Hamilton's equations

Intuition [Beginner]

You already know Newton's $F=ma$. It says: pick a force law, get a second-order differential equation for position, solve, predict the motion. The Lagrangian reformulation rewrites this as a single principle — paths that nature picks minimise (or extremise) the action, a number you compute by adding up "kinetic minus potential" along a path — and recovers the same equations. Hamilton's equations are the third move in this sequence, and they change which variables you track.

Instead of describing a particle by its position $q$ and velocity $\dot{q}$, describe it by its position $q$ and its momentum $p=m\dot{q}$. The two pieces of data live on equal footing — neither one is derived from the other. The space whose points are pairs $(q,p)$ is called phase space, and a state of the system is a single point in phase space.

The total energy, written as a function of $q$ and $p$ rather than $q$ and $\dot{q}$, is the Hamiltonian $H(q,p)$. For a particle in a potential,

$$H(q,p)=\frac{p^2}{2m}+V(q),$$

which you can read off as kinetic-plus-potential, just with $\frac{1}{2}m{\dot{q}}^2$ rewritten as $p^2/(2m)$ using $p=m\dot{q}$.

Hamilton's equations are then a surprisingly symmetric pair. In words: the rate of change of position equals "the slope of $H$ in the momentum direction", and the rate of change of momentum equals "minus the slope of $H$ in the position direction". Two first-order equations replace Newton's one second-order equation. The minus sign is load-bearing: it's what makes the flow conserve energy and what makes phase space behave like an incompressible fluid rather than a draining sink.

The formal statement, in symbols, lives in the Intermediate-tier Formal definition section. The two slopes-of-$H$ rules above are all you need at the Beginner tier.

Why bother? Three reasons that compound as you go deeper.

First, conservation laws fall out automatically. If $H$ does not depend on $q$, then $\dot{p}=0$ — momentum is conserved. If $H$ does not depend on time, then $H$ itself is conserved — that's energy conservation. Symmetries of the Hamiltonian and conserved quantities are paired one-to-one.

Second, the framework is the right shape for quantum mechanics. The Schrödinger equation is Hamilton's equations with operators in place of $q$ and $p$ and a commutator in place of the Poisson bracket. Every quantum textbook starts by promoting some classical Hamiltonian to an operator. You cannot build that bridge from Newton's $F=ma$ directly.

Third, the geometry behind the symmetric pair is symplectic geometry, which turns out to be the natural geometric structure for any conservative mechanical system, classical or quantum. This is the perspective developed in the Master tier.

Visual [Beginner]

Picture the simplest possible mechanical system: a mass on a spring, with Hamiltonian $H(q,p)=\frac{1}{2}{p}^2+\frac{1}{2}{q}^2$ (units chosen so the spring constant and mass equal 1). Phase space is the $(q,p)$-plane. Level sets of $H$ are circles centred at the origin — one circle per energy.

Hamilton's equations, read off the picture, say: the motion in phase space is a rigid rotation. A point starting at $(1,0)$ — the spring fully stretched, at rest — moves clockwise: down to $(0,-1)$ at quarter-period, where the mass passes through the origin with maximum (negative) momentum; then to $(-1,0)$, fully compressed; then $(0,1)$, passing the origin again with positive momentum; then back to $(1,0)$.

The motion follows the level set of $H$. It does not cross level sets. Energy is conserved automatically, not as an extra rule you have to remember.

Now scale this picture up: the Hamiltonian is any function $H(q,p)$, the level sets are any nested family of curves (or surfaces in higher dimension), and the flow always tracks the level sets — never crossing them. The minus sign that appears between the two equations is exactly what makes "flow along level sets of $H$, not toward low values of $H$" come out right.

Worked example [Beginner]

Take the same harmonic oscillator, $H(q,p)=\frac{1}{2}{p}^2+\frac{1}{2}{q}^2$, and start the mass at $q(0)=1$, $p(0)=0$ — fully stretched, at rest.

Step 1. Read off the two slopes of $H$. Treating $q$ as a constant and varying $p$, the slope of $H$ in the $p$-direction is $p$. Treating $p$ as a constant and varying $q$, the slope of $H$ in the $q$-direction is $q$.

Step 2. Write Hamilton's equations (rate of $q$ = first slope; rate of $p$ = minus second slope):

$$\dot{q}=p,\dot{p}=-q.$$

Step 3. Differentiate the first equation in time and substitute the second:

$$\ddot{q}=\dot{p}=-q.$$

This is the familiar second-order equation $\ddot{q}+q=0$, with general solution $q(t)=A\cos t+B\sin t$. The initial conditions $q(0)=1$ and $\dot{q}(0)=p(0)=0$ pin $A=1$, $B=0$, so

$$q(t)=\cos t,p(t)=\dot{q}(t)=-\sin t.$$

Step 4. Check energy conservation:

$$H(q(t),p(t))=\frac{1}{2}{\sin }^{2}t+\frac{1}{2}{\cos }^{2}t=\frac{1}{2}.$$

Constant in time, as required.

