(a) Define an inertial frame.

(b) Specify three different types of Galilean transformation on inertial frames whose space coordinates are $\mathbf{x}$ and whose time coordinate is $t$.

(c) State the Principle of Galilean Relativity.

(d) Write down the equation of motion for a particle in one dimension $x$ in a potential $V(x)$. Prove that energy is conserved. A particle is at position $x_{0}$ at time $t_{0}$. Find an expression for time $t$ as a function of $x$ in terms of an integral involving $V$.

(e) Write down the $x$ values of any equilibria and state (without justification) whether they are stable or unstable for:

(i) $V(x)=\left(x^{2}-4\right)^{2}$

(ii) $V(x)=e^{-1 / x^{2}}$ for $x \neq 0$ and $V(0)=0$.