Classical Mechanics/Lagrangian Exercises

Exercises in setting up Lagrange functions and deriving the equations of motion

I recommend going through every exercise below (unless you know at once how to solve each of them). These exercises are not difficult but will give you experience in dealing with Lagrange functions. You do not really understand the Lagrangian formalism unless you can solve these standard problems without much effort.

Each of the following exercises poses the same questions: a) Introduce generalized coordinates and write down a Lagrange function for the system described below. b) Derive the Euler-Lagrange equations of motion. (It is not necessary to solve them!) Every situation is set up in the gravitational field near the Earth. All the mentioned sticks are massless and completely rigid. All the mentioned springs are massless and perfectly elastic (without friction) and have spring constant $k$ . The lengths of springs at rest are given in the exercises.

1. A point mass $m$ hangs from the ceiling on a stick of length $l$ . It can move only in the $x-z$ vertical plane.

2. Two point masses $m$ hang from the ceiling (far from each other) on two sticks of length $l$ each. They can both move only in the $x-z$ vertical plane.

3. Two point masses $m$ hang from the ceiling on two sticks of length $l$ each. They can both move only in the $x-z$ vertical plane. In addition, there is a spring with rest length $l$ connecting the two point masses. The points where the sticks are attached to the ceiling are at a distance $l$ from each other. (You may find this illustration useful.)

4. A point mass $m$ hangs from the ceiling on a stick of length $l$ . Another point mass $m$ is attached to the first one with a stick of length $L$ . Both masses can move only in the $x-z$ vertical plane.

5. A point mass $m$ hangs from the ceiling on a stick of length $l$ . Another point mass $m$ is attached to the first one with a spring of rest length $L$ . Both masses can move only in the $x-z$ vertical plane.

6. Two point masses $m$ are attached to the two ends of a stick of length $2l$ . The midpoint of the stick is attached to the ceiling by a stick of length $l$ . The entire arrangement can move only in the $x-z$ vertical plane.

7. A point mass $m$ is attached to the ceiling by a stick of length $l$ . The point mass can move in all directions.

8. A point mass $m$ is attached to the ceiling by a spring of rest length $l$ . The point mass can move in all directions.

9. Two identical carts of mass $m$ can roll without friction on a rail along the $x$ axis. They are connected by a spring of rest length $l$ .

10. A cart of mass $m$ can roll without friction on a rail along the $x$ axis. A pendulum, consisting of a stick of length $l$ and a point mass $m$ , is mounted rigidly on the cart and can move freely within the $x-z$ vertical plane.