This course develops the foundational language and core techniques of symplectic geometry, the study of even-dimensional spaces equipped with a closed, nondegenerate 2-form. It begins at the linear level, where symplectic structures have no local invariants, and then moves to manifolds, where the same algebraic ideas control geometry, dynamics, and topology. Along the way, the course connects the abstract theory to concrete examples such as cotangent bundles, Hamiltonian systems, and geometric structures arising from physics and mechanics.

The main themes are the passage from linear symplectic algebra to global geometry, the special role of Hamiltonian vector fields and Poisson brackets, and the rigidity phenomena that distinguish symplectic geometry from ordinary differential topology. After establishing local normal forms through Darboux’s theorem and Moser’s method, the course studies symplectic and Hamiltonian isotopies, contact-type hypersurfaces, moment maps, and symplectic reduction. These ideas are then applied to toric and other global examples, before concluding with deeper topics such as rigidity beyond Darboux and integrable systems with action-angle coordinates. Each chapter builds on the previous one, moving from basic structures to powerful global tools and culminating in the interplay between symmetry, reduction, and dynamics.

# Introduction

Symplectic geometry is the geometry of spaces that carry a nondegenerate closed $2$-form. It grew from Hamiltonian mechanics, where the same structure records position, momentum, and the evolution of a physical system, but its modern form is a central part of differential geometry and topology. This course develops the subject from its linear algebraic core to Hamiltonian dynamics, moment maps, and symplectic reduction. The guiding theme is the tension between local flexibility and global rigidity: locally every symplectic manifold has the same model, while globally volume, cohomology, and holomorphic-curve phenomena impose strong constraints.

The prerequisites are smooth manifolds, tangent and cotangent bundles, differential forms, de Rham cohomology, basic Lie groups and Lie algebras, multilinear algebra, and elementary classical mechanics. The notes will use these tools without rebuilding them from first principles, but the first lectures will recall the pieces of linear algebra and differential forms that symplectic geometry uses in a distinctive way.

## The Geometry Behind Hamiltonian Mechanics

What geometric structure is preserved by Hamiltonian evolution, and why is it different from the metric geometry of lengths and angles? Classical phase space has coordinates $(q_1,\dots,q_n,p_1,\dots,p_n)$, where $q_i$ records position and $p_i$ records momentum. The fundamental object is not a Riemannian metric but the $2$-form

\begin{align*} \omega_0 = \sum_{i=1}^n dq_i \wedge dp_i. \end{align*}

This form pairs position directions with momentum directions and gives the coordinate-free language for Hamiltonian equations.

[motivation] ### Phase Space Rather Than Configuration Space A mechanical system with $n$ degrees of freedom is usually described by a configuration manifold $Q$. The actual state of the system also includes momentum, so the natural phase space is the cotangent bundle $T^*Q$. A point of $T^*Q$ is a pair $(q,p)$ with $q \in Q$ and $p \in T_q^*Q$, and the cotangent bundle carries a canonical $2$-form independent of coordinate choices. ### Conservation As Geometry Hamiltonian evolution is generated by a function $H:T^*Q \to \mathbb R$, the Hamiltonian. Instead of measuring the gradient of $H$ using a metric, symplectic geometry converts $dH$ into a vector field by contracting with the symplectic form. The resulting flow preserves the symplectic form, and hence preserves the associated phase-space volume. ### Why Closed Nondegenerate Two-Forms Nondegeneracy says that every covector can be represented by contraction with a unique vector. Closedness is the infinitesimal condition that makes the local normal form and Hamiltonian conservation laws work. Together these two requirements produce a geometry with no local curvature invariant but many global constraints. [/motivation]

The first use of the theory is therefore explanatory: it gives a coordinate-free account of Hamiltonian mechanics. The same language also creates questions with no mechanical origin, such as which manifolds admit symplectic forms and which embeddings preserve symplectic structure.

## Linear Algebra As The Local Model

What must be understood before putting a symplectic form on a manifold? At each point of a symplectic manifold there is a tangent space equipped with a nondegenerate alternating [bilinear form](/page/Bilinear%20Form). The hypotheses cannot be weakened without changing the geometry: a degenerate alternating form leaves some nonzero directions invisible, while a symmetric nondegenerate form measures lengths and signs rather than paired position-momentum directions. The first chapter studies this purely linear situation because every local and infinitesimal argument later depends on it.

