Question

Solve the following problems:

Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential energy, 1. 2. Identify the number of degrees of freedom Obtain the expression for the total kinetic and total potential energy, and demonstrate that the Lagrangian of the system can be written as 3. Write Lagrange's equations of motion 4. Find the canonical momenta pr pi, and use them to rewrite the Lagrangian. 5. Demonstrate that the Hamiltonian for this system can be written as pipr 2m1 2 7n 2 6. Write Hamilton's equations of motion and compare these with the results obtained in point 3. 7. Identify the constants of motion (i.e., conserved quantities) 8. Consider now the same problem in a horizontal plane, neglecting the effects of gravity. Does the number of constants of motion change?

Answer 1

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