Lagrangian Review Problem 1. Consider a point mass m in a gravitational field g in the -z direction.

1. Using as coordinates the Cartesian coordinates (x,y,z) of the mass, write down T, U, and L.
2. Find the Euler-Lagrange equations of motion for the three coordinates and give their general solutions (x(t), y(t), z(t)).
3. Identify the cyclic coordinates of the problem and find the corresponding conserved momenta.

Lagrangian Review Problem 2. Consider two railway cars of mass m ₁ and m₂ moving without friction on a straight level track, connected by a spring of constant k and unstretched length b. Let x₁ and x₂ be the distances of cars 1 and 2, respectively, from a fixed reference point on the track.

1. Using x₁ and x₂ as coordinates, write down T, U and L. Write the Euler-Lagrange equations of motion for x₁ and x₂. Are there any cyclic variables?
2. Now change to the generalized coordinates {q_j} = {X,x}, where X is the coordinate of the center of mass, and x = x₂ - x₁ is the relative coordinate. Find the expressions for T, U and L in terms of these coordinates.
3. Identify the cyclic coordinate and find the correspoinding conserved momentum. State the interpretation of the conserved momentum.
4. For the non-cyclic coordinate, find the Euler-Lagrangian equation of motion and interpret it in terms of an equivalent one-body problem.