6 · symplectic methods

differential forms, exterior differentiation and stokes' theorem

A differential $k$-form $\alpha$ on a manifold $M$ is a smooth skew-symmetric multilinear field on the tangent spaces. Exterior differentiation $$ d:\Omega^k(M)\to\Omega^{k+1}(M) $$ satisfies $d^2=0$ and is uniquely characterized by its action on functions and by the graded Leibniz rule. Stokes' theorem $$ \int_M d\alpha=\int_{\partial M}\alpha $$ links the local calculus of forms to global topological invariants and supplies the analytic foundation for all integral conservation laws that follow.

definition (differential $k$-form). a smooth section of $\bigwedge^k T^*M$ is a differential $k$-form. in local coordinates, $$ \alpha=\sum_{i_1<\cdots<i_k}\alpha_{i_1\dots i_k}(x)\,dx^{i_1}\wedge\cdots\wedge dx^{i_k}. $$

definition (exterior derivative on functions and products). on functions one sets $df(X)=X(f)$. the operator $d$ is required to satisfy $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^{\deg\alpha}\alpha\wedge d\beta$ and to be local. these axioms determine $d$ uniquely on all forms.

theorem ($d^2=0$). for every smooth form $\alpha$, $d(d\alpha)=0$.

proof. the claim is local. for a function $f$, $$ d(df)=\sum_{i,j}\partial_i\partial_j f\,dx^i\wedge dx^j=0 $$ because mixed partials commute and $dx^i\wedge dx^j$ is skew. if the identity holds for $\alpha$ and $\beta$, the Leibniz rule yields $$ d^2(\alpha\wedge\beta) =d^2\alpha\wedge\beta +(-1)^{\deg\alpha}d\alpha\wedge d\beta +(-1)^{\deg\alpha}d\alpha\wedge d\beta +\alpha\wedge d^2\beta =0 $$ because the cross terms cancel and $d^2\alpha=d^2\beta=0$ by induction on form degree (every form is locally a sum of products $f\,dx^{i_1}\wedge\cdots\wedge dx^{i_k}$). $\square$

theorem (stokes). if $M$ is an oriented manifold with boundary and $\alpha$ is a compactly supported $(n-1)$-form (or $M$ is compact), then $$ \int_M d\alpha=\int_{\partial M}\alpha. $$

proof (reduction to the standard half-space). by partitions of unity it is enough to treat forms supported in a single positively oriented chart. after a coordinate change the claim is the classical Stokes formula on a half-ball in $\mathbb{R}^n$, proved by integrating the definition of $d$ coordinatewise and using the fundamental theorem of calculus in each variable (boundary terms survive only on $\partial M$). $\square$

symplectic manifolds and hamiltonian vector fields

A symplectic manifold is a pair $(M,\omega)$ where $\omega$ is a closed, non-degenerate two-form. Non-degeneracy means that the map $$ v\mapsto\iota_v\omega $$ is an isomorphism between vector fields and one-forms. Given a smooth function $H$ (the Hamiltonian), the corresponding Hamiltonian vector field $X_H$ is defined by $$ \iota_{X_H}\omega=-dH. $$ The flow $\varphi_t$ of $X_H$ automatically preserves $\omega$ (Cartan's magic formula and $d\omega=0$), and is therefore a one-parameter group of symplectomorphisms. This is the geometric content of Hamilton's equations.

definition (symplectic manifold). a pair $(M,\omega)$ with $\omega\in\Omega^2(M)$ closed ($d\omega=0$) and non-degenerate: for each $x$, if $\omega_x(v,w)=0$ for all $w$ then $v=0$. necessarily $\dim M$ is even.

definition (hamiltonian vector field). for $H\in C^\infty(M)$, the unique field $X_H$ satisfying $\iota_{X_H}\omega=-dH$ (equivalently $\omega(X_H,\cdot)=-dH$) is the Hamiltonian vector field of $H$.

theorem (hamiltonian flow is symplectic). if $\varphi_t$ is the (local) flow of $X_H$, then $\varphi_t^*\omega=\omega$ wherever the flow is defined.