What this tells us: Hamilton's equations recover the familiar oscillator motion, but they get it as a flow in the plane of $(q,p)$ pairs — rotating clockwise around the origin with unit angular speed. The level set $H=\frac{1}{2}$ is the unit circle, and the motion stays on it forever.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $Q$ be the configuration manifold of a mechanical system — the space whose points are the system's spatial configurations. For a single particle in ${\mathbb{R}}^3$, $Q={\mathbb{R}}^3$; for a rigid body, $Q=\mathrm{SO}(3)\times {\mathbb{R}}^3$; for $N$ particles, $Q={\mathbb{R}}^{3N}$ modulo constraints. A point in $Q$ is denoted $q$; a smooth curve $t\mapsto q(t)$ has tangent vector $\dot{q}(t)\in T_{q(t)}Q$.

A Lagrangian system 05.00.01 is governed by a smooth function $L:TQ\times \mathbb{R}\to \mathbb{R}$ (the Lagrangian), and the equations of motion are the Euler-Lagrange equations

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}^i}-\frac{\partial L}{\partial q^i}=0.$$

The Hamiltonian formulation replaces the tangent bundle $TQ$ by the cotangent bundle $T^{\ast }Q$ via the Legendre transform 05.00.03. The conjugate momentum of $q^i$ is

$$p_i:=\frac{\partial L}{\partial {\dot{q}}^i}.$$

Assuming the Hessian condition $\det ({\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j)\ne 0$ (the Lagrangian is regular; if this holds globally we call $L$ hyper-regular), the fibre Legendre transform $\mathbb{F}L:TQ\to T^{\ast }Q$, $(q,\dot{q})\mapsto (q,p)$, is a local diffeomorphism. The Hamiltonian is then the Legendre transform of $L$:

$$H(q,p,t):=(p_i{\dot{q}}^i-L(q,\dot{q},t)){|}_{\dot{q}=\dot{q}(q,p,t)},$$

where $\dot{q}(q,p,t)$ is obtained by solving $p_i=\partial L/\partial {\dot{q}}^i$ for $\dot{q}$. We adopt the summation convention throughout: repeated upper-lower index pairs are summed over $i=1,\dots ,n$ where $n=\dim Q$.

Hamilton's equations are the first-order system

$$\boxed{{\dot{q}}^i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q^i},\frac{\partial L}{\partial t}=-\frac{\partial H}{\partial t}}$$

The third equation governs explicit time dependence; for time-independent Lagrangians it is vacuous.

In coordinates the system has $2n$ first-order ODEs in place of $n$ second-order Euler-Lagrange equations. The space $T^{\ast }Q$ with coordinates $(q^i,p_i)$ is phase space; a state of the system is a point in $T^{\ast }Q$.

The Hamiltonian formulation requires fixing the sign convention $p_i=+\partial L/\partial {\dot{q}}^i$ and the sign convention in the Legendre transform $H=p\dot{q}-L$. Other conventions (replacing $H$ by $-H$ or $L$ by $-L$, or relabelling $q$ and $p$ symmetrically) appear in the literature and exchange the signs in Hamilton's equations. The convention adopted here is the one used by Arnold, Marsden-Ratiu, Taylor, and Tong; Landau-Lifshitz and Goldstein use the same convention. State the convention before computing.

The Poisson bracket 05.02.02 of two smooth functions $f,g:T^{\ast }Q\to \mathbb{R}$ is

$$\{f,g\}:=\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q^i}.$$

It is bilinear, antisymmetric, satisfies the Leibniz rule $\{f,gh\}=\{f,g\}h+g\{f,h\}$, and satisfies the Jacobi identity $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$. The pair $(T^{\ast }Q,\{\cdot ,\cdot \})$ is a Poisson manifold; the bracket is the algebraic shadow of the canonical symplectic form $\omega =d{p}_i\wedge d{q}^i$ developed in the Master tier.

The evolution of an observable $f(q,p,t)$ along a Hamiltonian trajectory is

$$\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}.$$

Hamilton's equations themselves are the special case $f=q^i$ (${\dot{q}}^i=\{q^i,H\}$) and $f=p_i$ (${\dot{p}}_i=\{p_i,H\}$), using the canonical Poisson relations

$$\{q^i,q^j\}=0,\{p_i,p_j\}=0,\{q^i,p_j\}={\delta }_j^i.$$

A function $f$ with $\{f,H\}=0$ and $\partial f/\partial t=0$ is a conserved quantity (a constant of motion); its value is preserved along every trajectory.

