---
tag: 6VR8GASZJY
kind: concept
name: Mutual energy
msc: [31B15, 31A15]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Summary

The mutual energy of two positive measures is the double integral of a potential-theoretic kernel against one measure in each variable.
It is the interaction energy of two charge distributions, and it equals the integral of the potential of one measure against the other.
The [[energy of a measure|SFSRQCM3EG]] is the mutual energy of the measure with itself.
So the mutual energy is the bilinear form whose quadratic form is the energy.
Its [[positive definiteness|PNZT35FFN2]] turns measures of finite energy into a pre-Hilbert space, which is the basis of the energy method in potential theory.

## Explanations

### L1 human

Take two clouds of charge.
Their mutual energy adds up the repulsion between each bit of the first cloud and each bit of the second.
For two point charges at $a$ and $b$ in space, it is $1/\abs{a - b}$, which grows as the charges approach each other.