The n-Category Café

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:

$$p = M v$$

$M$ diagonal matrix 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 coenergy should be energy regarded as a function on the cotangent 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.