In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and French-Italian mathematician Joseph-Louis Lagrange in the 1750s.
Because a differentiable functional is stationary at its local maxima and minima, 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, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the action of the system. In classical mechanics, it is equivalent to Newton's laws of motion, but 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.
- 1 History
- 2 Statement
- 3 Examples
- 4 Variations for several functions, several variables, and higher derivatives
- 4.1 Single function of single variable with higher derivatives
- 4.2 Several functions of one variable
- 4.3 Single function of several variables
- 4.4 Several functions of several variables
- 4.5 Single function of two variables with higher derivatives
- 4.6 Several functions of several variables with higher derivatives
- 5 Generalization to Manifolds
- 6 See also
- 7 Notes
- 8 References
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.
- is the function to be found:
- such that is differentiable, , and ;
- ; is the derivative of :
- TX being the tangent bundle of X defined by
The Euler–Lagrange equation, then, is given by
where Lx and Lv denote the partial derivatives of L with respect to the second and third arguments, respectively.
If the dimension of the space X is greater than 1, this is a system of differential equations, one for each component:
Derivation of one-dimensional Euler–Lagrange equation
We wish to find a function which satisfies the boundary conditions , , and which extremizes the functional
We assume that has continuous first partial derivatives. A weaker assumption can be used, but the proof becomes more difficult.
If extremizes the functional subject to the boundary conditions, then any slight perturbation of that preserves the boundary values must either increase (if is a minimizer) or decrease (if is a maximizer).
Let be the result of such a perturbation of , where is small and is a differentiable function satisfying . Then define
We now wish to calculate the total derivative of with respect to ε.
It follows from the total derivative that
When ε = 0 we have gε = f, Fε = F(x, f(x), f'(x)) and Jε has an extremum value, so that
The next step is to use integration by parts on the second term of the integrand, yielding
Using the boundary conditions ,
Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation
Alternate derivation of one-dimensional Euler–Lagrange equation
Given a functional
on with the boundary conditions and , we proceed by approximating the extremal curve by a polygonal line with segments and passing to the limit as the number of segments grows arbitrarily large.
Divide the interval into equal segments with endpoints and let . Rather than a smooth function we consider the polygonal line with vertices , where and . Accordingly, our functional becomes a real function of variables given by
Extremals of this new functional defined on the discrete points correspond to points where
Evaluating this partial derivative gives
Dividing the above equation by gives
and taking the limit as of the right-hand side of this expression yields
The left hand side of the previous equation is the functional derivative of the functional . A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation.
the integrand function being L(x, y, y′) = √ evaluated at (x, y, y′) = (x, f(x), f′(x)).
The partial derivatives of L are:
By substituting these into the Euler–Lagrange equation, we obtain
that is, the function must have constant first derivative, and thus its graph is a straight line.
To find the equations of motions for a given system (whose potential energy is time-independent), one only has to follow these steps:
- From the kinetic energy , and the potential energy , compute the Lagrangian .
- Compute .
- Compute and from it, . It is important that be treated as a complete variable in its own right, and not as a derivative.
- Equate . This is the Euler–Lagrange equation.
- Solve the differential equation obtained in the preceding step. At this point, is treated "normally". Note that the above might be a system of equations and not simply one equation.
Particle in a conservative force field
The motion of a single particle in a conservative force field (for example, the gravitational force) can be determined by requiring the action to be stationary, by Hamilton's principle. The action for this system is
where x(t) is the position of the particle at time t. The dot above is Newton's notation for the time derivative: thus ẋ(t) is the particle velocity, v(t). In the equation above, L is the Lagrangian (the kinetic energy minus the potential energy):
- m is the mass of the particle (assumed to be constant in classical physics);
- vi is the i-th component of the vector v in a Cartesian coordinate system (the same notation will be used for other vectors);
- U is the potential of the conservative force.
In this case, the Lagrangian does not vary with its first argument t. (By Noether's theorem, such symmetries of the system correspond to conservation laws. In particular, the invariance of the Lagrangian with respect to time implies the conservation of energy.)
By partial differentiation of the above Lagrangian, we find:
where the force is F = −∇U (the negative gradient of the potential, by definition of conservative force), and p is the momentum. By substituting these into the Euler–Lagrange equation, we obtain a system of second-order differential equations for the coordinates on the particle's trajectory,
which can be solved on the interval [t0, t1], given the boundary values xi(t0) and xi(t1). In vector notation, this system reads
or, using the momentum,
which is Newton's second law.
Variations for several functions, several variables, and higher derivatives
Single function of single variable with higher derivatives
The stationary values of the functional
can be obtained from the Euler–Lagrange equation
under fixed boundary conditions for the function itself as well as for the first derivatives (i.e. for all ). The endpoint values of the highest derivative remain flexible.
Several functions of one variable
If the problem involves finding several functions () of a single independent variable () that define an extremum of the functional
then the corresponding Euler–Lagrange equations are
Single function of several variables
A multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, then
is extremized only if f satisfies the partial differential equation
Several functions of several variables
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:
where are indices that span the number of variables, that is they go from 1 to 2. Here summation over the indices is implied according to Einstein notation.
Several functions of several variables with higher derivatives
If there is 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 summation over the is implied according to Einstein notation. This can be expressed more compactly as
Generalization to Manifolds
Let be a smooth manifold, and let denote the space of smooth functions . Then, for functionals of the form
where is the Lagrangian, the statement is equivalent to the statement that, for all , each coordinate frame trivialization of a neighborhood of yields the following equations:
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- Lagrangian mechanics
- Hamiltonian mechanics
- Analytical mechanics
- Beltrami identity
- Functional derivative
- Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.
- A short biography of Lagrange
- Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics: Vol I, Interscience Publishers, New York.
- Weinstock, R., 1952, Calculus of Variations With Applications to Physics and Engineering, McGraw-Hill Book Company, New York.
- Hazewinkel, Michiel, ed. (2001), "Lagrange equations (in mechanics)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Euler-Lagrange Differential Equation", MathWorld.
- Calculus of Variations at PlanetMath.org.
- Gelfand, Izrail Moiseevich (1963). Calculus of Variations. Dover. ISBN 0-486-41448-5.
- Roubicek, T.: Calculus of variations. Chap.17 in: Mathematical Tools for Physicists. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, ISBN 978-3-527-41188-7, pp.551-588.