Longitudinal and transverse relativistic dynamics

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 p=γmv\mathbf{p}=\gamma m \mathbf{v}, we can write dpdt=dγdtmv+γmdvdt, \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 v\mathbf{v} dvdt=aL+aT, \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 dγdtmv=v2c2γ3maL, \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 γmdvdt=γm(aL+aT). \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 dpdt=γ3maL+γmaT. \frac{d\mathbf{p}}{dt}=\gamma^3m \mathbf{a}_L+\gamma m \mathbf{a}_T. ...

July 24, 2024 · 1 min

The Kapitza pendulum

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 ϕ\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 ϕ=π\phi = \pi, and unstable in the upright position ϕ=0\phi = 0. ...

February 11, 2024 · 1 min