s1116 - m 1 ~ r 1 × ~v 1 m 2 ~ r 2 × ~v 2 = m 1 μ m 1 ~...

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Solution 16 set 5. 1. Yes, the Lagrangian does not depend on the time. 2. Solving for ~ r 1 : M ~ R = m 1 ~ r 1 + m 2 ~ r 2 , ~ r = ~ r 1 - ~ r 2 , m 1 ~ r 1 = M ~ R - m 2 ~ r 2 = M ~ R + m 2 ~ r - m 2 ~ r 1 , ~ r 1 = ~ R + m 2 M ~ r , ~ r 2 = ~ R - m 1 M ~ r . 3. Using μ we can write ~ r 1 = μ m 1 ~ r , ~ r 2 = - μ m 2 ~ r . Therefore, T cm = 1 2 μ 2 m 1 + μ 2 m 2 ! ~v 2 = 1 2 μ 2 ( m 1 + m 2 ) m 1 m 2 ~v 2 = 1 2 μ ~v 2 4. Angular momentum:
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Unformatted text preview: m 1 ~ r 1 × ~v 1 + m 2 ~ r 2 × ~v 2 = m 1 μ m 1 ~ r × ~v 1-m 2 μ m 2 ~ r × ~v 2 = μ ~ r × ( ~v 1-~v 2 ) = μ ~ r × ~v ....
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