Exercises

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

2. 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 $n\geq 3$.

3. A particle of mass $m$ is constrained to move in one dimension such that its instantaneous displacement is $x$. The particle is released at rest from $x=b$, and is subject to a force of the form $f(x) = - k\,x^{-2}$. Show that the time required for the particle to reach the origin is
   
   $$\pi\left(\frac{m\,b^3}{8\,k}\right)^{1/2}.$$
   
   

4. A body of uniform cross-sectional area $A$ and mass density $\rho$ floats in a liquid of density $\rho_0$ (where $\rho<\rho_0$), and at equilibrium displaces a volume $V$. Show that the period of small oscillations about the equilibrium position is
   
   $$T = 2\pi\,\sqrt{\frac{V}{g\,A}}.$$
   
   

5. Using the notation of Section 2.9, show that the total momentum and angular momentum of a two-body system take the form
   

   $${\bf P} = M\,{\bf r}_{cm},$$

   $${\bf L} = M\,{\bf r}_{cm}\times \dot{\bf r}_{cm} + \mu\,{\bf r}\times \dot{\bf r},$$

   
   
   respectively, where $M=m_1+m_2$.