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.

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
Let

X = ½ ...

Solution Summary

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.

Four massless rods of length L are hinged together at their ends to form a rhombus. a particle of mass M is attached at each joint. the opposite corners of the rhombus are joined by springs, each with a spring constant k. In the equilibrium (square) configuration, the springs are unstretched. The motion is confined to a plane, a

I need some assistance for a 3 spring problem: horizontal configuration, frictionless surface. Spring constant is k for all three springs and mass = m for both weights ie configuration is:
wall| spring - weigh t- spring - weight - spring |wall.
Meed solution using eigenvalues and Lagrangian to obtain equations of motions.

Consider two massless springs connected in series. Spring 1 has a spring constant k_1, and spring 2 has a spring constant k_2. A constant force of magnitude F is being applied to the right. When the twosprings are connected in this way, they form a system equivalent to a single spring of spring constant k.
a) Find the sprin

The mass m1 moves on a smooth horizontal plane, m2 moves vertically under the force of gravity and the spring. Using polar coordinates r, theta for m1, l for m2 and taking b for the total length of the string plus the unstretched length of the spring, find: (diagram attached in file)
a. the Langrangian of the system
b. the equ

An electromagnetic field is given by the potential:
phi = 0 and A = ay(z-hat) + bt(x-hat)
with a and b constant where 'x-hat?'z-hat?are unit vectors along the x and z directions respectively.
a. Write the Lagrangian for a particle of charge q moving in this field.
b. Identify any constants of the motion
c. Write the

A system with one degree of freedom has a Hamiltonian
see attached
where A and B are certain functions of the coordinate q and p is the momentum conjugate to q.
a) Find the velocity q(dot)
b) Find the Lagrangian L(q q(dot)) (note variables)

Consider a simple plane pendulum consisting of a mass m connected to a string of length L.
After the pendulum is set in motion , the length of the string is shortened at a constant rate:
dL/dt = -k
The suspension point remains fixed.
Compute the following:
a) The Lagrangian and Hamiltonian functions
b) Compare

Three identical masses m are connected to 2 identical springs of stiffness k and equilibrium length b, as shown in the figure above. The masses are free to oscillate in one dimension along an axis that runs through all three. They lie on a level, frictionless, horizontal surface. Introduce coordinates, x1, x2 and x3 to measure t

Problem:
A) Write the Lagrangian for a simple pendulum consisting of a mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation.
B) Assume the massless string can stretch with a restoring force F=-k(r