Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 4 / 3A (Dynamics and Relativity)

An inertial frame is a reference frame in which a force-free particle moves with constant velocity, so Newton's first law holds.

The basic Galilean transformations include spacetime translations ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{a}$, $t^{\prime }=t-b$; spatial rotations ${\mathbf{x}}^{\prime }=R\mathbf{x}$, $t^{\prime }=t$; and boosts ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{v}t$, $t^{\prime }=t$, where $R$ is a rotation matrix and $\mathbf{v}$ is constant.

The Principle of Galilean relativity states that the laws of mechanics have the same form in every inertial frame.

The equation of motion isMultiplication by $\dot{x}$ givesso the mechanical energy $E=\frac{1}{2}m{\dot{x}}^2+V(x)$ is conserved. On a segment where the direction of motion is fixed,

For $V=(x^2-4{)}^2$, $V^{\prime }=4x(x^2-4)$. Thus $x=\pm 2$ are stable equilibria at the two potential minima, while $x=0$ is unstable at the local maximum.

For $x\ne 0$, $V^{\prime }(x)=2x^{-3}{e}^{-1/x^2}$ never vanishes. At $x=0$, the smooth flat extension has $V^{\prime }(0)=0$, and $V(x)>V(0)=0$ for every $x\ne 0$. Hence $x=0$ is the unique equilibrium and is stable.