Hamilton's Equations

Canonical evolution of coordinates and conjugate momenta. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Hamilton's equations describe the evolution of a system using canonical coordinates and conjugate momenta. After the Hamiltonian $H$ has been built, the state is no longer read as position and velocity, but as a point in phase space formed by $q$ and $p$. The physical question changes: one does not only seek a trajectory, but the field that indicates where each phase point moves.

This leaf naturally follows the Legendre transformation. There, the passage from velocities to momenta was justified; here, the result is used to generate motion. The Hamiltonian is not only a quantity to compute: its slopes produce the rates of change of coordinate and momentum.

🟢 Essential level

The essential idea is that the state moves in a phase plane. One component of motion says how the coordinate changes; the other says how momentum changes. These two components are not invented separately: they come from slopes of the Hamiltonian taken in different directions.

The slope in the momentum direction governs coordinate advance. The slope in the coordinate direction governs momentum change, but with opposite sign. This sign asymmetry is the mark of canonical structure: the equations are not two identical derivatives, but a rotation of the energy gradient that produces phase flow.

At the essential level, it is useful to imagine the Hamiltonian as a landscape and the state as a point that does not simply slide downhill. Canonical structure rotates that information and turns it into motion around level curves. This image explains why continuous evolution can occur even while the energy value remains fixed.

A practical way to consolidate Hamilton's Equations is to separate the physical system, the independent variables, and the mathematical representation before calculating. Canonical evolution of coordinates and conjugate momenta. The student should be able to explain what information the analytical description preserves, what it replaces from the Newtonian treatment, and which observable consequence would reveal an inconsistent choice.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

The first Hamilton equation defines the canonical velocity of the coordinate:

First Hamilton equation

Real student questions

A common question is why the second equation has a minus sign. That sign is not decorative convention; it makes the flow canonical and, in conservative models, makes motion proceed along level curves of $H$.

Another doubt is whether constant $H$ means nothing changes. The answer is no. A point can move continuously along a level curve while keeping the same Hamiltonian value. This difference is essential for understanding oscillations, orbits, and stability.

Final synthesis

Hamilton's equations turn the Hamiltonian into an evolution field for coordinate and momentum. The slope in $p$ produces $\dot{q}$, the slope in $q$ produces $\dot{p}$ with opposite sign, and explicit time dependence decides whether $H$ is conserved. The final goal is to read dynamics as flow in phase space.