The First-Order System
Newton's equations are 2nd-order. Hamilton's Equations break them into two 1st-order equations:
\[\dot{q} = \frac{\partial H}{\partial p}\]
\[\dot{p} = -\frac{\partial H}{\partial q}\]
These equations describe how position and momentum "dance" together. Notice the perfect symmetry, except for a single minus sign.
Worked Examples
Example 1: The Oscillator
\(H = \frac{p^2}{2m} + \frac{1}{2}kx^2\).
- \(\dot{x} = \frac{\partial H}{\partial p} = p/m\).
- \(\dot{p} = -\frac{\partial H}{\partial x} = -kx\).
- Combining these gives \(m\ddot{x} = -kx\), which is the familiar spring equation.
The Bridge to Quantum Mechanics
Hamilton's equations are the classical limit of the Heisenberg Equations of Motion. In the "Heisenberg Picture" of Quantum Mechanics, we treat the operators \(\hat{x}\) and \(\hat{p}\) as things that change over time, and their derivatives are given by their "commutators" with the Hamiltonian. The structure is identical to Hamilton's equations. This symmetry proves that the laws of quantum change are just a more refined version of the laws of classical change.