### Coenergy

#### Posted by John Baez

It’s Newton’s birthday! Google has a nice homepage today, which shows an apple dropping…

In honor of Sir Isaac, I thought I’d ask a question that’s related to classical mechanics: *what’s coenergy?*

I’m doing a lot of reading on electrical circuits these days — and most recently, the Hamiltonian and Lagrangian approaches to electrical circuits. And the word ‘coenergy’ keeps cropping up, but I’ve never seen it defined.

For example:

The function $\hat{E} (\dot{q})$ is the sum of the electric coenergy of the capacitors in the tree $\Gamma_1$ and the magnetic coenergy of the inductors in the cotree $\Lambda_2$…

from *Dissipative Systems: Analysis and Control* by Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland. Or:

CO-ENERGY (AGAIN)

In the linear case, energy and co-energy are numerically equal — the value of distinguishing between them may not be obvious.

Why bother with co-energy at all?

from *Coenergy: Again*, which seems to be notes from a course at MIT. Some other MIT course notes are more illuminating, but still awfully terse:

The constitutive equation for kinetic energy storage (inertia) is:

$p = M v$

$M$

diagonalmatrix of inertial parameters (masses, moments of inertia, e.g. about mass centers)Kinetic co-energy is the dual of kinetic energy:

$E_k^* = \int p^t d v = \frac{1}{2} v^t M v = E_k^*(v)$

Thus, by definition:

$p = \frac{\partial E_k^*}{\partial v}$

These are from Neville Hogan’s notes on *Inertial Mechanics*.

Now, I know that we can often think of kinetic energy either as a function of velocity or of momentum. In the first case, it’s a function on the tangent bundle $T Q$ of configuration space $Q$ (some manifold or other). In the second case, it’s a function on the cotangent bundle $T^* Q$. These two formulations are related by the ‘Legendre transform’.

Am I right in guessing from the above equations that ‘kinetic coenergy’ simply means kinetic energy regarded as a function of *velocity*? And maybe more generally ‘coenergy’ is energy regarded as a function on the tangent bundle? That seems perversely backwards to me — I’d say *co*energy should be energy regarded as a function on the *co*tangent bundle. But who am I to say?

And I’m just guessing, anyway. I find it a bit frustrating that people would prefer to write $E_k^* = E_k^*(v)$ rather than say ‘kinetic coenergy is just kinetic energy regarded as a function of velocity’ — if that’s what they actually mean. But I’m used to it.

## Re: Coenergy

Principles of analytical system dynamics by Richard A. Layton tells us that “energy varies with momentum; coenergy varies with flow” and that kinetic coenergy is “the proper form to use in the classical Lagrangian” but kinetic energy is right for the classical Hamiltonian (p. 14).

There’s a nice table on the following page contrasting energy and coenergy of ideal kinetic stores.