Energy

The term energy has an important physical meaning in physics and is an extremely useful concept. A much more abstract mathematical generalization is defined as follows. Let $\Omega$ be a Space with Measure $\mu\geq 0$ and let $\Phi(P,Q)$ be a real function on the Product Space $\Omega\times\Omega$. When

$$(\mu,nu) = \int\!\!\!\int \Phi(P,Q)\,d\mu(Q)\,d\nu(P)$$

$$= \int \Phi(P,\mu)\,d\nu(P)$$

exists for measures $\mu,\nu\geq 0$, $(\mu,\nu)$ is called the Mutual Energy and $(\mu,\mu)$ is called the Energy.

See also Dirichlet Energy, Mutual Energy

References

Iyanaga, S. and Kawada, Y. (Eds.). ``General Potential.'' §335.B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1038, 1980.