calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaextrema, the Euler–Lagrange equation is useful for solving

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, stating that at any point where a differentiable function attains a local extremum its

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

is zero. In

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...

, according to

Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, ...

of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

, it is equivalent to

Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...

; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has the advantage that it takes the same form in any system of

generalized coordinate In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...

s, and it is better suited to generalizations. In

classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...

there is an analogous equation to calculate the dynamics of a field.

History

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

, which led to the formulation of

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...

. Their correspondence ultimately led to the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, a term coined by Euler himself in 1766.A short biography of Lagrange

Statement

Let

(X,L)

be a mechanical system with

n

degrees of freedom. Here

X

is the configuration space and

L=L(t,\boldsymbol q, \boldsymbol v)

the ''Lagrangian'', i.e. a smooth real-valued function such that

\boldsymbol q \in X,

and

\boldsymbol v

is an

n

-dimensional "vector of speed". (For those familiar with

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

,

X

is a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

, and

L : _t \times TX \to ,

where

TX

is the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

of

X).

Let

(a,b,\boldsymbol x_a,\boldsymbol x_b)

be the set of smooth paths

\to X

for which

\boldsymbol q(a) = \boldsymbol x_a

and

\boldsymbol q(b) = \boldsymbol x_.

The action functional

S : (a,b,\boldsymbol x_a,\boldsymbol x_b) \to \mathbb

is defined via

= \int_a^b L(t,\boldsymbol q(t),\dot(t))\, dt.

A path

\boldsymbol q \in (a,b,\boldsymbol x_a,\boldsymbol x_b)

is a

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...

of

S

if and only if Here,

\dot(t)

is the time derivative of

\boldsymbol q(t).

Example

A standard example is finding the real-valued function ''y''(''x'') on the interval 'a'', ''b'' such that ''y''(''a'') = ''c'' and ''y''(''b'') = ''d'', for which the path

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

along the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

traced by ''y'' is as short as possible. :

\text = \int_^ \sqrt = \int_^ \sqrt\,\mathrmx,

the integrand function being . The partial derivatives of ''L'' are: :

\frac = \frac \quad \text \quad
\frac = 0.

By substituting these into the Euler–Lagrange equation, we obtain :

\begin
\frac \frac &= 0 \\ 
\frac &= C = \text \\
\Rightarrow y'(x)&= \frac =: A \\
\Rightarrow y(x) &= Ax + B
\end

that is, the function must have a constant first derivative, and thus its graph is a

straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

.

Generalizations

Single function of single variable with higher derivatives

The stationary values of the functional :

= \int_^ \mathcal(x, f, f', f'', \dots, f^)~\mathrmx ~;~~ f' := \cfrac, ~f'' := \cfrac, ~ f^ := \cfrac

can be obtained from the Euler–Lagrange equation :

\cfrac - \cfrac\left(\cfrac\right) + \cfrac\left(\cfrac\right) - \dots +
  (-1)^k \cfrac\left(\cfrac\right)  = 0

under fixed boundary conditions for the function itself as well as for the first

k-1

derivatives (i.e. for all

f^, i \in \

). The endpoint values of the highest derivative

f^

remain flexible.

Several functions of single variable with single derivative

If the problem involves finding several functions (

f_1, f_2, \dots, f_m

) of a single independent variable (

x

) that define an extremum of the functional :

= \int_^ \mathcal(x, f_1, f_2, \dots, f_m, f_1', f_2', \dots, f_m')~\mathrmx ~;~~ f_i' := \cfrac

then the corresponding Euler–Lagrange equations are :

\begin
     \frac - \frac\left(\frac\right) = 0 ; \quad i = 1, 2, ..., m
   \end

Single function of several variables with single derivative

A multi-dimensional generalization comes from considering a function on n variables. If

\Omega

is some surface, then :

= \int_ \mathcal(x_1, \dots , x_n, f, f_, \dots , f_)\, \mathrm\mathbf\,\! ~;~~ f_ := \cfrac

is extremized only if ''f'' satisfies the

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

:

\frac - \sum_^ \frac\left(\frac\right) = 0.

When ''n'' = 2 and functional

\mathcal I

is the

energy functional The energy functional is the total energy of a certain system, as a functional of the system's state. In the energy methods of simulating the dynamics of complex structures, a state of the system is often described as an element of an appropriate ...

