This course develops the mathematical framework of classical mechanics, treating mechanics not just as a collection of equations of motion but as a geometric theory of configuration, motion, and symmetry. It begins with the basic language of configuration spaces and variational principles, then moves to Lagrangian mechanics, constraints, and the passage from velocities to momenta via the Legendre transform. From there, the course builds the Hamiltonian formalism and the symplectic viewpoint, where phase space becomes the central object and the structure of the equations of motion is expressed through geometry.

The main themes are variational methods, symplectic geometry, conservation laws, symmetry reduction, and qualitative dynamics. Noether’s theory and momentum maps explain how symmetries produce conserved quantities, while symplectic reduction shows how to simplify systems with symmetry in a systematic way. Later chapters develop Hamilton-Jacobi theory, generating functions, stability analysis, periodic motion, normal forms, and the theory of integrable systems with action-angle coordinates. The final part turns to scattering, adiabatic invariants, and perturbative ideas, showing how the same geometric framework applies both to idealized exactly solvable systems and to more realistic systems with slow variation or weak interaction.

The chapters are arranged to move from foundations to advanced applications in a logical progression. The early material establishes the variational and geometric language needed to formulate mechanics cleanly; the middle chapters translate that language into Hamiltonian dynamics and symmetry-based methods; and the later chapters use those tools to study structure, stability, and long-time behavior. By the end of the course, the student should be able to move fluently between Lagrangian and Hamiltonian descriptions, exploit symmetry, and recognize the geometric mechanisms underlying classical phenomena.

# Introduction

This opening chapter sets the course in motion by fixing the viewpoint from which classical mechanics will be studied. Instead of treating mechanics as a catalogue of force laws, we regard it as a chain of geometric structures: configuration spaces describe possible positions, variational principles select physical paths, phase spaces organise evolution, and symmetries produce conserved quantities. The aim is to make familiar examples such as the pendulum, central force motion, and rigid body dynamics serve as prototypes for broader ideas in symplectic geometry and dynamical systems.

The course assumes multivariable calculus, linear algebra, ordinary differential equations, basic real analysis, smooth manifolds, differential forms, and elementary Lie theory. These prerequisites enter because the natural home of mechanics is rarely a single Euclidean coordinate chart: a particle on a sphere, a rigid body, or a system with symmetry asks for manifolds, tangent bundles, cotangent bundles, and group actions. The chapter therefore records the organising questions and the recurring objects that will appear throughout the lectures.

The failures to keep in mind are concrete. Coordinates may introduce artificial singularities, constraint forces may obscure the true degrees of freedom, and a conserved numerical quantity may be invisible if the symmetry producing it is not identified. The structures introduced here are therefore not decorative abstractions: each is designed to remove a specific ambiguity or obstruction that appears in direct Newtonian coordinates.

## What Is Classical Mechanics Trying To Describe?

The first question is not how to solve a differential equation, but what data a mechanical model should contain. A system has a space of possible positions, a rule assigning an energy or action to paths, and an evolution law that turns initial data into motion. The course studies how these pieces fit together and how much of the motion is forced by geometry rather than by coordinates.

[definition: Configuration Space] A configuration space is a smooth manifold $Q$ whose points represent possible positions of a mechanical system. [/definition]

The configuration space is the stage for positions, but motion needs velocities. For a path $q: I \to Q$, its velocity at time $t \in I$ lies in $T_{q(t)}Q$, and the collection of all such position-velocity pairs is the tangent bundle $TQ$. This motivates the Lagrangian notion of state, where position and velocity are recorded together.

[definition: State In Lagrangian Mechanics] A state in Lagrangian mechanics is a point $(q, v) \in TQ$, where $q \in Q$ and $v \in T_qQ$. [/definition]

This distinction matters because the same position can support many possible velocities. Newton's second-order equations become first-order equations on $TQ$, and the Lagrangian formalism expresses those equations through a scalar function on $TQ$. The basic Euclidean particle is the reference model that shows how this abstract language matches the classical force law.

[example: Particle In Euclidean Space] For a particle moving in $\mathbb R^n$, the configuration space is $Q=\mathbb R^n$, and each tangent space is canonically $T_q\mathbb R^n\cong \mathbb R^n$. Hence \begin{align*} TQ\cong \mathbb R^n\times \mathbb R^n, \end{align*} where a point of $TQ$ is written $(q,v)$ with $q=(q_1,\dots,q_n)$ and $v=(v_1,\dots,v_n)$. For a path $q:I\to\mathbb R^n$, the velocity is \begin{align*} \dot q(t)=(\dot q_1(t),\dots,\dot q_n(t)). \end{align*} The Lagrangian \begin{align*} L(q,v)=\frac{1}{2}m|v|^2-V(q) \end{align*} means explicitly \begin{align*} L(q,v)=\frac{1}{2}m\sum_{j=1}^n v_j^2-V(q_1,\dots,q_n). \end{align*} Applying the *Euler-Lagrange equations from stationary action*, we compute the two derivatives in the $i$th coordinate. Since $q_i$ occurs only in the potential term, \begin{align*} \frac{\partial L}{\partial q_i}(q,v)=-\frac{\partial V}{\partial q_i}(q). \end{align*} Since $v_i$ occurs in the kinetic term as $\frac{1}{2}m v_i^2$, \begin{align*} \frac{\partial L}{\partial v_i}(q,v)=m v_i. \end{align*} Along the path, $v_i=\dot q_i(t)$, so \begin{align*} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i}(q(t),\dot q(t))\right)=\frac{d}{dt}(m\dot q_i(t))=m\ddot q_i(t). \end{align*} The Euler-Lagrange equation therefore becomes \begin{align*} m\ddot q_i(t)-\left(-\frac{\partial V}{\partial q_i}(q(t))\right)=0. \end{align*} Equivalently, \begin{align*} m\ddot q_i(t)=-\frac{\partial V}{\partial q_i}(q(t)). \end{align*} Putting the $n$ scalar equations together gives \begin{align*} m\ddot q(t)=-\nabla V(q(t)). \end{align*} Thus the variational equation for kinetic minus potential energy recovers Newton's force law with conservative force $F(q)=-\nabla V(q)$. [/example]

This example is deliberately familiar: it shows that the geometric language is not replacing Newtonian mechanics, but reframing it in a way that survives beyond Cartesian coordinates. The next object records the principle that physical paths are selected by varying a number attached to the whole path.

## Why Variational Principles Come First

The next problem is to replace local force balance by a global rule for choosing paths. In many systems, the equation of motion is not introduced directly; it is derived from stationarity of an action functional. This is the bridge from [calculus of variations](/page/Calculus%20of%20Variations) to mechanics.

[definition: Lagrangian] A Lagrangian on a configuration space $Q$ is a smooth function $L: TQ \to \mathbb R$. [/definition]

A Lagrangian assigns an instantaneous value to each position-velocity pair. To compare entire candidate motions, the course needs a path-level quantity obtained by accumulating this value over time. This motivates the action functional.

[definition: Action Functional] Let $Q$ be a configuration space, let $I=[t_0,t_1]$, and let $L:TQ\to\mathbb R$ be a Lagrangian. For fixed endpoints $q_0,q_1\in Q$, let \begin{align*} \mathcal A(q_0,q_1)=\{q\in C^\infty(I,Q):q(t_0)=q_0,\ q(t_1)=q_1\}. \end{align*} The action functional is the map $S:\mathcal A(q_0,q_1)\to\mathbb R$ defined by \begin{align*} S[q] = \int_{t_0}^{t_1} L(q(t), \dot q(t))\,dt. \end{align*} [/definition]