Euler–Lagrange Equation

Equivalent forms

\delta S = \delta\!\int L\,dt = 0

\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0

Newton's F = ma, Fermat's least time, and the field equations of physics all fall out of one principle: make the action stationary.

Symbol	Name	SI unit	Dimension	Range
$R$	Ramp height Drop height for the racing beads	1	1	0.4 – 1
$g$	Gravity Gravitational acceleration (m/s²)	m/s^2	L T^-2	5 – 12
$t_{brach}$	Brachistochrone timeoutput Time on the least-time curve	s	T	0 – 3
$t_{line}$	Straight-line timeoutput Time on the straight ramp	s	T	0 – 3

Unit systems

Symbol	SI	CGS	Imperial
$g$	m/s^2	cm/s^2	ft/s^2
$t_{brach}$	s	—	—

Where it holds

Applies to any functional

\int L(q

,q̇,t)dt with fixed endpoints and smooth L. Extends to fields (multiple integrals) and constraints (Lagrange multipliers).

Dimensional analysis

d/dt(\partial L/\partial q

̇)

\partial L/\partial q

\partial L/\partial q

̇ has dimension [L]/[q̇

]=

action/q; d/dt of it matches

\partial L/\partial q = [L]/[q]

. Both terms share dimension energy/q.

\checkmark

Discovery

Euler & Lagrange · 1755

Euler derived it geometrically in 1744; the teenage Lagrange sent Euler a purely analytic 'variational' derivation in 1755, and Euler generously let the younger man's method take the spotlight, naming the field 'calculus of variations'. It became the foundation of Lagrangian and Hamiltonian mechanics and, later, field theory.

Try this

Of all the paths a ball, a light ray, or a planet could take, nature always picks the one that makes a certain total — the action — stationary. The Euler–Lagrange equation is the rule that turns 'pick the best path' into a differential equation.

Race two beads to the bottom: one on the straight ramp, one on the brachistochrone curve the equation predicts. The curved path wins — least time, not least distance.

Research status: stable

Real-world applications

Lagrangian and Hamiltonian mechanics
Geodesics in general relativity
Optical ray paths (Fermat's principle)
Optimal control and shape optimization

Common misconceptions

Nature minimizes action — it makes it stationary (a saddle is common), not necessarily minimal.
The shortest path is always the fastest — the brachistochrone shows otherwise.
It only works in time — the same equation optimizes any integral (geodesics, minimal surfaces, optics).

Experimental verification

Bead-on-a-wire brachistochrone demos reproduce the predicted least-time win of the cycloid; every Lagrangian-mechanics prediction (pendulums, orbits, coupled oscillators) that agrees with experiment is a test of this equation.

Derivation

Vary

q(t)\to q+\varepsilon \eta

with

\eta

vanishing at the endpoints.

\delta S = \int (\partial L/\partial q \eta + \partial L/\partial q

̇

\eta

̇)dt; integrate the second term by parts (boundary term dies) to factor

out \eta

:

\int (\partial L/\partial q - d/dt \partial L/\partial q

̇

)\eta dt = 0

.

Since

\eta is

arbitrary, the bracket must vanish everywhere.

Limiting cases

L independent of q⟶

\partial L/\partial q

̇ = constA conserved momentum — Noether's theorem in miniature.

L independent of t⟶ energy conservedThe Hamiltonian (Beltrami identity) is constant.

straight vs cycloid⟶ cycloid is fasterLeast distance ≠ least time; the brachistochrone is a cycloid.

What if…

What if there are several coordinates q_i?

You get one Euler–Lagrange equation per coordinate — a coupled system, the full equations of motion.

What if L has no explicit q (cyclic coordinate)?

$\partial L/\partial q$ ̇ is conserved — a symmetry gives a conservation law, the seed of Noether's theorem.

1

Recover Newton's law

Given ·

L:: $\tfrac{1}{2}m q$ ̇ $^{2} - V(q)$

Find · equation of motion

Steps

$\partial L/\partial q$ ̇ $= m q$ ̇
d/dt gives m q̈
$\partial L/\partial q = -V$ '; set difference to 0

Result · m q̈

= -V'(q)

2

Shortest path is a straight line

Given ·

f:: $\sqrt{1+y'^{2}}$

Find · extremal y(x)

Steps

f has no explicit y, $so \partial f/\partial y$ '= const
$y'/\sqrt{1+y'^{2}}$ =const $\Rightarrow y$ '=const
Constant slope = straight line

Result ·

y = mx + b

hamilton principle mathfunctional derivativeFundamental Theorem of Calculus→noether symmetry conservation