---
tag: PPTX47481A
kind: definition
name: Field of extremals
msc: [49K05]
defines: K4DFT3H9BJ
uses: [JRQZ803Q5Y, PS0VV149ET, 0VW90BBSJ4]
sources:
  - url: https://en.wikipedia.org/wiki/Calculus_of_variations
---

## Definition

Let $L(x, y, p)$ be a $C^2$ function on $\R \times \R \times \R$, with [[integral functional|JRQZ803Q5Y]] $J(y) = \int L(x, y, y') \, dx$.
Let $D \subseteq \R^2$ be a simply connected open set.
A *field of extremals* of $J$ on $D$ is a family of functions $y = \varphi(x, c)$, with $c$ in an open interval $I$, such that:

1. for each $c \in I$, the function $x \mapsto \varphi(x, c)$ solves the [[Euler–Lagrange equation|PS0VV149ET]] of $J$;
2. through each point $(x, y) \in D$ passes exactly one curve $y = \varphi(x, c)$ of the family;
3. the parameter $c(x, y)$ of that curve is a [[continuously differentiable|0VW90BBSJ4]] function on $D$.

The *slope function* of the field is $p(x, y) = \partial_x \varphi(x, c(x, y))$.
An extremal $y_0$ is *embedded* in the field if $y_0 = \varphi(\cdot, c_0)$ for some $c_0 \in I$.

## Remarks

Example: for $L = \sqrt{1 + p^2}$ the extremals are straight lines.
The family $\varphi(x, c) = x + c$, $c \in \R$, is a field on $D = \R^2$, with $c(x, y) = y - x$ and slope function $p = 1$.

Non-example: the family $\varphi(x, c) = c x$, $c \in \R$, of lines through the origin is not a field on $\R^2$.
All its curves pass through $(0, 0)$, and no curve passes through the points $(0, y)$ with $y \ne 0$.
On the half-plane $x > 0$ it is a field, with $c = y / x$; it is then a *central field*, whose curves all start at one point on the boundary.

Let an extremal $y_0$ on $[a, b]$ satisfy $L_{pp} > 0$ along it, with no point conjugate to $a$ in $(a, b]$.
Then it can be embedded in a field on a neighbourhood of its graph.