Hamiltonian

Reexpression of Newtonian (and Lagrangian) mechanics that takes positions and momenta and gives you energy. Characterises A Hamiltonian system is usefully well-behaved in a variety of ways - eigenvalues all have mirrors, the system is integrable, independence from positions gives you invariance in moments (this last seems useful as hell when thinking about curried expressions / logic but I haven't figured out how yet. SICM has a whole section on Hamiltonians in classical mechanics. They are also applicable in optimal control theory, as developed by Lev Pontryagin as part of Pontryagin's maximum principle. Source:

The Hamiltonian in control theory seems like a related but wholly distinct beast, similarly derivable from the (related but distinct) control theory Lagrangian. Here, the configuration space is not subdivided into position and derivative (or equivalently into manifold and tangent bundle) but into state and control variables. Can control variables be treated as a tangent bundle to state? The analogy seems compelling but I think there are important differences – specifically, I want to say state is dependent on prior state and also on control variables (meaning I can define a rate of change but it has to flow from both of these things).

So what is a dynamicla system really? Here's probably where I pause and ask Anish about it.

In November 2025 the reason I care about this is because I want to understand control systems so I can analogize them better with game systems, following on the work of Jeff Shamma.