Exercises

1. Consider an isolated system of $N$ point objects interacting via gravity. Let the mass and position vector of the $i$th object be $m_i$ and ${\bf r}_i$, respectively. What is the vector equation of motion of the $i$th object? Write expressions for the total kinetic energy, $K$, and potential energy, $U$, of the system. Demonstrate from the equations of motion that $K+U$ is a conserved quantity.

2. Consider a function of many variables $f(x_1,x_2,\cdots,x_n)$. Such a function which satisfies
   
   $$f(t\,x_1, t\,x_2,\cdots,t\,x_n) = t^a\,f(x_1,x_2,\cdots,x_n)$$
   
   
   for all $t>0$, and all values of the $x_i$, is termed a homogenous function of degree $a$. Prove the following theorem regarding homogeneous functions:
   
   $$\sum_{i=1,n} x_i\,\frac{\partial f}{\partial x_i} = a\,f$$
   
   

3. Consider an isolated system of $N$ point objects interacting via attractive central forces. Let the mass and position vector of the $i$th object be $m_i$ and ${\bf r}_i$, respectively. Suppose that magnitude of the force exerted on object $i$ by object $j$ is $k_i\,k_j\,\vert{\bf r}_i-{\bf r}_j\vert^{-n}$. Here, the $k_i$ measure some constant physical property of the particles (e.g., their electric charges). Write an expression for the total potential energy $U$ of the system. Is this a homogenous function? If so, what is its degree? Write the equation of motion of the $i$th particle. Use the mathematical theorem from the previous exercise to demonstrate that
   
   $$\frac{1}{2}\frac{d^2 I}{dt^2} = 2\,K + (n-1)\,U,$$
   
   
   where $I=\sum_{i=1,N} m_i\, r_i^{\,2}$, and $K$ is the kinetic energy. This result is known as the virial theorem. Demonstrate that there are no bound steady-state equilibria for the system (i.e., states in which the global system parameters do not evolve in time) when $n\geq 3$.

4. A star can be through of as a spherical system that consists of a very large number of particles interacting via gravity. Show that, for such a system, the virial theorem, introduced in the previous exercise, implies that
   
   $$\frac{d^2 I}{dt^2} = -2\,U + c,$$
   
   
   where $c$ is a constant, and the $r_i$ are measured from the geometric center. Hence, deduce that the angular frequency of small amplitude radial pulsations of the star (in which the radial displacement is directly proportional to the radial distance from the center) takes the form
   
   $$\omega = \left(\frac{\vert U_0\vert}{I_0}\right)^{1/2},$$
   
   
   where $U_0$ and $I_0$ are the equilibrium values of $U$ and $I$. Finally, show that if the mass density within the star varies as $r^{-\alpha}$, where $r$ is the radial distance from the geometric center, and where $\alpha<5/2$, then
   
   $$\omega = \left(\frac{5-\alpha}{5-2\,\alpha}\,\frac{G\,M}{R^{\,3}}\right)^{1/2},$$
   
   
   where $M$ and $R$ are the stellar mass and radius, respectively.

5. Consider a system of $N$ point particles. Let the $i$th particle have mass $m_i$, electric charge $q_i$, and position vector ${\bf r}_i$. Suppose that the charge to mass ratio, $q_i/m_i$, is the same for all particles. The system is placed in a uniform magnetic field ${\bf B}$. Write the equation of motion of the $i$th particle. You may neglect any magnetic fields generated by the motion of the particles. Demonstrate that the total momentum ${\bf P}$ of the system precesses about ${\bf B}$ at the frequency ${\mit\Omega} = q_i\,B/m_i$. Demonstrate that $L_\parallel + {\mit\Omega}\,I_\parallel/2$ is a constant of the motion. Here, $L_\parallel$ is the total angular momentum of the system parallel to the magnetic field, and $I_\parallel$ is the moment of inertia of the system about an axis parallel to ${\bf B}$ which passes through the origin.