just an additive constant in the Lagrangian that we will ignore from now on. In spherical coordinates the components of r are x = r sin θ cos φ, (6) y = r sin θ sin (7) z = r cos θ, (8) where r = | r | , θ is the polar angle between r and the z axis, and φ is the azimuthal angle measured from the x axis in the x - y plane. After transforming into spherical coordinates the Lagrangian reads L = μ 2 [ · r 2 + r 2 · θ 2 + r 2 · φ 2 sin 2 θ ] V ( r ) . (9) Next, using the prescription, p j = ∂L/∂ · q j ,we f nd the following results for the three conjugate momenta p r , p θ ,and p φ : p r = μ · r, (10) 1

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p θ = μr 2 · θ, (11) and p φ = μr 2 · φ sin 2 θ. (12) Next we construct the energy function h ,de f ned here as h = · r ∂L · r + · θ · θ + · φ · φ L.
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