Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations[1] 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.
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 classical mechanics,[2] 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.
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.
Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.
The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs in mathematics.
It relies on the fundamental lemma of calculus of variations.
extremizes the functional subject to the boundary conditions, then any slight perturbation of
The next step is to use integration by parts on the second term of the integrand, yielding
Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation
, we proceed by approximating the extremal curve by a polygonal line with
Extremals of this new functional defined on the discrete points
of the right-hand side of this expression yields
The left hand side of the previous equation is the functional derivative
A standard example[citation needed] 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.
is some surface, then is extremized only if f satisfies the partial differential equation When n = 2 and functional
is the energy functional, this leads to the soap-film minimal surface problem.
If there are several unknown functions to be determined and several variables such that the system of Euler–Lagrange equations is[5] 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[5] which can be represented shortly as: wherein
in order to avoid counting the same partial derivative multiple times, for example
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 where
are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is where the summation over the
The vector field generating time translations is denoted by
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.