proof. Cartan's magic formula reads $\mathcal{L}_X=d\,\iota_X+\iota_X\,d$ for the Lie derivative of forms. thus $$ \frac{d}{dt}\varphi_t^*\omega =\varphi_t^*(\mathcal{L}_{X_H}\omega) =\varphi_t^*\bigl(d\iota_{X_H}\omega+\iota_{X_H}d\omega\bigr). $$ now $d\omega=0$ and $\iota_{X_H}\omega=-dH$, so $d\iota_{X_H}\omega=-d(dH)=0$. therefore $\frac{d}{dt}\varphi_t^*\omega=0$, and $\varphi_0^*\omega=\omega$ yields $\varphi_t^*\omega=\omega$. $\square$

remark (hamilton equations in Darboux coordinates). when $\omega=\sum_i dp_i\wedge dq^i$ the defining relation $\iota_{X_H}\omega=-dH$ becomes $\dot q=\partial H/\partial p$, $\dot p=-\partial H/\partial q$.

liouville, poisson brackets and the lie algebra of hamiltonian fields

Because a symplectomorphism preserves $\omega$, it preserves the Liouville volume form $\omega^n/n!$. Consequently every Hamiltonian flow is incompressible (Liouville's theorem). The Poisson bracket $$ \{F,G\}=\omega(X_F,X_G) $$ turns the space of smooth functions into a Lie algebra, and the map $H\mapsto X_H$ is a Lie-algebra anti-homomorphism into the Lie algebra of vector fields. Vanishing of the Poisson bracket is the geometric expression of the conservation of $G$ along the flow of $H$.

definition (liouville volume). on a $2n$-dimensional symplectic manifold the volume form $\mu=\omega^n/n!$ is nowhere zero and orientation-compatible with any Darboux chart.

theorem (liouville). if $\varphi_t$ is a Hamiltonian flow, then $\varphi_t^*\mu=\mu$. in particular the flow is volume-preserving.

proof. pullback is a ring homomorphism, so $\varphi_t^*(\omega^n)=(\varphi_t^*\omega)^n=\omega^n$ by the previous theorem. dividing by $n!$ gives $\varphi_t^*\mu=\mu$. $\square$

definition (poisson bracket). $\{F,G\}:=\omega(X_F,X_G)=-X_F(G)=X_G(F)$.

theorem (poisson bracket is a lie algebra). $\{\cdot,\cdot\}$ is bilinear, skew-symmetric, and satisfies the Jacobi identity. moreover $$ X_{\{F,G\}}=-[X_F,X_G] $$ (anti-homomorphism of Lie algebras). in particular $G$ is constant along the flow of $X_H$ if and only if $\{H,G\}=0$.

proof. bilinearity and skew-symmetry are immediate from those of $\omega$. the identity $X_F(G)=dG(X_F)=-\omega(X_G,X_F)=\omega(X_F,X_G)=\{F,G\}$ gives $\{H,G\}=X_H(G)$, so conservation along $X_H$ is exactly $\{H,G\}=0$. to compare brackets of fields, evaluate for an arbitrary test function $K$: $$ [X_F,X_G](K) =X_F(X_G K)-X_G(X_F K) =X_F(\{G,K\})-X_G(\{F,K\}) =\{\{G,K\},F\}-\{\{F,K\},G\}. $$ the Jacobi identity for functions (expand $\mathrm{Jac}(F,G,K)=0$ identically, a local coordinate computation or the closedness of $\omega$ via Cartan) rearranges this as $-\{\{F,G\},K\}=-X_{\{F,G\}}(K)$. hence $[X_F,X_G]=-X_{\{F,G\}}$. Jacobi for vector fields then yields Jacobi for $\{\cdot,\cdot\}$. $\square$

theorem (energy conservation). $\{H,H\}=0$, so $H$ is constant along its own Hamiltonian flow.

proof. skew-symmetry of $\{\cdot,\cdot\}$. $\square$

poincare-cartan invariant, hamilton-jacobi and generating functions

On the extended phase space the one-form $$ \theta=p_i\,dq^i-H\,dt $$ (the Poincare-Cartan form) has exterior derivative whose kernel defines the equations of motion. Its integral over any closed curve is invariant under the flow. A generating function $S(q,P,t)$ produces a canonical transformation via $$ p=\partial S/\partial q,\qquad Q=\partial S/\partial P,\qquad K=H+\partial S/\partial t. $$ When the new Hamiltonian $K$ vanishes, $S$ satisfies the Hamilton-Jacobi equation $$ \partial S/\partial t+H(q,\partial S/\partial q,t)=0, $$ reducing integration of the flow to the determination of a complete integral of a first-order PDE.