Counterexamples to common slips

• A non-hyper-regular Lagrangian — one for which the Hessian ${\partial }^{2}L/\partial {\dot{q}}^i\partial {\dot{q}}^j$ is singular at some $(q,\dot{q})$ — does not admit a well-defined Hamiltonian via the recipe above. Gauge theories and the relativistic point particle in their natural Lagrangian forms have this problem and are handled by the Dirac-Bergmann constraint algorithm, not by the textbook Legendre transform.
• The momentum $p_i=\partial L/\partial {\dot{q}}^i$ is not in general equal to $m_i{\dot{q}}^i$. For a particle in an electromagnetic field with $L=\frac{1}{2}m{\dot{\mathbf{q}}}^2+e\mathbf{A}(\mathbf{q})\cdot \dot{\mathbf{q}}-eV(\mathbf{q})$, the canonical momentum is $\mathbf{p}=m\dot{\mathbf{q}}+e\mathbf{A}$ — the kinetic momentum $m\dot{\mathbf{q}}$ and the canonical (conjugate) momentum $\mathbf{p}$ differ by $e\mathbf{A}$. This is what makes gauge transformations of $\mathbf{A}$ a genuine transformation of phase space rather than a relabelling.
• The Hamiltonian is not in general equal to the total energy. The two coincide when $L=T-V$ with $T$ a homogeneous quadratic in $\dot{q}$ and $V$ time-independent. In rotating frames or for explicitly-time-dependent constraints $H\ne T+V$, even though Hamilton's equations still apply.
• Hamilton's equations are first-order; the apparent doubling of equation count from $n$ to $2n$ is balanced by the halving of order from $2$ to $1$. The total dimension of the initial-value problem — $2n$ scalars — is the same as in the Lagrangian formulation.

Key theorem with proof [Intermediate+]

Theorem (Equivalence of Hamilton's equations and the Euler-Lagrange equations under hyper-regularity). *Let $L:TQ\times \mathbb{R}\to \mathbb{R}$ be a hyper-regular Lagrangian, $\mathbb{F}L:TQ\to T^{\ast }Q$ the fibre Legendre transform, and $H:T^{\ast }Q\times \mathbb{R}\to \mathbb{R}$ the associated Hamiltonian. Then a smooth curve $\gamma (t)=(q(t),\dot{q}(t))$ in $TQ$ satisfies the Euler-Lagrange equations for $L$ if and only if the curve $\sigma (t):=\mathbb{F}L(\gamma (t))=(q(t),p(t))$ in $T^{\ast }Q$ satisfies Hamilton's equations for $H$.*

Proof. The Hamiltonian is defined by $H(q,p,t)=p_i{\dot{q}}^i-L(q,\dot{q},t)$ where $\dot{q}=\dot{q}(q,p,t)$ is determined by inverting $p_i=\partial L/\partial {\dot{q}}^i$.

Differentiate $H$ with respect to $p_i$, treating $\dot{q}$ as a function of $(q,p,t)$:

$$\frac{\partial H}{\partial p_i}={\dot{q}}^i+p_j\frac{\partial {\dot{q}}^j}{\partial p_i}-\frac{\partial L}{\partial {\dot{q}}^j}\frac{\partial {\dot{q}}^j}{\partial p_i}.$$

The last two terms cancel because $p_j=\partial L/\partial {\dot{q}}^j$, leaving

$$\frac{\partial H}{\partial p_i}={\dot{q}}^i.\text{\hspace{1em}(∗)}$$

Differentiate $H$ with respect to $q^i$:

$$\frac{\partial H}{\partial q^i}=p_j\frac{\partial {\dot{q}}^j}{\partial q^i}-\frac{\partial L}{\partial q^i}-\frac{\partial L}{\partial {\dot{q}}^j}\frac{\partial {\dot{q}}^j}{\partial q^i}=-\frac{\partial L}{\partial q^i},\text{\hspace{1em}(∗∗)}$$

again using $p_j=\partial L/\partial {\dot{q}}^j$ to cancel two terms. Differentiating with respect to $t$ gives, similarly, $\partial H/\partial t=-\partial L/\partial t$.

Now suppose $\gamma$ satisfies the Euler-Lagrange equations $\frac{d}{dt}(\partial L/\partial {\dot{q}}^i)=\partial L/\partial q^i$. Along $\sigma =\mathbb{F}L\circ \gamma$ the conjugate momentum is $p_i(t)=\partial L/\partial {\dot{q}}^i{|}_{\gamma (t)}$, so

$${\dot{p}}_i(t)=\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}^i}=\frac{\partial L}{\partial q^i}\stackrel{(\ast \ast )}{=}-\frac{\partial H}{\partial q^i},$$

which is the second Hamilton equation. Equation ($\ast$) gives ${\dot{q}}^i=\partial H/\partial p_i$ identically along $\sigma$ — this is the first Hamilton equation, true by definition of $\dot{q}(q,p,t)$ in terms of $(q,p)$. So $\sigma$ satisfies Hamilton's equations.

