Lagrangian Analysis of Two-Body Systems and Spring Dynamics

Posted by on Aug 2, 2025 in Physical Engineering | 0 comments

Problem 1: Two Masses in a Uniform Gravitational Field

Two masses m1 and m2 move in a uniform gravitational field g and interact via a potential energy U(r).

  1. Show that the Lagrangian can be decomposed as L = LCM + Lrel.
  2. Write down the Lagrange equations for the CM coordinates X, Y, Z and describe the motion of the CM. Then write the Lagrange equations for the relative coordinates and show that the motion of r is the same as that of a particle of reduced mass μ, position r, and potential energy U(r)
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