papers/hanzo-hmm/sections/18-hamiltonian-derivation-details.tex

\section{Hamiltonian Derivation Details}

\label{app:derivation}

Starting from the Lagrangian formulation, the market Lagrangian is:
\begin{equation}
\mathcal{L}(\bm{q}, \dot{\bm{q}}) = \frac{1}{2}\sum_i \mu_i \dot{q}_i^2 - V(\bm{q}),
\end{equation}
where $V(\bm{q}) = \sum_i w_i \kappa / q_i + (\lambda/2)\sum_i w_i(q_i - \bar{q}_i)^2/\bar{q}_i^2$ is the potential energy. The conjugate momentum is $p_i = \partial\mathcal{L}/\partial\dot{q}_i = \mu_i\dot{q}_i$, and the Legendre transform yields:
\begin{equation}
\mathcal{H} = \sum_i p_i\dot{q}_i - \mathcal{L} = \sum_i \frac{p_i^2}{2\mu_i} + V(\bm{q}).
\end{equation}

In the limit $\mu_i \to 0$ (zero inertia), we recover the basic HMM Hamiltonian (\ref{eq:hamiltonian}) after the identification $p_i \to w_i q_i p_i^{\text{credit}} / \kappa$.