For the converse, suppose $\sigma (t)=(q(t),p(t))$ satisfies Hamilton's equations. Equation ($\ast$) along $\sigma$ gives ${\dot{q}}^i=\partial H/\partial p_i$, consistent with the first Hamilton equation. Differentiate $p_i(t)=\partial L/\partial {\dot{q}}^i$ in time:

$${\dot{p}}_i=\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}^i}\stackrel{\text{Hamilton}}{=}-\frac{\partial H}{\partial q^i}\stackrel{(\ast \ast )}{=}\frac{\partial L}{\partial q^i},$$

so the Euler-Lagrange equations hold along $\gamma =\mathbb{F}{L}^{-1}\circ \sigma$. ∎

Corollary. *Under the same hypotheses, the action $S[\gamma ]=\int Ldt$ on paths in $Q$ equals the action $\tilde{S}[\sigma ]=\int (p_i{\dot{q}}^i-H)dt$ on paths in $T^{\ast }Q$, up to boundary terms, and the variational principle $\delta \tilde{S}=0$ with $q$ fixed at endpoints (free variations of $p$) is equivalent to Hamilton's equations.* The substitution $H=p_i{\dot{q}}^i-L$ is exactly Legendre conjugation; the variational principle on $T^{\ast }Q$ is the modified Hamilton principle (Hamilton-Pontryagin, or Cartan's variational principle in the symplectic case).

The role of hyper-regularity is to make the Legendre transform globally invertible. For regular Lagrangians (Hessian invertible locally, not globally), the equivalence is local; the global picture requires either restricting to a region or augmenting with the inverse Legendre transform on each chart.

Worked example: the Kepler problem at intermediate level

A particle of mass $m$ in the gravitational potential $V(r)=-k/r$ where $k=GMm>0$ and $r=|\mathbf{q}|$ has Lagrangian $L=\frac{1}{2}m|\dot{\mathbf{q}}{|}^2+k/r$. In spherical coordinates $(r,\theta ,\phi )$, $L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2+r^2{\sin }^2\theta {\dot{\phi }}^2)+k/r$.

The conjugate momenta: $$ p_r = m\dot r, \quad p_\theta = mr^2 \dot\theta, \quad p_\varphi = mr^2 \sin^2\theta ,\dot\varphi. $$

Inverting and substituting into $H=p_i{\dot{q}}^i-L$:

$$H(r,\theta ,\phi ,p_r,p_{\theta },p_{\phi })=\frac{p_r^2}{2m}+\frac{p_{\theta }^2}{2m{r}^2}+\frac{p_{\phi }^2}{2m{r}^2{\sin }^2\theta }-\frac{k}{r}.$$

Two cyclic coordinates: $\phi$ does not appear in $H$, so ${\dot{p}}_{\phi }=0$ and $p_{\phi }$ is conserved. Restrict to the equatorial plane $\theta =\pi /2$ ($p_{\theta }=0$) to reduce to the planar problem. The remaining Hamiltonian is $H=p_r^2/(2m)+p_{\phi }^2/(2m{r}^2)-k/r$ on a two-dimensional phase space $(r,p_r)$ at fixed $\ell :=p_{\phi }$, with the centrifugal ${\ell }^2/(2m{r}^2)$ now in the effective potential — the textbook reduction Hamilton's equations make automatic.

Exercises [Intermediate+]

Geometric formulation: Hamilton's equations on a symplectic manifold [Master]

The intermediate-tier statement of Hamilton's equations privileges Darboux coordinates $(q^i,p_i)$; the underlying object is coordinate-free. Let $Q$ be a smooth manifold of dimension $n$ and let $T^{\ast }Q$ be its cotangent bundle, with projection $\pi :T^{\ast }Q\to Q$.

There is a canonical 1-form on $T^{\ast }Q$, the Liouville (tautological) 1-form $\theta \in {\Omega }^1(T^{\ast }Q)$, defined intrinsically by

$${\theta }_{\alpha }(v):=\alpha (d{\pi }_{\alpha }(v))\text{for }\alpha \in T^{\ast }Q,v\in T_{\alpha }(T^{\ast }Q).$$

In Darboux coordinates $(q^i,p_i)$ on $T^{\ast }Q$ where $p_i$ are the fibre coordinates dual to $d{q}^i$ on the base, $\theta =p_{i}d{q}^i$. The canonical symplectic 2-form is

$$\omega :=-d\theta =d{q}^i\wedge d{p}_i,$$

a closed ($d\omega =0$) and non-degenerate 2-form on $T^{\ast }Q$ 05.02.05. Note the sign convention is fixed: $\omega =-d\theta$ with $\theta =p_{i}d{q}^i$ gives $\omega =d{q}^i\wedge d{p}_i$; the opposite sign convention $\omega =d\theta =d{p}_i\wedge d{q}^i$ appears in some sources (Marsden-Ratiu; we follow the same convention here, having earlier written $\omega =d{p}_i\wedge d{q}^i$ — both differ from Arnold's by an overall sign, exchanging $X_H$ with $-X_H$ but leaving Hamilton's equations as stated invariant under the simultaneous swap of conventions). Within a single document, fix the convention and stay with it. Throughout this unit, $\omega =d{q}^i\wedge d{p}_i$ and Hamilton's equations are as stated above.

Darboux's theorem 05.01.04 guarantees that any 2$n$-dimensional symplectic manifold $(M,\omega )$ admits, around any point, local coordinates $(q^1,\dots ,q^n,p_1,\dots ,p_n)$ in which $\omega =d{q}^i\wedge d{p}_i$. So the cotangent-bundle picture is locally the universal example; abstract symplectic manifolds (e.g., $\mathbb{C}{P}^n$ with the Fubini-Study form, or coadjoint orbits) are not cotangent bundles globally, but Hamilton's equations look the same in any Darboux chart.

