Solution Found!
(a) Write down the Lagrangian L (x1, x2, xl, i2) for two
Chapter 7, Problem 7.8(choose chapter or problem)
(a) Write down the Lagrangian L (x1, x2, xl, i2) for two particles of equal masses, m1 = m2 = m, confined to the x axis and connected by a spring with potential energy U = -1kx2. [Here x is the extension of the spring, x = (x1 x2 1), where 1 is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.] (b) Rewrite L in terms of the new variables X = i (x1 + x2) (the CM position) and x (the extension), and write down the two Lagrange equations for X and x. (c) Solve for X (t) and x (t) and describe the motion.
Questions & Answers
QUESTION:
(a) Write down the Lagrangian L (x1, x2, xl, i2) for two particles of equal masses, m1 = m2 = m, confined to the x axis and connected by a spring with potential energy U = -1kx2. [Here x is the extension of the spring, x = (x1 x2 1), where 1 is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.] (b) Rewrite L in terms of the new variables X = i (x1 + x2) (the CM position) and x (the extension), and write down the two Lagrange equations for X and x. (c) Solve for X (t) and x (t) and describe the motion.
ANSWER:Step 1 of 4
The following are given by the question:
The masses of the particle,
Unstarched length of the spring 
The kinetic energy of the particle, 
Here, 
Total kinetic energy of the particles,
The potential energy of the particle
The Lagrangian of the system is given by
Substitute the values and solve as
Hence the Lagrangian of the system is 