, this leads to the soap-film minimal surface problem.

Several functions of several variables with single derivative

If there are several unknown functions to be determined and several variables such that :

= \int_ \mathcal(x_1, \dots , x_n, f_1, \dots, f_m, f_, \dots , f_, \dots, f_, \dots, f_) \, \mathrm\mathbf\,\! ~;~~ f_ := \cfrac

the system of Euler–Lagrange equations is :

\begin
    \frac - \sum_^ \frac\left(\frac\right) &= 0_1 \\
    \frac - \sum_^ \frac\left(\frac\right) &= 0_2 \\
    \vdots \qquad \vdots \qquad &\quad \vdots  \\
    \frac - \sum_^ \frac\left(\frac\right) &= 0_m.
  \end

Single function of two variables with higher derivatives

If there is a single unknown function ''f'' to be determined that is dependent on two variables ''x''₁ and ''x''₂ and if the functional depends on higher derivatives of ''f'' up to ''n''-th order such that :

& = \int_ \mathcal(x_1, x_2, f, f_, f_, f_, f_, f_, \dots, f_)\, \mathrm\mathbf \\ & \qquad \quad f_ := \cfrac \; , \quad f_ := \cfrac \; , \;\; \dots \end

then the Euler–Lagrange equation is :

\begin
    \frac
    & - \frac\left(\frac\right)
      - \frac\left(\frac\right) 
      + \frac\left(\frac\right)
      + \frac\left(\frac\right)
      + \frac\left(\frac\right) \\
    & - \dots
      + (-1)^n \frac\left(\frac\right) = 0
  \end

which can be represented shortly as: :

\frac +\sum_^n \sum_ (-1)^j \frac \left( \frac\right)=0

wherein

\mu_1 \dots \mu_j

are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the

\mu_1 \dots \mu_j

indices is only over

\mu_1 \leq \mu_2 \leq \ldots \leq \mu_j

in order to avoid counting the same partial derivative multiple times, for example

f_ = f_

appears only once in the previous equation.

Several functions of several variables with higher derivatives

If there are ''p'' unknown functions ''f''_i to be determined that are dependent on ''m'' variables ''x''₁ ... ''x''_m and if the functional depends on higher derivatives of the ''f''_i up to ''n''-th order such that :

& = \int_ \mathcal(x_1, \ldots, x_m; f_1,\ldots,f_p; f_,\ldots, f_; f_,\ldots, f_;\ldots; f_, \ldots, f_)\, \mathrm\mathbf \\ & \qquad \quad f_ := \cfrac \; , \quad f_ := \cfrac \; , \;\; \dots \end

where

\mu_1 \dots \mu_j

are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is :

\frac +\sum_^n \sum_ (-1)^j \frac \left( \frac\right)=0

where the summation over the

\mu_1 \dots \mu_j

is avoiding counting the same derivative

f_ = f_

several times, just as in the previous subsection. This can be expressed more compactly as :

\sum_^n \sum_ (-1)^j \partial_^j \left( \frac\right)=0

Generalization to manifolds

Let

M

be a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

, and let

C^\infty(,b

denote the space of smooth functions

f\colon,b to M

. Then, for functionals

\to \mathbb

of the form :

\int_a^b (L\circ\dot)(t)\,\mathrm t

where

L\colon TM\to\mathbb

is the Lagrangian, the statement

\mathrm S_f=0

is equivalent to the statement that, for all

t\in,b /math>, each coordinate frame trivialization (x^i,X^i) of a neighborhood of \dot(t) yields the following \dim M equations:
: \forall i:\frac\frac\bigg, _=\frac\bigg, _.

Notes

References

* * * * * Roubicek, T.:
Calculus of variations
Chap.17 in
Mathematical Tools for Physicists
(Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. {{DEFAULTSORT:Euler-Lagrange Equation Ordinary differential equations Partial differential equations Calculus of variations Articles containing proofs Leonhard Euler

History

Statement

Example

Generalizations

Single function of single variable with higher derivatives

Several functions of single variable with single derivative

Single function of several variables with single derivative

Several functions of several variables with single derivative

Single function of two variables with higher derivatives

Several functions of several variables with higher derivatives

Generalization to manifolds

See also

Notes

References