Hamiltonian vector field 05.02.01. For $H\in C^{\infty }(M)$ on a symplectic manifold $(M,\omega )$, the Hamiltonian vector field $X_H$ is the unique vector field satisfying

$$\boxed{i_{X_H}\omega =dH.}$$

Non-degeneracy of $\omega$ makes the equation $i_X\omega =dH$ uniquely solvable for $X$, so $X_H$ exists and is unique. In Darboux coordinates, writing $X_H=A^i{\partial }_{q^i}+B_i{\partial }_{p_i}$ and computing $i_{X_H}\omega =i_{X_H}(d{q}^i\wedge d{p}_i)=A^{i}d{p}_i-B_{i}d{q}^i$, the equation $i_{X_H}\omega =dH=(\partial H/\partial q^i)d{q}^i+(\partial H/\partial p_i)d{p}_i$ matches coefficients to give $A^i=\partial H/\partial p_i$ and $B_i=-\partial H/\partial q^i$, i.e.,

$$X_H=\frac{\partial H}{\partial p_i}\frac{\partial }{\partial q^i}-\frac{\partial H}{\partial q^i}\frac{\partial }{\partial p_i}.$$

Hamilton's equations are the integral-curve equations of $X_H$: $\dot{\gamma }(t)=X_H(\gamma (t))$.

Symplectic-form preservation. By Cartan's magic formula ${\mathcal{L}}_{X_H}\omega =d{i}_{X_H}\omega +i_{X_H}d\omega =d(dH)+0=0$, using $d^2=0$ and the closedness of $\omega$. So the flow ${\varphi }_t$ of $X_H$ preserves $\omega$:

$${\varphi }_t^{\ast }\omega =\omega \text{for all }t\in \mathbb{R}.$$

A diffeomorphism $\psi :M\to M$ with ${\psi }^{\ast }\omega =\omega$ is a symplectomorphism (also called a canonical transformation). The Hamiltonian flow is therefore a one-parameter family of symplectomorphisms.

Liouville's theorem 05.02.07. The top exterior power ${\omega }^n/n!$ is a volume form on $M^{2n}$ (the Liouville volume). Since $\omega$ is preserved, so is ${\omega }^n/n!$ — phase-space volume is conserved under Hamiltonian flow. This is Liouville's theorem on phase-space volume: incompressibility of the Hamiltonian flow regarded as a fluid on $M$.

The symplectic preservation ${\varphi }_t^{\ast }\omega =\omega$ is strictly stronger than volume preservation: it preserves $\omega$, ${\omega }^2/2!$, …, ${\omega }^n/n!$ — the entire family of symplectic invariants — and additionally fixes the pairing structure on tangent vectors. Symplectic flows are volume-preserving, but volume-preserving flows on $M^{2n}$ for $n\ge 2$ are not in general symplectic; Eliashberg's symplectic-rigidity theorem (1989) is the deep statement that distinguishes them at $C^0$.

Poisson bracket as the symplectic shadow. The Poisson bracket 05.02.02 is constructed from $\omega$ via

$$\{f,g\}:=\omega (X_f,X_g)=X_f(g)=-X_g(f).$$

The Leibniz rule for $\{\cdot ,\cdot \}$ is built into the vector-field interpretation; antisymmetry comes from antisymmetry of $\omega$; the Jacobi identity is equivalent to $d\omega =0$ via the calculation

$$\{f,\{g,h\}\}+\text{cyclic}=-d\omega (X_f,X_g,X_h)=0.$$

So closedness of $\omega$ is exactly the Jacobi identity for $\{\cdot ,\cdot \}$. This is the structural reason classical mechanics has a Lie-algebra-of-observables structure under the Poisson bracket — and the structural reason quantum mechanics, which replaces $\{\cdot ,\cdot \}$ by $(i\hslash {)}^{-1}[\cdot ,\cdot ]$, must have $[\hat{A},\hat{B}]=i\hslash \widehat{\{A,B\}}+O({\hslash }^2)$ at the semiclassical level. The Jacobi identity for commutators is automatic for associative algebras; the corresponding classical statement requires the additional input $d\omega =0$.

Hamilton's equations as a moment map. A more abstract restatement: the time-1 flow of $X_H$ is the unique symplectomorphism whose generating function (in the sense developed in the next unit) is $H$. This perspective routes Hamilton's equations through the variational principle on $T^{\ast }Q$ — the modified Hamilton principle — and is the entry point to canonical transformations as the symplectomorphism group of $(T^{\ast }Q,\omega )$ and Hamilton-Jacobi theory as the analysis of distinguished generating functions.

