---
tag: Q12AB1MJ64
kind: concept
name: Energy functional on paths
msc: ["58E10", 53C22]
sources:
  - url: https://en.wikipedia.org/wiki/Geodesic
---

## Summary

The energy of a path in a [[Riemannian manifold|7CBT4SM83Y]] is half the integral of the squared speed.
Among paths with fixed ends, its critical points are exactly the [[geodesics|NSYMQMHPDN]] with constant speed.
In variational work it is preferred to the length.
It is not invariant under reparametrization, so its critical points are better behaved and its Hessian is easier to compute.
It is the main example of a function to which Morse theory on infinite-dimensional manifolds applies.

## Explanations

### L1 human

The length of a path does not care how fast it is travelled; the energy does.
The energy is smallest, for a given length and travel time, when the speed is constant.
So minimizing energy finds the shortest path and also a uniform way to travel along it.
For example, in the plane the path $t \mapsto (t, 0)$ for $0 \le t \le 1$ has energy $\frac{1}{2}$, while $t \mapsto (t^2, 0)$ covers the same segment with energy $\frac{2}{3}$.