Hamilton's Canonical Equations

Equivalent forms

\dot{q} = \frac{p}{m}, \quad \dot{p} = -kq \;\text{(SHO)}

\dot{z} = \{z, H\}

One asymmetric minus sign generates all of conservative dynamics — and makes phase-space flow incompressible.

Symbol	Name	SI unit	Dimension	Range
$T_period$	Orbit periodoutput Time for one loop of the phase-space ellipse	s	T	0 – 20
$m$	Mass Oscillator mass	kg	M	0.1 – 5
$k$	Spring constant Stiffness of ½kq²	N/m	MT^-2	1 – 50
$A$	Initial displacement Starting position q(0) with p(0) = 0	m	L	0.1 – 2

Unit systems

Symbol	SI	CGS	Imperial
$T_period$	s	s	s
$m$	kg	g	lb
$k$	N/m	dyn/cm	lbf/ft
$A$	m	cm	ft

Where it holds

Equivalent to the Euler–Lagrange equations wherever the Legendre transform is invertible; the natural language for chaos theory, perturbation theory, and canonical quantization.

Dimensional analysis

q̇

\to [LT^{-1}]

; ṗ

\to [MLT^{-2}]

\partial H/\partial p \to [ML^{2}T^{-2}]/[MLT^{-1}] = [LT^{-1}]

;

\partial H/\partial q \to [ML^{2}T^{-2}]/[L] = [MLT^{-2}]

Velocity and force — both match.

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Discovery

William Rowan Hamilton · 1835

In 'Second Essay on a General Method in Dynamics' (1835), Hamilton split Lagrange's second-order equations into twin first-order ones. The symmetric, almost-but-not-quite mirrored pair (note the minus sign!) revealed phase space's hidden symplectic geometry — the stage on which chaos theory and quantum mechanics would later play.

Try this

Replace one second-order ODE with two first-order ones and motion becomes an incompressible river flowing through phase space.

For H = p²/2m + ½kq², follow the flow q̇ = p/m, ṗ = −kq around its phase-space ellipse.

Research status: stable

Real-world applications

Weather and climate models (geophysical fluid dynamics is Hamiltonian at its core)
Particle accelerator lattice design
Molecular dynamics with symplectic integrators
Optimal control (Pontryagin's maximum principle mirrors the canonical equations)

Common misconceptions

The two equations are symmetric — the minus sign breaks the symmetry and encodes all the physics
They're just E–L rewritten with no benefit — first-order form unlocks phase portraits, Liouville's theorem, and chaos analysis
q and p are independent physical things — they're coupled coordinates of a single state point

Experimental verification

Symplectic integrators built directly on these equations track the solar system for billions of simulated years with bounded energy error — a stringent structural test; accelerator beams follow the predicted phase-space flow turn after turn.

Derivation

Take dH of

H = \Sigma p_iq

̇_

i - L

:

dH = \Sigma (q

̇_i dp_i + p_i dq̇_

i) - \Sigma (\partial L/\partial q_i)dq_i - \Sigma (\partial L/\partial q

̇_i)dq̇_

i - (\partial L/\partial t)dt

.

The dq̇ terms cancel by definition of p.

Comparing with

dH = \Sigma (\partial H/\partial q_i)dq_i + \Sigma (\partial H/\partial p_i)dp_i + (\partial H/\partial t)dt

, and using ṗ_

i = \partial L/\partial q_i (E

–L), read off the canonical pair.

Limiting cases

\partial H/\partial q = 0

(free particle)⟶ p constant, q grows linearlyStraight horizontal flow lines in phase space.

H independent of t⟶ Flow stays on one contour of HEnergy surfaces are invariant — conservation as geometry.

Any Hamiltonian flow⟶

\nabla \cdot (q

̇, ṗ

) = 0

\partial ^{2}H/\partial q\partial p - \partial ^{2}H/\partial p\partial q = 0

: phase volume is incompressible (Liouville).

What if…

What if I run the flow backwards?

$t \to -t$ with $p \to -p$ retraces every orbit — Hamiltonian dynamics is time-reversible.

What if I track a blob of initial conditions instead of a point?

Its shape distorts but its area never changes — Liouville's theorem.

What if H depends on time?

The flow pattern itself shifts each instant and energy is pumped in or out — but the equations still hold.

1

SHO phase flow

Given ·

m:: 1
k:: 10
A:: 1

Find · q(t), p(t) and the period

Steps

q̇ $= \partial H/\partial p = p/m$ and ṗ $= -\partial H/\partial q = -kq$
Differentiate the first and substitute: q̈ $= -(k/m)q$
$q = A \cos (\sqrt{k/m} t)$ ; $T = 2\pi \sqrt{m/k} = 2\pi \sqrt{1/10} \approx 1.99\,\mathrm{s}$

Result ·

q = A \cos (\omega t)

,

p = -m\omega A \sin (\omega t)

,

\omega = \sqrt{10}

;

T = 2\pi /\omega \approx 1.99\,\mathrm{s}

2

Free fall in phase space

Given ·

m:: 1
k:: 0

Find · Flow for

H = p^{2}/2m + mgz

Steps

$\partial H/\partial p = p/m$ gives ż
$-\partial H/\partial z = -mg$ gives ṗ $= -mg$ (constant force)
p falls linearly, z traces a parabola — projectile motion, first-order style

Result · ż

= p/m

, ṗ

= -mg

: parabolic flow lines

The Hamiltonian (Legendre Transform)→Poisson Bracket & Liouville Flow→Euler–Lagrange Equation→Simple Harmonic Motion Period→Double Pendulum (Lagrangian Chaos)→