Physics equations in Markdown to PDF

# Equations of physics

## Classical mechanics

Newton's second law and the gravitational force, then the Lagrangian form:

$ \mathbf{F} = m\,\ddot{\mathbf{r}} \qquad \mathbf{F} = -\frac{G M m}{r^{2}}\,\hat{\mathbf{r}} \qquad \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 $

A simple pendulum of length $\ell$ has period $T = 2\pi\sqrt{\ell / g}$ for small angles.

## Electromagnetism

Maxwell's equations in differential form:

$ \begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned} $

The same in integral form, and the Lorentz force:

$ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = -\frac{\mathrm{d}}{\mathrm{d}t} \iint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} \qquad \mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) $

## Quantum mechanics

$ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \left[ -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) \right] \Psi(\mathbf{r}, t) $

$ \left( i\gamma^{\mu} \partial_{\mu} - m \right) \psi = 0 \qquad [\hat{x}, \hat{p}] = i\hbar \qquad \Delta x \, \Delta p \geq \frac{\hbar}{2} \qquad \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle $

The hydrogen energy levels and the Planck relation:

$ E_n = -\frac{m_e e^4}{8 \varepsilon_0^2 h^2} \cdot \frac{1}{n^2} \approx -\frac{13.6\ \text{eV}}{n^2} \qquad E = h\nu = \hbar\omega $

## Relativity

A boost along $x$ with $\beta = v/c$ and $\gamma = 1/\sqrt{1 - \beta^2}$:

$ \begin{pmatrix} ct' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} $

$ R_{\mu\nu} - \tfrac{1}{2} R\, g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \qquad E^2 = (pc)^2 + (m c^2)^2 $

## Fluids and heat

Navier-Stokes for an incompressible fluid, the heat equation, and Planck's law of black-body radiation:

$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}, \qquad \nabla \cdot \mathbf{v} = 0 $

$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u \qquad B_\nu(T) = \frac{2 h \nu^3}{c^2} \cdot \frac{1}{e^{h\nu / k_B T} - 1} \qquad S = k_B \ln \Omega $

| Constant | Symbol | Value |
|:---------|:------:|:------|
| Speed of light | $c$ | $2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}$ |
| Planck constant | $h$ | $6.626 \times 10^{-34}\ \mathrm{J\,s}$ |
| Gravitational constant | $G$ | $6.674 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ |
| Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}\ \mathrm{J\,K^{-1}}$ |