Lagrangian and Hamiltonian's Mechanics:
Write down the Lagrangian L for two particles of equal mass m1 = m2 = m, confined to the x-axis and connected by a spring with potential energy U = ½ k x^2. (Here x is the extension of the spring, x = (x1- x2 -l), where l is the spring's outstretched length, and that mass l remains to the right of mass 2 at all times.] (b) Rewrite L in terms of the new variables X = ½(x1+x2) (the CM position) and x (the extension) and write down the two Lagrange equation of X and x (c) Solve for X(t) and x(t) and describe the motion.
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Potential energy V = ½ k (x1 - x2 - l)2
Kinetic energy T = ½ m1 2 + ½ m2 2
L = T - V = ½ m1 2 + ½ m1 2 - ½ k (x1 - x2 - l)2
X = ½ ...
Equations of motion of two particles connected by a spring, which is confined to move along x-axis, have been found. I have derived the equations of motion using Lagrangian mechanics. This 2-page word document shows the step-by-step, clear and detailed solution to this classical mechanics problem.