definition (poincare-cartan form). on $T^*Q\times\mathbb{R}$ with coordinates $(q,p,t)$ and Hamiltonian $H(q,p,t)$, set $\Theta=p_i\,dq^i-H\,dt$.

theorem (characteristic equations from $d\Theta$). a curve $\gamma(s)=(q(s),p(s),t(s))$ is an integral curve of the characteristic distribution of $d\Theta$ (annihilator of maximal rank of $d\Theta$) precisely when, after reparametrization with $t$ as time, Hamilton's equations hold for $(q,p)$.

proof. compute $$ d\Theta =dp_i\wedge dq^i -dH\wedge dt =\bigl(dp_i+\partial_{q^i}H\,dt\bigr)\wedge dq^i -\partial_{p_i}H\,dp_i\wedge dt $$ (summation implied). imposing $\iota_{\dot\gamma}d\Theta=0$ with $\dot t\neq 0$ and reading coefficients of the basis forms yields $\dot q=\partial_p H$, $\dot p=-\partial_q H$, which is Hamilton's system. $\square$

remark (generating functions). a function $S(q,P,t)$ with $\det\partial^2 S/\partial q\,\partial P\neq 0$ defines an exact symplectic relation $p\,dq-P\,dQ=dS-K\,dt$ after $K=H+\partial_t S$, i.e. a time-dependent canonical transformation. if $K=0$ then $S$ solves Hamilton-Jacobi and the new momenta $P$ are constant of motion.

action-angle variables, averaging and KAM

A completely integrable Hamiltonian system on a $2n$-dimensional symplectic manifold admits $n$ independent integrals in involution. Under suitable compactness and non-degeneracy assumptions the common level sets are Lagrangian tori, and there exist action-angle coordinates $(I,\vartheta)$ in which $$ H=H(I),\qquad \dot{\vartheta}=\partial H/\partial I,\qquad \dot I=0. $$ When a small perturbation $\varepsilon H_1(I,\vartheta)$ is added, the averaging principle replaces $H_1$ by its average over the torus. Kolmogorov-Arnold-Moser theory asserts that those tori whose frequency vectors are sufficiently irrational persist as slightly deformed invariant tori for all sufficiently small $\varepsilon$. The surviving tori form a set of positive measure, thereby establishing the stability of a large portion of phase space.

definition (integrals in involution). smooth functions $F_1,\dots,F_n$ are in involution if $\{F_i,F_j\}=0$ for all $i,j$. they are independent if $dF_1\wedge\cdots\wedge dF_n\neq 0$ on a dense open set.

theorem (liouville-arnold, statement). let $F_1,\dots,F_n$ be independent integrals in involution on a $2n$-dimensional symplectic manifold, and let $c$ be a regular value such that the level set $M_c=\{F=c\}$ is compact and connected. then $M_c$ is diffeomorphic to a torus $\mathbb{T}^n$, and there exist action-angle coordinates $(I,\vartheta)$ in a neighborhood of $M_c$ putting the flow of any Hamiltonian of the form $H=h(F)$ into the linear form $\dot\vartheta=\omega(I)$, $\dot I=0$.

remark. the proof constructs the angle coordinates from the joint flow of the commuting Hamiltonian fields $X_{F_i}$ (chapter 5 orbits on compact abelian Lie groups are tori) and defines actions as integrals of the Liouville one-form over a basis of cycles. full technical details (properness, exactness of periods) are classical; the geometric content is that complete integrability linearizes the dynamics on Lagrangian tori.

remark (KAM). for real-analytic Hamiltonians $H_0(I)+\varepsilon H_1(I,\vartheta)$ with nondegenerate frequency map $I\mapsto\partial H_0/\partial I$, Diophantine tori survive as invariant Lagrangian tori for small $\varepsilon$. the proof is a rapidly convergent Newton scheme for conjugacy to a linear flow and will not be reproduced here; the theorem is the precise persistence statement of classical perturbation theory.

darboux-weinstein, moment maps and slices

Darboux's theorem states that every symplectic manifold is locally isomorphic to $(\mathbb{R}^{2n},\sum dp_i\wedge dq^i)$. The Weinstein tubular neighborhood theorem extends this linearization to a neighborhood of any Lagrangian submanifold.

