The Euler-Lagrange equation

To extremise $J[y] = \int_a^b L(x, y, y’)\,dx$ subject to boundary conditions $y(a)=y_a$, $y(b)=y_b$:

\[\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} = 0\]

Example: geodesics on $\mathbb{R}^2$ minimise arc length $J = \int\sqrt{1+y’^2}\,dx$, giving $y’’ = 0$ — straight lines.

Hamilton’s principle

Classical mechanics follows from: the physical trajectory $q(t)$ extremises the action:

\[S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t)\,dt\]

where $L = T - V$ (kinetic minus potential energy). The Euler-Lagrange equation gives Newton’s laws.

Variational inference

In Bayesian ML, the posterior $p(\theta

x)$ is approximated by $q(\theta)$ from a tractable family by minimising $D_{KL}(q|p)$. The ELBO (Evidence Lower BOund) is the variational objective.