please provide an explanation and solution for everything! thank you.

1. Two objects of masses mi and my are connected by a linear spring of natural length L. They slide without friction along a horizontal floor. 1.1. Write down the Lagrangian for the system. 1.2. Rewrite the Lagrangian in the center-of-mass and relative coordinates 1.3. Derive the equations of motion for the system. 2. Consider a pendulum with string length varying according to ( (t) = 1(1-e-2 ) where L and 2 are constants. 2.1. Write down the Lagrangian. 2.2. Identify and describe the constraint. 2.3. Derive the equation of motion and find out what happens as / approaches infinity. 3. The Lagrangian of a certain particle of mass m is given by L=-mc 1-4r.r-U(r) 3.1. Find the equation of motion. 3.2. What quantity is conserved when U is a constant