# Euler-Lagrange differential equation (advanced)

Let $M$ and $N$ be ${C}^{2}$ manifolds^{}. Let $L:M\times TN\to \mathbb{R}$ be twice differentiable^{}. Define a functional $F:D\subset {C}^{2}(M,N)\to \mathbb{R}$ as

$$F(q)={\int}_{M}L(x,q(x),\mathbf{D}q(x)){d}^{m}x$$ |

where $D$ is the subset of http://planetmath.org/node/5555${\mathrm{C}}^{\mathrm{2}}\mathtt{}\mathrm{(}\mathrm{M}\mathrm{,}\mathrm{N}\mathrm{)}$ for which this integral converges.

Note that if $f\in D$ and $g\in {C}^{2}(M,N)$ and the set $\{x\in M\mid f(x)\ne g(x)\}$ is compact, then $g\in D$. We may impose a topology on $D$ as follows: Suppose that $f\in D$, that $K\subset M$ is compact, and that ${U}_{0}\subset {C}^{2}(K,N)$ is open. Then we define an open set $U\subset D$ as the set of all functions $g\in D$ such that $f(x)=g(x)$ when $x\notin K$ and such that the restriction of $g$ to $K$ lies in ${U}_{0}$.

It is not hard to show that the functional $F$ is continuous^{} in this topology, and hence it makes sense to speak of local extrema of $F$. Suppose that ${q}_{0}\in {C}^{2}(M,N)$ is a local extremum. Furthermore, suppose that $f:M\times [-1,+1]\to N$ is twice differentiable and $f(x,0)={q}_{0}(x)$ for all $x\in {q}_{0}$ and $f(x,y)={q}_{0}(x)$ for all $y\in [-1,+1]$ when $x$ does not lie in a certain compact subset $K\subset M$. Then, viewed as a map from $[-1,+1]$ to $D$, $f$ will be continuous. Therefore, since ${q}_{0}$ is a local extremum of $F$, $0$ wil be a local extremum of the function $y\mapsto F(f(\cdot ,y))$. Because the function $y\mapsto F(f(\cdot ,y))$ is differentiable, it will be the case that

$${\frac{d}{d\lambda}F(f(\cdot ,\lambda ))|}_{\lambda =0}=0$$ |

It can be shown (see the addendum to this entry) that this condition will be satisfied if and only if ${q}_{0}$ is a solution of the following differential equation^{}:

$$dL-d\left(\frac{\partial L}{\partial (dq)}\right)=0.$$ | (1) |

This differential equation is known as the *Euler-Lagrange differential equation ^{}* (or Euler-Lagrange condition).

The Euler-Lagrange equation can only be used to investigate local extrema which are smooth functions. To a certain extent, this limitation can be ameliorated — one can study piecewise smooth functions by supplementing the Euler-Lagrange equation with auxiliary conditions at discontinuities and, in some cases, one can consider non-smooth solutions as weak solutions of the Euler-Lagrange equation.

In the special cases $dL=0$, the Euler-Lagrange equation can be replaced by the Beltrami identity^{}.

Title | Euler-Lagrange differential equation (advanced) |
---|---|

Canonical name | EulerLagrangeDifferentialEquationadvanced |

Date of creation | 2013-03-22 14:45:32 |

Last modified on | 2013-03-22 14:45:32 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 16 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 47A60 |

Synonym | Euler-Lagrange condition |

Defines | Euler-Lagrange differential equation |