Euler–Lagrange equations

Euler–Lagrange equations. In the calculus of variations, a central result provides the fundamental differential equation that any extremal of a functional must satisfy. Developed independently in the 1750s by Leonhard Euler and later refined by Joseph-Louis Lagrange, these equations form the cornerstone for deriving the equations of motion in Lagrangian mechanics and have profound applications across theoretical physics and geometry. Their derivation stems from considering the first variation of an action integral and applying the fundamental lemma of the calculus of variations.

Statement of the equation

For a functional \( J \) defined by an integral of a Lagrangian function \( L \), the Euler–Lagrange equations yield the necessary condition for an extremum. Consider the standard problem of finding a function \( y(x) \) that extremizes the functional \( J[y] = \int_{a}^{b} L(x, y(x), y'(x)) \, dx \), where the endpoints \( y(a) \) and \( y(b) \) are fixed. Provided that \( L \) is continuously differentiable with respect to its arguments, the Euler–Lagrange equation is the ordinary differential equation given by \( \frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0 \). This formulation was famously published by Lagrange in his Mécanique Analytique, building upon earlier work by Euler in Methodus Inveniendi Lineas Curvas. For systems with multiple dependent variables, such as in describing the path of a particle in three dimensions, a set of coupled equations arises, one for each generalized coordinate.

Derivation

The classical derivation applies the calculus of variations to the action integral. One considers a one-parameter family of comparison functions \( y(x) + \epsilon \eta(x) \), where \( \eta(x) \) is an arbitrary differentiable function vanishing at the boundaries \( a \) and \( b \), and \( \epsilon \) is a small parameter. The condition for an extremum is that the first variation \( \delta J \) vanishes for all such \( \eta \). Computing the derivative of \( J \) with respect to \( \epsilon \) at \( \epsilon = 0 \), integrating the term involving \( \eta' \) by parts, and applying the fundamental lemma of the calculus of variations yields the equation. This process mirrors techniques used by Bernoulli family mathematicians in solving the brachistochrone problem. The derivation assumes sufficient smoothness of the Lagrangian and the admissible functions, a point later rigorously addressed in the work of David Hilbert and others in the context of the direct method in the calculus of variations.

Basic examples

A canonical example is the derivation of the equations of motion for a particle in classical mechanics. For a particle of mass \( m \) moving in a potential \( V(\mathbf{r}) \), the Lagrangian is \( L = T - V = \frac{1}{2} m \dot{\mathbf{r}}^2 - V(\mathbf{r}) \). Applying the equations for each component recovers Newton's second law: \( m \ddot{\mathbf{r}} = -\nabla V \). Another foundational example is finding the shortest path between two points in a plane, leading to a straight line. Here, the Lagrangian is the arc length element \( L = \sqrt{1 + (y')^2} \), and the equations reduce to \( y'' = 0 \). The brachistochrone curve, solved by Johann Bernoulli, is the path of fastest descent under gravity and yields a cycloid as the solution. In geometric optics, Fermat's principle of least time leads to the Eikonal equation governing light rays.

Generalizations

Several important generalizations extend the basic framework. For functionals depending on higher-order derivatives, such as in the theory of elastic beams, the equation becomes \( \frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) + \frac{d^2}{dx^2} \left( \frac{\partial L}{\partial y''} \right) - \cdots = 0 \). In classical field theory, where the Lagrangian density depends on fields and their spacetime derivatives, the result is the Euler–Lagrange equation for fields, a set of partial differential equations like the Klein–Gordon equation or Maxwell's equations. The Hamilton's principle provides the foundational variational principle for these field equations. Furthermore, the theory can be extended to problems with constraints using Lagrange multipliers, a technique pivotal in analytical mechanics and optimal control theory.

Applications

The equations are indispensable in theoretical physics. In Lagrangian mechanics, they form the basis for deriving the equations of motion for complex systems, from a double pendulum to celestial bodies governed by the n-body problem. They are central to the formulation of quantum field theory via the path integral formulation developed by Richard Feynman. In general relativity, the Einstein–Hilbert action yields the Einstein field equations through variational principles. Beyond physics, they appear in optimal control theory, such as in the Pontryagin's maximum principle, and in differential geometry for finding geodesics on manifolds, a concept generalized in the calculus of variations on manifolds. The formalism also underpins modern numerical methods like finite element analysis for solving boundary value problems.

Category:Differential equations Category:Calculus of variations Category:Equations