Serial of year 32

Before we dive into the art of analytical mechanics, we should brush up on classical mechanics on the following series of problems.

1. A homogenous marble with a very small radius sits on top of a crystal sphere. After being granted an arbitrarily small speed, the marble starts rolling down the sphere without slipping. Where will the marble separate and fall of the sphere?
2. Instead of the sphere from the previous problem, the marble now sits on a crystal paraboloid given by the equation $y = c - ax^2$. Again, where will the marble separate from the paraboloid?
3. A cyclist going at the speed $v$ takes a sharp turn to a road perpendicular to his original direction. During the turn, he traces out a part of a circle with radius $r$. How much does the cyclist have to lean into the turn? You may neglect the moment of inertia of the wheels and approximate the cyclist as a mass point.
   Bonus: Do not neglect the moment of intertia of the wheels.

1. Suppose we have a dumbbell consisting of two mass points with masses $m$ and $M$ connected via a massless rod. This dumbbell is in a free fall. Write a constraint function and Lagrangian equations of the first kind for this object.
   
2. Suppose we have a triangular prism with mass $M$ on a horizontal platform as in the picture. A mass point with the mass $m$ is sliding down a side of the prism. The angle between said side and the platform is $\alpha $. You may neglect friction.
   

• Set up Lagrangian equations of the first kind for this situation.
• Show that, for zero initial speed of the mass point, the total momentum of this system in the direction of $x$ axis is zero.
• Solve the system of (Lagrangian) equations and find the time-dependent equations for the speeds of the prism and the mass point.
• Find the ratio between these two speeds.

1. Set up Lagrangian equations of the first kind for a simple pendulum. Show that the law of conservation of energy holds for this situation.

1. Suppose we have a horizontal plane with a small hole. Through this hole goes a rope with length $l$ on which a weight of mass $M$ is hung. You may consider the weight to be a mass point. One the other end of the rope there is a second mass point with mass $m$. The rope between them is stretched thanks to the weight of mass $M$. Initially, the whole setup is in rest while the part of the rope below the plane is vertical. Then we grant the mass point on the plane velocity $v$ in a horizontal direction perpendicular to the rope as we let the system go free. Neglect all friction in this problem. Choose appropriate coordinates and find the Lagrangian for this situation.
   
2. Suppose we have an iron rod bent to a shape of a parabola given by the equation $y = x^2$. The gravity of Earth points in the negative direction of the $y$ axis. A mass point of mass $M$ can move freely along the parabola. A second mass point with mass $m$ is connected to the first by a rigid rod of length $l$. This way we have created a pendulum with a hinge sliding along the rod. The system can move only in the plane of the parabola. Find appropriate generalized coordinates and the Lagrangian for this situation.
   
3. Suppose we have line along which slides a mass point with mass $m$ (without friction). The angle between the line and the horizontal plane is $\alpha $. Find appropriate generalized coordinates and the Lagrangian for this situation. Then set up Lagrangian equations of the second kind, double-integrate them and find the solution. Do not forget about the constants of integration and explain their physical meaning. What will be their values if the mass point starts at rest at the height $h$?

1. Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane. Instructions:: Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality.
2. Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
3. Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius. Hint:: This problem can be easily solved even without the knowledge of spherical trigonometry.

1. Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius $1 \mathrm{ly}$. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically.
2. Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass $m$ in a spherically symmetrical central-force field. \[\begin{equation*} \dot {r}^2 = \frac {2}{m} \(E - V(r) - \frac {l^2}{2mr^2}\) \end {equation*}\] Where $r$ is the length of the radius vector, $E$ is the total energy, $l$ is angular momentum, and $V(r)$ is the potential energy of the mass point.
3. Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.

1. Suppose we have a simple pendulum with a mass point of mass $m$. The initial angular displacement is $120\dg $. Set up Lagrangian equations of the first kind for this pendulum and find the angular displacement for which the force acting on the pendulum rod is strongest.
2. Suppose we have a simple pendulum hanging from the mass point of another simple pendulum. The rods of both pendulums are of the same length. Set up the Lagrangian and Lagrangian equations of the second kind for this system.
3. Consider a mass point free to move along the $x$ axis, from which is hanging a simple pendulum. Find the Lagrangian of this system and using the Hamiltonian principle find the respective kinematic equations by setting the Gateaux differentials with respect to each variable equal to zero. Each Gateaux differential will then yield one kinematic equation. Compare the resultant equations to the ones you would obtain by solving this problem utilizing Lagrangian equations of the second kind.