Classical Mechanics

A very simple calculation in relativistic dynamics show an impressive aspect that differentiates the momentum change due to a force in the direction of movement of the body and the transverse ones. By using the definition of relativistic momentum $\mathbf{p}=\gamma m \mathbf{v}$, we can write $$\frac{d\mathbf{p}}{dt} = \frac{d\gamma}{dt} m \mathbf{v} + \gamma m \frac{d\mathbf{v}}{dt},$$ by defining the longitudinal and transverse acceleration, with respect to the vector $\mathbf{v}$ $$\frac{d\mathbf{v}}{dt} = \mathbf{a}_L +\mathbf{a}_T,$$ it is simple to show that the first term in the first equation is only longitudinal $$\frac{d\gamma}{dt} m \mathbf{v} = \frac{v^2}{c^2} \gamma^3 m \mathbf{a}_L ,$$ and the second term is composed of longitudinal and transverse parts $$\gamma m \frac{d\mathbf{v}}{dt}=\gamma m (\mathbf{a}_L+\mathbf{a}_T).$$ Summing the contributions, we can separate the Newton equation with respect to transverse and longitudinal forces, obtaining $$\frac{d\mathbf{p}}{dt}=\gamma^3m \mathbf{a}_L+\gamma m \mathbf{a}_T.$$ ...

The Kapitza pendulum is a model in classical mechanics that exhibits counterintuitive behaviour. It is a pendulum with a pivot point that oscillates vertically. If $\phi$ is the angle the pendulum have with respect to the vertical downward position, for some values of the driving frequency and amplitude, it can be stable in the inverted position $\phi = \pi$, and unstable in the upright position $\phi = 0$. ...