Hamiltonian Mechanics Applied to Markets

This note applies the formal structure of Hamiltonian systems in contact geometry to model market dynamics, where the action represents accumulated money.

The Market Phase Space

In Hamiltonian mechanics, we work on a phase space $(q, p)$ where:

$q$ : generalized coordinates
$p$ : generalized momentum conjugate to $q$
Action: $S = \int p d q - H d t$

For markets, we reinterpret this as:

Hamiltonian Mechanics	Market Interpretation
$q^{a}$	Inventory level (kg, units of goods)
$p_{a}$	Shadow price or marginal value (€/unit) — what one additional unit contributes to the system
$S = \int (p_{a} d q^{a} d t)$	Total accumulated value (opportunity cost and holding cost) through evolution

The Conjugate Pair: Value and Quantity

The pairing $p \cdot d q$ represents the change in total value:

Value change = \int p_{a} d q^{a}

Here, $p$ is not the literal market price you pay, but the marginal value or opportunity cost: how much the system's total worth changes with each additional unit of $q$ . Under a change of units (covector transformation), both $p$ and $q$ scale such that their product (total value contribution) remains invariant — analogous to the shop example's price-quantity duality, but here $p$ is the system's internal valuation, not the store's asking price.

Global Inventory Case

At the scale of entire human economic systems:

$Q (t) = (Q_{food}, Q_{energy}, Q_{capital}, \dots)$ : global aggregate inventory by sector
$P_{a} (t)$ : global shadow price for each good—the marginal contribution of one more unit to humanity's total welfare or constraint satisfaction

The global Hamiltonian would encode

H_{global} (Q, P, t) = production constraints + environmental costs + scarcity rents