Calculus of variations

Calculus of variations is the field that searches for maxima and minima of Functionals, rather than the more elementary case of functions from

R

R

Table of contents
- Functional Calculus of variations
- Euler-Lagrange equation Calculus of variations
  - Equations of motion Euler-Lagrange equation

Functional

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A function that takes input function and outputs a real number.

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given

F

, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that

F

such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙} = 0

(2)

\frac{\partial F}{\partial q}

and

\frac{\partial F}{\partial q ˙}

we simply mean "the partial derivative of

F

with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where

q

is the unknown, since:

the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q}$ is a function of $t, q (t), \overset{q}{˙} (t)$
the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙}$ is a function of $t, q (t), \overset{q}{˙} (t)$ . And so it's derivative with respect to time will contain only up to $\overset{q}{¨}$

Now let's think about the multi-dimensional case. Instead of having

q : R \to R

, we now have

q : R \to R^{n}

. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case,

F

is going to be a function of 5 variables

F : R^{5} \to R

rather than 3 as in the one dimensional case, and the functional looks like:

I (q) = \int_{t_{0}}^{t} F (t, q_{1} (t), q_{2} (t), \overset{q_{1}}{˙} (t), \overset{q_{2}}{˙}) (t) d t

(3)

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions

q_{1}

and

q_{2}

of order up to 2 in both variables:

\frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{2} ˙} = 0

(4)

At this point, notation is getting a bit clunky, so people will often condense the

q_{i}

vector

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2} ˙} = 0

(5)

or just omit the arguments of

F

entirely:

\frac{\partial F}{\partial q _{i}} - \frac{d}{d t} \frac{\partial F}{\partial q _{i} ˙} = 0

(6)

Video 1.

Calculus of Variations ft. Flammable Maths by vcubingx (2020)

Source.

Equations of motion

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These are the final equations that you derive from the Lagrangian via the Euler-Lagrange equation which specify how the system evolves with time.

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Maxima and minima

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Calculus of variations by

Codex  0 Created 2026-09-24 Updated 2026-11-05

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The calculus of variations studies functionals whose inputs are functions, curves or fields. A variational problem specifies an admissible class, including endpoint or other boundary conditions, and asks for stationary points or minimizers. The first variation gives an equation for stationary candidates; a second variation, convexity or a direct comparison is needed to decide minimality.

For example, minimize

E [u] = \frac{1}{2} \int_{0}^{1} u' (x)^{2} d x, u (0) = a, u (1) = b .

(1)

A zero-endpoint variation

u + ϵη

gives

δ E [u; η] = \int_{0}^{1} u' η'

. If

u

is twice differentiable, integration by parts and the fundamental lemma of the calculus of variations yield

u'' = 0

. The candidate is the straight line

ℓ (x) = a + (b - a) x

It is also the unique global minimizer, not merely stationary. Write any admissible

u = ℓ + w

with zero endpoint values for

w

. The cross term is

(b - a) \int_{0}^{1} w' = 0

, so

E [ℓ + w] - E [ℓ] = \frac{1}{2} \int_{0}^{1} w'^{2} d x \geq 0.

(2)

Equality forces

w = 0

. This argument extends to Sobolev spaces with a meaningful endpoint trace and illustrates the Dirichlet principle. In more complicated problems, the direct method in the calculus of variations establishes existence without first solving the stationarity equation.

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