Classical Mechanics

How are the ions in a solution distributed in a region of space subjected to an external electric field? The Poisson-Boltzmann equation gives us a way to answer this question. The equation is basically the Poisson equation from electrostatics $$ \nabla^2 \phi = -\frac{\rho}{\varepsilon} , , $$ where $\varepsilon$ is the dielectric constant of the solvent, but with a suitable choice of the charge density, in a self-consistent way. The charge density is given by the requirement that the solution is thermalized, and the ions distribute in the potential in a way described by the Boltzmann distribution. Assuming that the electrostatic potential is the dominant part of the energy, we have $$ n_i = a_i\exp[-\beta z_i\phi] , . $$ where $\beta = 1/k_BT$ is the inverse temperature, $z_i$ is the charge of the charge of the $i$-species present, and $a_i$ are a proportionality constant, such that $$ \sum_i \int dV \ n_i = N , , $$ the total number of ions. Because of charge neutrality, $$ \sum_i a_i z_i = 0 , . $$ We take the charge density to be $$ \rho = \sum_i z_i n_i , , $$ and therefore the Poisson-Boltzmann equation reads $$ \nabla^2 \phi = \frac{1}{\varepsilon} \sum_i z_i a_i \exp[-\beta z_i \phi] , . $$ Aha! A nice nonlinear equation! How to solve it? We can use perturbation methods. ...

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 $\phi = 0$. ...