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Wikipedia / Euler–Lagrange equation

In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.

Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle 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, it is equivalent to Newton's laws of motion; 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 coordinates, and it is better suited to generalizations. In classical field theory 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, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.

Statement

be a real dynamical system is the configuration space the Lagrangian, i.e. a smooth real-valued function is the time derivative. (For those familiar with differential geometry is a smooth manifold is the tangent bundle).

be the set of smooth paths.

The action functional

is a stationary point if and only if

. See proofs below for more rigorous detail.

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 along the curve traced by y is as short as possible.

.

The partial derivatives of L are:

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

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

Canonical momenta and constants of motion

The conjugate momentum for a generalized coordinate

An important special case of the Euler–Lagrange equation occurs when L does not contain a generalized coordinate that is, the conjugate momentum is a constant of the motion.

In such cases, the coordinate is called a cyclic coordinate. For example, if we use polar coordinates , , to describe the planar motion of a particle, and if does not depend on , the conjugate momentum is the conserved angular momentum.

Generalizations

Single function of single variable with higher derivatives

The stationary values of the functional

can be obtained from the Euler–Lagrange equation

remain flexible.

Several functions of single variable with single derivative

) of a single independent variable) that define an extremum of the functional

then the corresponding Euler–Lagrange equations are

Single function of several variables with single derivative

is some surface, then

is extremized only if f satisfies the partial differential equation

When n is the energy functional, 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

the system of Euler–Lagrange equations is

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 x1 and x2 and if the functional depends on higher derivatives of f up to n-th order such that

then the Euler–Lagrange equation is

which can be represented shortly as:

in order to avoid counting the same partial derivative appears only once in the previous equation.

Several functions of several variables with higher derivatives

If there are p unknown functions fi to be determined that are dependent on m variables x1 ... xm and if the functional depends on higher derivatives of the fi up to n-th order such that

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

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

Field theories

Generalization to manifolds

be a smooth manifold denote the space of smooth functions of the form

, each coordinate frame trivialization equations:

Euler–Lagrange equations can also be written in a coordinate-free form as

is the canonical momenta 1-form. The vector field and the Lie derivative and use coordinate expressions for the Lie derivative to see equivalence with coordinate expressions of the Euler Lagrange equation. The coordinate free form is particularly suitable for geometrical interpretation of the Euler Lagrange equations.

See also

Notes

References