Canonical transformations and generating functions [Master]

A diffeomorphism $\Phi :T^{\ast }Q\to T^{\ast }Q$ with ${\Phi }^{\ast }\omega =\omega$ is a canonical transformation (synonym: symplectomorphism). Hamiltonian flows are a one-parameter subgroup of the canonical-transformation group; abstract canonical transformations are the discrete cousins.

Hamilton's equations are covariant under canonical transformations in a precise sense: if $\Phi (q,p)=(Q(q,p),P(q,p))$ is canonical, and $K(Q,P):=H({\Phi }^{-1}(Q,P))$ is the pulled-back Hamiltonian, then a trajectory of $X_H$ on the source side is mapped to a trajectory of $X_K$ on the target side. Choosing $\Phi$ wisely — to bring $K$ into a form with as many cyclic coordinates as possible — is the strategic content of Hamilton-Jacobi theory: find a canonical transformation under which $K$ depends only on the new momenta $P$, in which case the new equations of motion reduce to $\dot{P}=0$, $\dot{Q}=\partial K/\partial P=$ constant, and the system is solved by quadrature.

Canonical transformations are classified by generating functions, four standard types corresponding to which pair of $\{q,p,Q,P\}$ is taken as independent variables. The simplest is type-1: $F_1(q,Q,t)$ with $p_i=\partial F_1/\partial q^i$, $P_i=-\partial F_1/\partial Q^i$, $K=H+\partial F_1/\partial t$. The other three types differ by Legendre transforms in $(p,q)$ or $(P,Q)$. The Hamilton-Jacobi equation

$$H(q,\frac{\partial S}{\partial q},t)+\frac{\partial S}{\partial t}=0$$

is the equation for a type-2 generating function $S(q,P,t)$ that achieves the maximum-simplification target ($K\equiv 0$, all new momenta conserved). Its theory and its limit relation to the eikonal equation of geometric optics — and via the WKB approximation to the Schrödinger equation — are the subject of unit 09.05.01 pending.

Integrable systems and the Liouville-Arnold theorem [Master]

A Hamiltonian system on $(M^{2n},\omega )$ is integrable in the Liouville sense if it admits $n$ functionally independent conserved quantities $f_1=H,f_2,\dots ,f_n$ whose pairwise Poisson brackets all vanish ($\{f_i,f_j\}=0$). The conserved quantities are said to be in involution.

Liouville-Arnold theorem. Let $f_1,\dots ,f_n$ be functionally independent conserved quantities in involution on a $2n$-dimensional symplectic manifold $M$, with $f_1=H$. Fix a level set $M_c:=\{x\in M:f_i(x)=c_i\}$ for $c=(c_1,\dots ,c_n)\in {\mathbb{R}}^n$, and assume $M_c$ is compact and connected. Then:

(i) $M_c$ is diffeomorphic to the $n$-torus ${\mathbb{T}}^n$;

(ii) there exist coordinates $({\varphi }^1,\dots ,{\varphi }^n)(\mathrm{mod}2\pi )$ on $M_c$ in which the Hamiltonian flow is linear: ${\dot{\varphi }}^i={\omega }^i(c)$;

(iii) on a neighbourhood of $M_c$, there exist symplectic coordinates $(I_1,\dots ,I_n,{\varphi }^1,\dots ,{\varphi }^n)$ — action-angle coordinates — with $\omega =d{I}_i\wedge d{\varphi }^i$ and $H=H(I)$ depending only on the actions. Hamilton's equations are ${\dot{I}}_i=0$, ${\dot{\varphi }}^i=\partial H/\partial I_i$, integrable explicitly by quadrature.

The Liouville-Arnold theorem is proven by analysing the ${\mathbb{R}}^n$-action of the commuting flows of the $X_{f_i}$ on $M_c$ — the orbit-stabiliser theorem makes $M_c$ a quotient ${\mathbb{R}}^n/\Lambda$ for some lattice $\Lambda \cong {\mathbb{Z}}^n$, and the compactness hypothesis closes the argument 05.09.01. Proof details live in Arnold §49-50 and Abraham-Marsden §5.

Examples of integrable systems where this machinery applies in full: the harmonic oscillator in any dimension; the Kepler problem (Liouville-integrable on each energy surface, superintegrable with an extra conserved quantity — the Laplace-Runge-Lenz vector — that closes orbits); the spherical pendulum on each energy surface generically; the Euler top (free rigid body in ${\mathbb{R}}^3$, three conserved quantities $H,{\mathbf{L}}^2,L_z$); the Lagrange and Kovalevskaya tops; the Toda lattice. Beyond these the situation is generically non-integrable: typical Hamiltonian systems are chaotic, the KAM theorem governs the persistence of integrable tori under small perturbations 05.09.01, and Arnold diffusion describes the slow drift across resonances on long timescales.

