Paper 4, Section I, 3A

Dynamics and Relativity | Part IA, 2018

(a) Define an inertial frame.

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

(c) State the Principle of Galilean Relativity.

(d) Write down the equation of motion for a particle in one dimension xx in a potential V(x)V(x). Prove that energy is conserved. A particle is at position x0x_{0} at time t0t_{0}. Find an expression for time tt as a function of xx in terms of an integral involving VV.

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

(i) V(x)=(x24)2V(x)=\left(x^{2}-4\right)^{2}

(ii) V(x)=e1/x2V(x)=e^{-1 / x^{2}} for x0x \neq 0 and V(0)=0V(0)=0.

Typos? Please submit corrections to this page on GitHub.