When a Lie group $G$ acts on $(M,\omega)$ by symplectomorphisms and the action admits an equivariant moment map $$ \mu:M\to\mathfrak{g}^*, $$ the components of $\mu$ are the Noether conserved quantities associated with the infinitesimal generators. On a Kahler manifold the moment map of a holomorphic group action is often the gradient of a $G$-invariant function with respect to the Kahler metric. Coisotropic embeddings realize a given Poisson manifold as a coisotropic submanifold of a symplectic manifold, while symplectic slices provide normal forms for the neighborhood of a group orbit. These constructions are the local models on which singular reduction and the theory of symplectic stratifications are built.

theorem (darboux). about every point of a symplectic manifold $(M,\omega)$ there exist local coordinates $(q^1,\dots,q^n,p_1,\dots,p_n)$ in which $$ \omega=\sum_{i=1}^n dp_i\wedge dq^i. $$

proof. it suffices to work near $0\in\mathbb{R}^{2n}$ with a symplectic form $\omega$ that coincides with the standard form $\omega_0=\sum dp_i\wedge dq^i$ at the origin (apply a linear symplectic basis of $T_0 M$). set $\omega_t=(1-t)\omega_0+t\omega$ for $t\in[0,1]$. then $\omega_t$ is closed and nondegenerate near $0$ for all $t$ (nondegeneracy is an open condition and holds at $t=0$). hence $\omega-\omega_0=d\alpha$ for a one-form $\alpha$ with $\alpha_0=0$ (Poincare lemma). solve Moser's equation $$ \iota_{X_t}\omega_t=-\alpha $$ for a time-dependent field $X_t$ vanishing at $0$. let $\psi_t$ be its flow; then $$ \frac{d}{dt}(\psi_t^*\omega_t) =\psi_t^*\bigl(\mathcal{L}_{X_t}\omega_t+\partial_t\omega_t\bigr) =\psi_t^*\bigl(d\iota_{X_t}\omega_t+\omega-\omega_0\bigr) =\psi_t^*( -d\alpha+\omega-\omega_0)=0 $$ because $\partial_t\omega_t=\omega-\omega_0$ and $d\alpha=\omega-\omega_0$. thus $\psi_1^*\omega=\omega_0$. the map $\psi_1$ provides Darboux coordinates. $\square$

definition (equivariant moment map). a $G$-action by symplectomorphisms admits a moment map $\mu:M\to\mathfrak{g}^*$ if for every $\xi\in\mathfrak{g}$ the fundamental field $\xi_M$ is Hamiltonian with $\iota_{\xi_M}\omega=-d\langle\mu,\xi\rangle$, and $\mu$ is equivariant with respect to the coadjoint action on $\mathfrak{g}^*$.

theorem (noether from the moment map). if $H$ is $G$-invariant, $\mathcal{L}_{\xi_M}H=0$ for all $\xi$, then $\langle\mu,\xi\rangle$ is constant along the flow of $X_H$.

proof. $$ X_H\langle\mu,\xi\rangle =\{H,\langle\mu,\xi\rangle\} =\omega(X_H,\xi_M) =-dH(\xi_M) =-\xi_M(H)=0 $$ by invariance. $\square$

complete integrability, kostant-symes and hidden symmetries

A Hamiltonian system is completely integrable when its phase space is foliated by invariant Lagrangian tori. The Kostant-Symes lemma produces such systems on coadjoint orbits of Lie algebras by restricting a suitable invariant function; the resulting equations are solvable by factorization in the Lie group. Hidden symmetries (often realized by Lax pairs or by pseudodifferential operators) enlarge the obvious symmetry algebra and account for the unexpected integrability of systems of Calogero-Moser type.

definition (complete integrability). a Hamiltonian system $(M,\omega,H)$ of dimension $2n$ is completely integrable if there exist $n$ independent integrals in involution including $H$ (or any regular reparametrization of the energy).

remark (kostant-symes). if $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{n}$ is a vector space splitting of a Lie algebra into subalgebras, restriction of an $\mathrm{Ad}^*$-invariant function on $\mathfrak{g}^*$ to the dual of one factor yields a Poisson-commuting family on coadjoint orbits (after the appropriate identification). this is a Lie-algebraic engine for integrable systems; details live in the geometry of coadjoint orbits.

