# Euler-Lagrange differential equation (elementary)

Let $q(t)$ be a twice differentiable function from $\mathbb{R}$ to $\mathbb{R}$ and let $L$ be a twice differentiable function from ${\mathbb{R}}^{3}$ to $\mathbb{R}$. Let $\dot{q}$ denote $\frac{d}{dt}q$.

Define the functional $I$ as follows:

$$I(q)={\int}_{a}^{b}L(t,q(t),\dot{q}(t))\mathit{d}t$$ |

Suppose we regard the function $L$ and the limits of integratiuon $a$ and $b$
as fixed and allow $q$ to vary. Then we could ask for which functions $q$ (if any) this integral attains an extremal (minimum or maximum) value. (Note: especially in Physics literature, the function $L$ is known as the *Lagrangian*.)

Suppose that a differentiable function ${q}_{0}:[a,b]\to \mathbb{R}$ is an extremum^{} of $I$. Then, for every differentiable function $f:[-1,+1]\times [a,b]\to \mathbb{R}$ such that $f(0,x)={q}_{0}(x)$, the function
$g:[-1,+1]\to \mathbb{R}$, defined as

$$g(\lambda )={\int}_{a}^{b}L(t,f(\lambda ,t),\frac{\partial f}{\partial t}(\lambda ,t))\mathit{d}t$$ |

will have an extremum at $\lambda =0$. If this function is differentiable^{}, then $dg/d\lambda =0$ when $\lambda =0$.

By studying the condition $dg/d\lambda =0$ (see the addendum to this entry for details), one sees that, if a function $q$ is to be an extremum of the integral $I$, then $q$ must satisfy the following equation:

$$\frac{\partial}{\partial q}L-\frac{d}{dt}\left(\frac{\partial}{\partial \dot{q}}L\right)=0.$$ | (1) |

This equation is known as the *Euler–Lagrange differential equation ^{}* or the Euler-Lagrange condition. A few comments on notation might be in . The notations $\frac{\partial}{\partial q}L$ and $\frac{\partial}{\partial \dot{q}}L$ denote the partial derivatives

^{}of the function $L$ with respect to its second and third arguments, respectively. The notation $\frac{d}{dt}$ means that one is to first make the argument a function of $t$ by replacing the second argument with $q(t)$ and the third argument with $\dot{q}(t)$ and secondly, differentiate the resulting function with respect to $t$. Using the chain rule

^{}, the Euler-Lagrange equation can be written as follows:

$$\frac{\partial}{\partial q}L-\frac{{\partial}^{2}}{\partial t\partial \dot{q}}L-\dot{q}\frac{{\partial}^{2}}{\partial q\partial \dot{q}}L-\ddot{q}\frac{{\partial}^{2}}{\partial {\dot{q}}^{2}}L=0$$ | (2) |

This equation plays an important role in the calculus of variations^{}. In using this equation, it must be remembered that it is only a necessary condition and, hence, given a solution of this equation, one cannot to the conclusion that this solution is a local extremum of the functional $F$. More work is needed to determine whether the solution of the Euler-Lagrange equation is an extremum of the integral $I$ or not.

In the special case $\frac{\partial}{\partial t}L=0$, the Euler-Lagrange equation can be replaced by the Beltrami identity^{}.

Title | Euler-Lagrange differential equation (elementary) |

Canonical name | EulerLagrangeDifferentialEquationelementary |

Date of creation | 2013-03-22 12:21:49 |

Last modified on | 2013-03-22 12:21:49 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 34 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 47A60 |

Classification | msc 70H03 |

Classification | msc 49K05 |

Synonym | Euler-Lagrange condition |

Related topic | CalculusOfVariations |

Related topic | BeltramiIdentity |

Related topic | VersionOfTheFundamentalLemmaOfCalculusOfVariations |

Defines | Euler-Lagrange differential equation |

Defines | Lagrangian |