Integrable systems represent one of the deepest and most beautiful intersections of geometry, dynamics, and analysis in mathematical physics. Rather than viewing these systems as exceptional cases, this course positions integrability as a fundamental organizing principle: many of the nonlinear equations that govern waves, fluids, and field theories possess hidden symmetries and conservation laws that make them exactly solvable. We begin with finite-dimensional Hamiltonian mechanics, where the Arnold-Liouville theorem reveals that integrable systems foliate phase space into invariant tori, allowing us to reduce their dynamics to simple flows on these geometric structures. This classical perspective provides both intuition and rigorous tools that we will extend into infinite dimensions.

The course then transitions to integrable partial differential equations—primarily nonlinear wave equations like the Korteweg-de Vries equation and sine-Gordon equation—where integrability manifests through remarkable properties: solitons that preserve their shape through collisions, exact solutions via algebraic methods, and a surprising connection to the scattering data of an associated linear operator. The Inverse Scattering Transform emerges as the fundamental technique, revealing that certain nonlinear PDEs can be linearized through a clever change of variables involving spectral data. From this vantage point, solving a nonlinear equation reduces to manipulating the spectral parameters of an associated Schrodinger-type problem.

Throughout, we emphasize that integrability is not a computational accident but a structural phenomenon rooted in deep algebraic and geometric principles: infinite-dimensional Hamiltonian structures, Lax representations, and symmetry groups. By the course's end, you will understand how these disparate tools—canonical transformations, Lie algebras, spectral theory, and differential geometry—unify under the umbrella of integrable systems theory.

The theory of ordinary differential equations reaches its most elegant form when a system possesses enough structure to be solved by pure integration rather than case-by-case analysis. This chapter develops the classical theory of integrable Hamiltonian systems, beginning with the geometry of flows generated by vector fields and culminating in the Arnold-Liouville theorem, which characterises exactly when a Hamiltonian system can be reduced to linear motion on a torus. The journey passes through the Poisson bracket formalism, the notion of first integrals, canonical coordinate transformations, and the action-angle variables that make explicit integration possible.

The central question is: when can we find coordinates in which Hamilton's equations become so simple that the solution is just a linear function of time? The answer, given by Arnold-Liouville, requires precisely $n$ independent first integrals in involution for a $2n$-dimensional system. Establishing this requires understanding both the algebraic structure of Hamiltonian flows and the topology of the level sets they foliate.

# 1. Integrability of ODE's

## Vector Fields and Flow Maps

A fundamental question in the theory of differential equations is whether a given initial value problem has a solution, whether that solution is unique, and how it depends on the initial data. For ODE systems of the form $\dot{x} = V(x)$, these questions are answered once we know the vector field $V$ is sufficiently regular.

An $m$-dimensional first-order ODE is specified by a vector field $V: \mathbb{R}^m \to \mathbb{R}^m$ and an initial condition $x_0 \in \mathbb{R}^m$. The task is to find $x: (a, b) \to \mathbb{R}^m$, defined on some open interval $(a, b)$ containing $0$, satisfying

[definition: Initial Value Problem] The **initial value problem** associated to a vector field $V: \mathbb{R}^m \to \mathbb{R}^m$ and initial condition $x_0 \in \mathbb{R}^m$ is the system \begin{align*} \dot{x} = V(x), \quad x(0) = x_0. \end{align*} A **solution** is a smooth map $x: (a, b) \to \mathbb{R}^m$ on some open interval $(a, b) \ni 0$ satisfying both conditions. [/definition]

Restricting to first-order systems is no restriction in practice: any $k$-th order ODE can be rewritten as a first-order system by introducing auxiliary variables for each derivative up to order $k - 1$. The course assumes throughout that vector fields are smooth enough for the following fundamental existence and uniqueness result to hold.

[quotetheorem:1331]

The proof of this result uses the Picard-Lindelöf fixed-point argument and belongs to the Part IB Differential Equations and Analysis courses; it is quoted here without proof.

The smoothness hypothesis on $V$ is essential. To see what can go wrong, consider the scalar ODE $\dot{x} = x^{2/3}$ on $\mathbb{R}$. The right-hand side is continuous but fails to be Lipschitz at $x = 0$. Starting from $x(0) = 0$, both $x(t) = 0$ and $x(t) = (t/3)^3$ are solutions — so uniqueness fails. Even when a local solution exists and is unique, it need not persist for all time: $\dot{x} = x^2$ with $x(0) = 1$ has the unique solution $x(t) = 1/(1-t)$, which blows up at $t = 1$. Global existence requires either a global Lipschitz bound or an a priori estimate preventing blow-up.

The smooth dependence on initial conditions is particularly significant: it allows us to package all solutions simultaneously into a single map.

[definition: Flow Map] Given a smooth vector field $V: \mathbb{R}^m \to \mathbb{R}^m$, the **flow map** is the family of smooth maps $g^t: \mathbb{R}^m \to \mathbb{R}^m$ defined by \begin{align*} g^t(x_0) = x(t), \end{align*} where $x(t)$ is the unique solution to $\dot{x} = V(x)$, $x(0) = x_0$. We often write $g^t x_0$ for $g^t(x_0)$. [/definition]

The flow map encodes the full dynamics: knowing $g^t$ for all $t$ is equivalent to knowing all solutions. Its algebraic properties reflect the deterministic and time-reversible character of smooth ODEs.

[quotetheorem:1351]

[citeproof:1351]

Together, properties (1)-(3) say that $\{g^t : t \in \mathbb{R}\}$ forms a one-parameter group of diffeomorphisms of $\mathbb{R}^m$. The vector field $V$ is called the **infinitesimal generator** of this group, because Taylor-expanding the flow gives