Classical Mechanics

Prerequisites: Newtonian Mechanics. Full path (5 pages)

• Intuitive
• Concrete
• Abstract
• Formulations
• Roadmaps
• History

Intuitive

Classical mechanics is the theory of how bodies move when forces act on them. It covers a thrown ball, a swinging pendulum, a planet around the Sun and a spinning top. The basic claim is that if you know where every part of a system is and how fast it moves at one instant, and you know the forces, then the whole future and past of the system are fixed.

A good picture is a billiard table. Once the cue ball leaves the cue, its path and every collision that follows are determined by its initial position and velocity and by the rules for how the balls bounce off each other and off the cushions. Nothing random enters. In practice small errors in the initial data can grow quickly, which is why long billiard shots or weather forecasts are hard, but in principle the outcome is set.

Classical mechanics works when objects move much slower than light and when the typical actions (energy times time, or momentum times distance) are large compared with Planck's constant, so quantum effects are negligible. Outside that range it is replaced by special relativity and quantum mechanics, and both reduce to classical mechanics in the appropriate limit.

The same physics can be written in several ways. Newton describes motion with forces. The Lagrangian and Hamiltonian formulations describe it with energies and are listed below. They make the same predictions but make different problems easy.

Concrete

In the Newtonian formulation a system of $N$ point particles with masses $m_i$ and positions $\vec x_i(t)$ obeys Newton's second law $$m_i \ddot{\vec x}_i = \vec F_i(\vec x_1,\dots,\vec x_N,\dot{\vec x}_1,\dots,\dot{\vec x}_N,t), \qquad i = 1,\dots,N.$$ These are $3N$ second-order ordinary differential equations. Given all initial positions and velocities, they have a unique solution, at least for some time interval, when the forces are well behaved (for example locally Lipschitz continuous). This is the mathematical form of determinism.

If the forces come from a potential, $\vec F_i = -\nabla_i V$, we can describe the same system with the Lagrangian $L = T - V$, where $T = \sum_i \frac12 m_i \dot{\vec x}_i^2$, in any set of generalized coordinates $q^a$. The equations of motion are the Euler-Lagrange equations $$\frac{d}{dt}\frac{\partial L}{\partial \dot q^a} - \frac{\partial L}{\partial q^a} = 0 .$$ With the canonical momenta $p_a = \partial L/\partial \dot q^a$ and the Hamiltonian $H = p_a \dot q^a - L$ (a Legendre transformation) the same content becomes Hamilton's equations $$\dot q^a = \frac{\partial H}{\partial p_a}, \qquad \dot p_a = -\frac{\partial H}{\partial q^a},$$ which are first order in time.

Example: the harmonic oscillator. A mass $m$ on a spring of stiffness $k$ has $F = -kx$. Newton gives $m\ddot x = -kx$. The Lagrangian is $L = \frac12 m\dot x^2 - \frac12 k x^2$, and the Euler-Lagrange equation reproduces $m\ddot x + kx = 0$. With $p = m\dot x$ the Hamiltonian is $H = \frac{p^2}{2m} + \frac12 k x^2$, and Hamilton's equations read $\dot x = p/m$, $\dot p = -kx$. All three give $x(t) = A\cos(\omega t + \varphi)$ with $\omega = \sqrt{k/m}$. In phase space $(x,p)$ the trajectories are ellipses of constant $H$.

The choice of formulation is practical. Newton's equations are direct when the forces are known. The Lagrangian form handles constraints and curvilinear coordinates easily, and its symmetries give conservation laws through Noether's theorem. The Hamiltonian form is the starting point for canonical transformations, Hamilton-Jacobi theory, statistical mechanics and quantization.

• No-Nonsense Classical Mechanics by Jakob Schwichtenberg, chapter "Bird's-Eye View of Classical Mechanics"

Abstract

The configuration space of a mechanical system is a smooth manifold $Q$ of dimension $n$, the number of degrees of freedom. For $N$ free particles $Q = \mathbb{R}^{3N}$; for a rigid body with a fixed point $Q = SO(3)$.

In the Lagrangian formulation the state is a point of the tangent bundle $TQ$, and the dynamics is fixed by a function $L: TQ \to \mathbb{R}$ (possibly time dependent). Physical motions are the curves $q:[t_0,t_1] \to Q$ that make the action $S[q] = \int_{t_0}^{t_1} L(q,\dot q)\,dt$ stationary under variations with fixed endpoints. The fiber derivative $\mathbb{F}L: TQ \to T^*Q$, $(q,\dot q) \mapsto (q, \partial L/\partial \dot q)$, is the Legendre transformation. When it is a diffeomorphism ($L$ hyperregular, as for $L = \frac12 g_{ab}(q)\dot q^a \dot q^b - V(q)$ with a Riemannian metric $g$) the Lagrangian and Hamiltonian pictures are equivalent. When it is not, the system has constraints and one uses Dirac's constraint analysis.

In the Hamiltonian formulation the state space is the cotangent bundle $T^*Q$ with its canonical symplectic form $\omega = dq^a \wedge dp_a$ (sign conventions differ between books). More generally the phase space is any symplectic manifold $(P,\omega)$. A function $H$ defines a vector field $X_H$ by $$\iota_{X_H}\omega = dH$$ (again up to a convention dependent sign), whose integral curves are the solutions of Hamilton's equations. The Poisson bracket $\{f,g\} = \omega(X_f, X_g)$ makes $C^\infty(P)$ a Lie algebra, and the time evolution of any observable is $\dot f = \{f, H\}$. The flow of $X_H$ preserves $\omega$ and therefore the Liouville volume $\omega^n/n!$, which is Liouville's theorem.

Symmetries are described by a Lie group $G$ acting on $P$ by symplectomorphisms. When the action is Hamiltonian it has a momentum map $\mu: P \to \mathfrak{g}^*$, and if $H$ is $G$-invariant, $\mu$ is conserved along the flow. This is the Hamiltonian form of Noether's theorem. Reducing by the symmetry (Marsden-Weinstein reduction, $\mu^{-1}(c)/G_c$ with $G_c$ the stabilizer of $c$) yields a smaller symplectic manifold when $\mu$ is equivariant, $c$ is a regular value and $G_c$ acts freely and properly on $\mu^{-1}(c)$, which is how one treats rigid bodies or the reduction of the two-body problem.

The other formulations fit into this picture. Koopman-von Neumann mechanics represents the Hamiltonian flow by a one-parameter unitary group on $L^2(P,\omega^n/n!)$. Quantum mechanics replaces the commutative Poisson algebra $C^\infty(P)$ by a noncommutative operator algebra, with the commutator $\frac{1}{i\hbar}[\cdot,\cdot]$ playing the role of the Poisson bracket to leading order in $\hbar$.

Formulations

There are different ways of describing classical mechanics. Each has its individual strength and weaknesses, but in terms of observable predictions, they are all equivalent.

• Newtonian Mechanics $\leftrightarrow$ classical mechanics in real space
• Lagrangian Mechanics $\leftrightarrow$ classical mechanics in configuration space
• hamiltonian $\leftrightarrow$ classical mechanics in phase space
• Koopman-von-Neumann Mechanics $\leftrightarrow$ classical mechanics in Hilbert space

Roadmaps

Traditional Roadmap

Reading Recommendations

• The best book that gives a grand tour of classical mechanics is The Lazy Universe by Coopersmith

History

• A History of Mechanics by Dugas

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus
5. Newtonian Mechanics