Integrable Systems

It is sometimes stated that there are only two exactly solvable problems in classical mechanics--the simple harmonic oscillator and the two-body central force problem. This is not strictly true (we shall consider systems other than these which admit exact solutions), but what is true is that the conditions under which one can find exact solutions to a system of differential equations describing a physical system are rarely satisfied. The point of this thesis is to describe what is necessary in order for such solutions to exist, and how to find the solutions when they do exist. The key to the approach lies in the notion that the symmetries of a system lead to conserved quantities (Noether's Theorem), which can be used to reduce the number of degrees of freedom of the system. With enough conserved quantities, we can reduce the number of degrees of freedom to zero and obtain a solution to the system. Such a system is said to be *integrable.* The precise formulation of this statement for systems of ordinary differential equations is known as the Arnold-Liouville Theorem, and it includes a method for constructing solutions explicitly. However, in practice the method outlined in the Arnold-Liouville Theorem can be difficult to implement, and it is better to use the existence statement in the Arnold-Liouville Theorem to ascertain whether or not it is worth attempting to solve a system, then find a solution using methods tailored to the problem rather than the general approach given in the Arnold-Liouville Theorem. Before we can state the Arnold-Liouville Theorem in precise terms, we need to introduce Hamiltonian mechanics, a formulation of classical mechanics in which the dynamics are determined by the properties of the *Hamiltonian,* a function of the dynamical variables and of time which is often, but not always, equal to the energy of the system. While equivalent to Newtonian and Lagrangian mechanics, Hamiltonian mechanics turns out to be much more useful from the standpoint of integrable systems. Its generalizations allow us to study a wide range of systems, including those described by partial differential equations. The content of the chapters is as follows: In Chapter 1, we introduce the basics of Hamiltonian mechanics, including the Hamiltonian, Hamilton's equations, and the Poisson bracket. We then proceed to formally define what an integrable system is and finish with a simple example known as the harmonic oscillator. Chapter 2 focuses on stating and proving the Arnold-Liouville Theorem, which as noted above is the cornerstone of the theory of integrability for ordinary differential equations. We also solve the two-body central force problem using the constructive method of the Arnold-Liouville Theorem. In Chapter 3, we explore the generalizations of Hamiltonian mechanics and the Poisson bracket, proving along the way statements which will come in handy for proving the integrability of the Toda Lattice and which lay the groundwork for understanding how we can generalize the theory to systems of partial differential equations. In Chapter 4, we introduce our most interesting and complex example, the Toda Lattice, which consists of a one-dimensional chain of particles which repel each other with a force falling off exponentially with distance. After proving the integrability of this system using the Arnold-Liouville Theorem, we briefly describe some of its solutions. We conclude in Chapter 5 by describing how we can generalize the Poisson bracket to partial differential equations, then giving a flavor of the rich hierarchies of integrable systems which exist by relating the Toda Lattice to certain partial differential equations. Finally, the reader should be aware that while we focus here on applications to physics, the mathematical tools we develop can in principle be applied to any system of differential equations which can be formulated in terms of Hamiltonian mechanics. For instance, one could apply the ideas presented here to financial or logistical modelling.