[definition: Symplectic Vector Space] A symplectic [vector space](/page/Vector%20Space) is a finite-dimensional real vector space $V$ equipped with a bilinear form $\omega:V \times V \to \mathbb R$ such that: 1. $\omega(v,w)=-\omega(w,v)$ for all $v,w \in V$; 2. if $v \in V$ satisfies $\omega(v,w)=0$ for all $w \in V$, then $v=0$. [/definition]

The alternating condition forces every vector to pair to zero with itself, so the information in $\omega$ is entirely in how different directions interact. Nondegeneracy is the replacement for invertibility: it lets the form identify vectors with covectors. This identification is the linear operation behind Hamiltonian vector fields.

[example: Standard Symplectic Vector Space] Let $V=\mathbb R^{2n}$ with coordinate functions $(q_1,\dots,q_n,p_1,\dots,p_n)$, and define \begin{align*} \omega_0=\sum_{i=1}^n dq_i\wedge dp_i. \end{align*} We verify directly that $\omega_0$ is alternating and nondegenerate. For tangent vectors \begin{align*} u=\sum_{i=1}^n a_i\frac{\partial}{\partial q_i}+b_i\frac{\partial}{\partial p_i} \end{align*} and \begin{align*} v=\sum_{i=1}^n c_i\frac{\partial}{\partial q_i}+d_i\frac{\partial}{\partial p_i}, \end{align*} the identity $(\alpha\wedge\beta)(u,v)=\alpha(u)\beta(v)-\alpha(v)\beta(u)$ gives \begin{align*} (dq_i\wedge dp_i)(u,v)=dq_i(u)dp_i(v)-dq_i(v)dp_i(u)=a_i d_i-c_i b_i. \end{align*} Therefore \begin{align*} \omega_0(u,v)=\sum_{i=1}^n(a_i d_i-b_i c_i). \end{align*} Interchanging $u$ and $v$ gives \begin{align*} \omega_0(v,u)=\sum_{i=1}^n(c_i b_i-d_i a_i)=-\sum_{i=1}^n(a_i d_i-b_i c_i)=-\omega_0(u,v), \end{align*} so $\omega_0$ is alternating. To prove nondegeneracy, suppose $\omega_0(u,v)=0$ for every $v\in V$. Taking $v=\partial/\partial p_j$ gives $c_i=0$ for all $i$, $d_j=1$, and $d_i=0$ for $i\ne j$, hence \begin{align*} 0=\omega_0\left(u,\frac{\partial}{\partial p_j}\right)=a_j. \end{align*} Taking $v=\partial/\partial q_j$ gives $c_j=1$, $c_i=0$ for $i\ne j$, and $d_i=0$ for all $i$, hence \begin{align*} 0=\omega_0\left(u,\frac{\partial}{\partial q_j}\right)=-b_j. \end{align*} Thus $a_j=b_j=0$ for every $j$, so $u=0$. Hence $(\mathbb R^{2n},\omega_0)$ is a symplectic vector space, and the formula shows explicitly that each $q_i$-direction pairs only with its corresponding $p_i$-direction. [/example]

The standard model shows what the desired coordinates should look like, but it does not yet explain whether an arbitrary nondegenerate alternating form can be put into that shape. The next theorem solves that classification problem and gives the normal form used in almost every later local argument.

[quotetheorem:3295]

[citeproof:3295]

This theorem explains why symplectic geometry has no pointwise eigenvalue-type invariant for the form itself. Both hypotheses are doing real work: if the form is alternating but degenerate, for instance the zero form on $\mathbb R^2$, no basis can make it look like $dq \wedge dp$ because some nonzero vector pairs to zero with every vector. Alternation also matters, since a symmetric nondegenerate form is classified by signature rather than by a universal standard symplectic matrix. The theorem is only a linear classification at one vector space; it does not choose bases smoothly over a manifold, nor does it compare different global symplectic forms. The difficulty of the subject begins when these pointwise normal forms have to be chosen coherently over a manifold and compared with global topology.