---
tag: D1EBAAFEWV
kind: concept
name: Hilbert's invariant integral
msc: [49K05, 49L99]
sources:
  - url: https://en.wikipedia.org/wiki/Calculus_of_variations
---

## Summary

Hilbert's invariant integral is a line integral attached to a [[field of extremals|K4DFT3H9BJ]] of a variational problem.
Its value depends only on the end points of the path, not on the path inside the field.
Along curves of the field it equals the functional itself.
Comparing it with the functional on an arbitrary curve gives the Weierstrass formula, which expresses $J(y) - J(y_0)$ as an integral of the [[Weierstrass excess function|R8TWRPSPNV]].

## Explanations

### L1 human

Inside a field of extremals, each point comes with a preferred slope, the slope of the field curve through it.
Hilbert built from these slopes a line integral that does not care which path one takes between two points.
On a field curve it gives exactly the value of the functional.
So to compare a competitor curve with the field curve, one compares the functional on the competitor with this path-independent integral.
The difference is an integral of the excess function, and if the excess is nonnegative, the field curve wins.