Connection to quantum mechanics [Master]

The semiclassical correspondence is realised via canonical quantisation, the map sending classical observables $f(q,p)$ to operators $\hat{f}$ on a Hilbert space, with the Poisson bracket sent to the commutator divided by $i\hslash$:

$$\{f,g\}⟼\frac{1}{i\hslash }[\hat{f},\hat{g}].$$

Hamilton's equation $df/dt=\{f,H\}$ becomes the Heisenberg equation $d\hat{f}/dt=(i\hslash {)}^{-1}[\hat{f},\hat{H}]$. The canonical Poisson relations $\{q^i,p_j\}={\delta }_j^i$ become the Heisenberg commutation relations $[{\hat{q}}^i,{\hat{p}}_j]=i\hslash {\delta }_j^i$. The Hamilton-Jacobi equation becomes the WKB limit of the Schrödinger equation. The Lagrangian-side variational principle becomes the Feynman path-integral. Symplectic geometry is the geometry of the classical limit of quantum mechanics, and the structural reason it appears at all is that the Lie algebra of self-adjoint operators on a Hilbert space is non-abelian and quantum observables form a non-commutative version of the Poisson algebra.

The correspondence is faithful for quadratic observables (the harmonic oscillator quantises exactly), and breaks at higher polynomial orders by operator-ordering ambiguities — Groenewold's theorem (1946) shows no homomorphism extends to all polynomial observables. The corrections to the naive quantisation rule are organised by formal deformation quantisation (Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer 1978) and the Kontsevich formality theorem (1997), which constructs a $\star$-product on any Poisson manifold.

For the QM side proper, see unit 12.02.01 pending (Hilbert space formalism) and 12.10.01 pending (path integrals), and the connections drawn there.

Lean formalization [Intermediate+]

Mathlib does not yet cover Hamiltonian mechanics. The closest layers are:

• Mathlib.LinearAlgebra.SymplecticGroup: skew-symmetric bilinear forms and the standard symplectic group $\mathrm{Sp}(2n,\mathbb{R})$ as a matrix subgroup.
• Mathlib.Geometry.Manifold.Tangent, Mathlib.Geometry.Manifold.Cotangent: smooth tangent and cotangent bundles as smooth bundles.
• Mathlib.Geometry.Manifold.Diffeomorph: smooth maps and diffeomorphisms.

There is no Mathlib definition of "Hamilton's equations on a symplectic manifold", no canonical 2-form on a cotangent bundle as a Mathlib structure, and no Poisson-bracket-from-symplectic-form correspondence. The formalisation pathway is laid out in lean_mathlib_gap in the frontmatter.

lean_status: none reflects this gap; no lean_module ships with this unit. Tyler's review attests intermediate-tier correctness. The aggregated lean_status: none units in the §09 chapter become a Mathlib contribution roadmap as that section grows.

Connections [Master]

• Legendre transform 05.00.03 is the bridge from the Lagrangian to the Hamiltonian formulation. The hyper-regularity condition there is what gives the Hessian-invertibility hypothesis in the Equivalence Theorem above.

• Hamiltonian vector field 05.02.01 is the coordinate-free object that Hamilton's equations describe. The relation $i_{X_H}\omega =dH$ is the defining equation; the coordinate Hamilton's equations are its expression in a Darboux chart.

• Poisson bracket 05.02.02 is the algebraic shadow of the symplectic form. The evolution equation $df/dt=\{f,H\}$ packages Hamilton's equations and the chain rule into one line.

• Cotangent bundle as canonical symplectic manifold 05.02.05 is where the symplectic form $\omega =d{q}^i\wedge d{p}_i$ comes from. Any configuration manifold's cotangent bundle is automatically symplectic; this is why Hamiltonian mechanics is the natural formulation on a manifold rather than only on ${\mathbb{R}}^{2n}$.

• Darboux theorem 05.01.04 says every symplectic manifold looks locally like ${\mathbb{R}}^{2n}$ with $\omega =d{q}^i\wedge d{p}_i$. This is what makes the coordinate expression of Hamilton's equations universal.

• Liouville volume and Liouville's theorem 05.02.07 is the immediate consequence of ${\varphi }_t^{\ast }\omega =\omega$: phase-space volume is preserved. Statistical-mechanics derivations of the microcanonical ensemble take this conservation law as a primitive.

• Noether's theorem 05.00.04 in its Hamiltonian form: continuous symmetries of $H$ are in one-to-one correspondence with conserved quantities, packaged via the moment map for the symmetry group's action on $T^{\ast }Q$.

• Hamilton-Jacobi equation 09.05.01 pending (pending) is the canonical-transformation strategy taken to its limit; the WKB approximation of the Schrödinger equation reduces to it in the classical limit.

• KAM theorem 05.09.01 describes what survives of Hamilton's-equation integrability when an integrable Hamiltonian is perturbed. The starting point is always a Hamiltonian flow on a symplectic manifold; the conclusion is about a specific class of invariant tori for the perturbed flow.

• Geometric mechanics 09.09.01 pending (pending) generalises Hamilton's equations to Poisson manifolds, coadjoint orbits, and the Lie-Poisson framework on duals of Lie algebras — where infinite-dimensional examples (ideal fluids via Arnold 1966, plasma physics, BBGKY hierarchies) live.

• Canonical quantisation and the path integral [12.02.01, 12.10.01] (pending) take this unit's Poisson algebra and Hamiltonian as the classical limit of the quantum-mechanical operator algebra and Hamiltonian.

