Exercises

1. Consider a system of $N$ mutually interacting point objects. Let the $i$th object have mass $m_i$ and position vector ${\bf r}_i$. Suppose that the $j$th object exerts a central force ${\bf f}_{ij}$ on the $i$th. In addition, let the $i$th object be subject to an external force ${\bf F}_i$. Here, $i$ and $j$ take all possible values. Find expressions for the rate of change of the total linear momentum ${\bf P}$ and the total angular momentum ${\bf L}$ of the system.

2. 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.

3. 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:
   
   $$\sum_{i=1,n} x_i\,\frac{\partial f}{\partial x_i} = a\,f$$
   
   

4. 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 problem 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-states for the system when $n\geq 3$.

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 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.