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.© BrainMass Inc. brainmass.com October 24, 2018, 10:49 pm ad1c9bdddf
Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank you for using Brainmass.
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.