Historical & philosophical context [Master]

Hamilton introduced his equations in two papers in Philosophical Transactions of the Royal Society, 1834 and 1835 ^{[Hamilton 1834]}, building on his earlier work on geometrical optics. The 1834 paper, On a general method in dynamics, derived the equations as a recasting of Lagrange's Mécanique analytique (1788) in which the action $S$, regarded as a function of endpoint and time, satisfies the partial differential equation now bearing his name. Jacobi extended the framework in lectures published posthumously in 1866 ^{[Jacobi 1866]}, emphasising the canonical-transformation viewpoint and the integration of mechanics by separation of variables in the Hamilton-Jacobi equation. The 19th century closed with the framework fully developed analytically; the 20th opened with Poincaré's Méthodes nouvelles de la mécanique céleste (1892-99) recasting the question as one of qualitative dynamics on phase space.

The geometric reformulation — Hamilton's equations as the integral curves of a Hamiltonian vector field on a symplectic manifold — emerged through the work of Cartan (the Leçons sur les invariants intégraux, 1922), with Weyl, Liouville, and Birkhoff contributing along the way. Arnold's Mathematical Methods of Classical Mechanics (1974, English translation 1978; second edition 1989) ^{[Arnold 1989]} consolidated the geometric viewpoint into a canonical reference. Marsden-Ratiu's Introduction to Mechanics and Symmetry (1994, second edition 1999) ^{[Marsden-Ratiu 1999]} extended it to the equivariant / Lie-group / momentum-map framework now standard in geometric mechanics. The Liouville-Arnold theorem on integrable systems was proved in Arnold's 1963 Russian Math. Surveys paper ^{[Arnold 1963]}; the KAM theorem, settled in Arnold's 1963 paper and Moser's 1962 smooth-twist version, characterises the persistence of integrability under perturbation 05.09.01.

Hamiltonian flow is the canonical object of classical determinism. The 1814 Essai philosophique sur les probabilités of Laplace ^{[Laplace 1814]} put the determinism thesis in its sharpest form: an intelligence that knew the initial $(q,p)$ of every particle and the laws of motion would know all of past and future. Mathematically, this is exactly the statement that a Hamiltonian flow on $T^{\ast }Q$ is uniquely determined by its initial point and the equation $\dot{\gamma }=X_H\circ \gamma$. The thesis was challenged on multiple fronts in the 20th century: by quantum mechanics (no joint sharp values of $q$ and $p$); by Poincaré's discovery of chaotic orbits in the three-body problem (sensitivity to initial conditions makes prediction infeasible even when the equations are deterministic); and by the singularity theorems of Hawking and Penrose in general relativity (deterministic evolution can break down at finite proper time). Hamilton's equations remain the cleanest setting in which to state the classical-determinism thesis precisely, and the cleanest contrast against the quantum-mechanical and chaotic and relativistic deviations.

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

• Hamilton, W. R., "On a general method in dynamics", Phil. Trans. Roy. Soc. 124 (1834), 247–308; "Second essay on a general method in dynamics", 125 (1835), 95–144. [Need to source — originator papers.]
• Jacobi, C. G. J., Vorlesungen über Dynamik (lectures of 1842-43, published 1866), ed. A. Clebsch.
• Lagrange, J. L., Mécanique analytique (1788).
• Laplace, P. S., Essai philosophique sur les probabilités (1814).
• Poincaré, H., Les méthodes nouvelles de la mécanique céleste, 3 vols. (1892, 1893, 1899).
• Arnold, V. I., "Proof of A. N. Kolmogorov's theorem on the conservation of conditionally periodic motions under a small perturbation of the Hamiltonian", Russian Math. Surveys 18:5 (1963), 9–36.
• Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer GTM 60, 1989).
• Marsden, J. E. & Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd ed. (Springer TAM 17, 1999).
• Abraham, R. & Marsden, J. E., Foundations of Mechanics, 2nd ed. (Addison-Wesley, 1978).
• Goldstein, H., Poole, C. P. & Safko, J., Classical Mechanics, 3rd ed. (Pearson, 2002).
• Landau, L. D. & Lifshitz, E. M., Mechanics, 3rd ed. (Course of Theoretical Physics Vol. 1, Pergamon, 1976).
• Taylor, J. R., Classical Mechanics (University Science Books, 2005).
• Susskind, L. & Hrabovsky, G., The Theoretical Minimum: Classical Mechanics (Basic Books, 2014).
• Tong, D., Classical Dynamics (DAMTP Cambridge lecture notes, §4 "The Hamiltonian Formulation").

---

Wave 1 physics seed unit, produced manually 2026-05-18 (per docs/plans/PHYSICS_PLAN.md §5 — manual-first pattern-setter for §09–13). All four cross-domain hooks_out targets are proposed; no chem/bio/phil seed unit yet exists to receive confirmed promotion. Status remains draft pending Tyler's review and the §11 Next-Actions retro per PHYSICS_PLAN.