contraction and deformation theory

Contractions of Lie algebras (Inonu-Wigner contractions) induce contractions of the associated symplectic homogeneous spaces. The Whitehead lemmas and the Hochschild-Serre spectral sequence control the deformation theory of these actions, explaining how the symplectic geometry of the Galilei group deforms into that of the Poincare group when the speed of light is restored to a finite value.

remark. a contraction is a continuous family of Lie brackets $[-,-]_\varepsilon$ converging to a new bracket as $\varepsilon\to 0$ (classically a rescaling of generators). the dual coadjoint geometry contracts accordingly; Galilei structures appear as $c\to\infty$ limits of Poincare structures. this is the infinitesimal counterpart of the passage between relativistic and nonrelativistic Hamiltonian geometries of chapter 1's Galilean modeling.

optical realization of symplectic geometry

The same symplectic structures govern the propagation of light.

• Gaussian (paraxial, rotationally symmetric) optics is described by matrices in $\operatorname{SL}(2,\mathbb{R})$.
• Linear optics with arbitrary apertures requires the full symplectic group $\operatorname{Sp}(2n,\mathbb{R})$, which preserves the optical form $\sum dp_i\wedge dq_i$.
• Geometrical optics in the short-wavelength limit is the theory of symplectic diffeomorphisms generated by eikonal functions (point characteristics).
• Wave optics at the Fresnel level lifts the symplectic group to its double cover, the metaplectic group $\operatorname{Mp}(2n,\mathbb{R})$, realized by unitary integral operators on $L^2(\mathbb{R}^n)$. The metaplectic correction supplies the phase factor $e^{-i\pi/4}$ at each passage through a caustic, resolving the topological Maslov index.

definition (linear symplectic group). $$ \operatorname{Sp}(2n,\mathbb{R}) =\{S\in\mathrm{GL}(2n,\mathbb{R}): S^T J S=J\}, $$ where $J$ is the standard matrix of $\omega_0$. for $n=1$, $\operatorname{Sp}(2,\mathbb{R})=\operatorname{SL}(2,\mathbb{R})$.

theorem (linear optics preserves $\omega_0$). a linear map $S:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ preserves $\omega_0(u,v)=u^T J v$ for all $u,v$ if and only if $S\in\operatorname{Sp}(2n,\mathbb{R})$.

proof. $\omega_0(Su,Sv)=(Su)^T J(Sv)=u^T(S^T JS)v$. this equals $u^T J v$ for all $u,v$ if and only if $S^T JS=J$. $\square$

In this way symplectic geometry unifies the classical mechanics of particles with the Hamiltonian theory of light, while the moment-map and integrability techniques of the abstract theory find concrete realization in both domains.

exercises

exercise 1 (harmonic oscillator). on $\mathbb{R}^2$ with $\omega=dp\wedge dq$ and $H=\frac12(p^2+q^2)$, compute $X_H$ and solve the flow explicitly. verify $\varphi_t^*\omega=\omega$ by direct differentiation and check that $H$ is constant.

exercise 2 (poisson algebra). show that $\{q^i,p_j\}=\delta^i_j$ and $\{q^i,q^j\}=\{p_i,p_j\}=0$ in standard Darboux coordinates, and compute $\{H,q\}$, $\{H,p\}$ for $H=\frac12 p^2+V(q)$ in one degree of freedom.

exercise 3 (liouville area). for a planar Hamiltonian flow, explain why $\varphi_t^*\omega=\omega$ is equivalent to area preservation. give a one-line proof that the flow of $X_H$ cannot have asymptotically stable equilibria in the plane.

exercise 4 (optical $\operatorname{SL}(2)$). take $S=\begin{pmatrix}1&d\\0&1\end{pmatrix}$ (a free drift of length $d$ in paraxial $2\times 2$ ray-transfer convention). verify $\det S=1$ and interpret the action on $(q,p)$.

exercise 5 (action-angle for the oscillator). for $H=\frac12(p^2+q^2)$, introduce polar coordinates $q=\sqrt{2I}\cos\vartheta$, $p=\sqrt{2I}\sin\vartheta$ (up to sign convention) and show that $(I,\vartheta)$ are action-angle coordinates: $\omega=dI\wedge d\vartheta$ and $H=H(I)$ only. read off the frequency $\dot\vartheta=\partial H/\partial I$.