Some random equations
Classical Mechanics
\[\begin{aligned} & S = \int_{t_1}^{t_2} L\left(q_i,\dot q_i,t\right)\,dt \\ & \delta S = 0 \end{aligned}\]Electrodynamics
\[\begin{aligned} &\nabla\cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\ &\nabla\times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla\cdot \mathbf{B} = 0 \\ &\nabla\times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}\]Statistical Mechanics
\[\begin{aligned} & S = - k_B \sum_i p_i \ln p_i \\ & \delta S = 0 \\ & dU = T\,dS - P\,dV + \mu\,dN \end{aligned}\]Quantum Mechanics
\[\begin{aligned} & |\psi\rangle \in \mathcal{H} \quad \hat{A} = \hat{A}^\dagger \\ & P(a_n) = |\langle a_n | \psi \rangle|^2 \\ & i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H}|\psi(t)\rangle \end{aligned}\]