Classical Mechanics

The foundation of physics describing motion and forces, from Newton's laws to the elegant formulations of Lagrange and Hamilton.

Classical mechanics is the physics of everyday motion — how forces, energy, and momentum dictate the paths of everything from thrown balls to orbiting planets. It is built in three layers: Newton’s force-based picture, the deeper energy-based formulations of Lagrange and Hamilton, and the modern geometric view that connects mechanics to chaos, quantum theory, and computation. Four ideas run through all of it:

Force changes motion. Newton’s $F = ma$ — objects keep their velocity unless a net force acts.
Symmetry conserves quantities. Energy, momentum, and angular momentum are conserved because of underlying symmetries (Noether’s theorem).
Action is extremized. A system follows the path that extremizes $S = \int L\,dt$.
Determinism has limits. Nonlinear systems can be chaotic — perfectly deterministic yet practically unpredictable.

Explore Classical Mechanics

The pages below build from the core force-based picture, through the energy-based and geometric formalisms, into the modern computational and nonlinear frontier, and finally to the applied subjects that classical mechanics feeds. Read them in order for a guided arc, or jump straight to the layer you need.

Core

Newtonian Mechanics & Conservation Laws

Newton's three laws, kinematics and dynamics, work and energy, conservation laws and Noether's theorem, rotational motion, and gravitation.

Oscillations & Waves

The simple and damped harmonic oscillator, coupled oscillators and normal modes, the wave equation, dispersion, and the first hints of nonlinearity.

Formalism

Lagrangian & Hamiltonian Mechanics

The principle of least action, Euler-Lagrange equations, generalized coordinates, Hamilton's equations, phase space, Poisson brackets, canonical transformations, and Hamilton-Jacobi theory.

Geometric Formalism

Symplectic geometry, phase-space flow, fiber bundles, geometric (Berry) phases, and the differential-forms language that unifies the formalisms.

Modern & Computational

Chaos & Nonlinear Dynamics

Nonlinear dynamics, sensitive dependence and Lyapunov exponents, KAM theory, the transition to chaos, and the frontiers of deterministic unpredictability.

Computational Methods

Symplectic and variational integrators, molecular dynamics, N-body methods, and the numerical analysis that keeps long simulations physically faithful.

Applications

Rigid Body Dynamics

The inertia tensor and principal axes, Euler's equations and Euler angles, the symmetric and asymmetric top, gyroscopic precession and nutation, and the tennis-racket theorem.

Thermodynamics

Where mechanical energy, work, and heat meet — the macroscopic laws that classical many-body motion ultimately obeys.

Fluid Mechanics

Continuum mechanics: Newton's laws applied to deformable matter, from the Euler and Navier-Stokes equations to turbulence.

Statistical Mechanics

Bridging Hamiltonian dynamics for enormous numbers of particles to the emergent laws of thermodynamics.

The Landscape of Classical Mechanics

Classical mechanics is not a single recipe but a family of equivalent viewpoints that grew more abstract and more powerful over three centuries. The map below shows how the three great formulations relate, what mathematical home each lives in, and where each one ultimately points — toward chaos, statistical mechanics, and quantum theory. Keep it in mind as a guide while reading: every section is a stop on this route.

graph TD
    N["Newtonian Mechanics<br/>F = ma<br/>(vectors, forces)"] --> L["Lagrangian Mechanics<br/>L = T - V<br/>(configuration space)"]
    L --> H["Hamiltonian Mechanics<br/>H = T + V<br/>(phase space)"]
    N -.->|same physics| L
    L -.->|same physics| H
    L --> NO["Noether's Theorem<br/>symmetry to conservation"]
    H --> HJ["Hamilton-Jacobi<br/>action as a field"]
    H --> CH["Chaos and<br/>nonlinear dynamics"]
    H --> SM["Statistical<br/>Mechanics"]
    HJ --> QM["Quantum Mechanics<br/>path integral"]
    classDef core fill:#e3f2fd,stroke:#1976d2,stroke-width:2px;
    classDef bridge fill:#fff3e0,stroke:#e65100,stroke-width:2px;
    class N,L,H core;
    class NO,HJ,CH,SM,QM bridge;

The Three Formulations at a Glance

All three describe the same physics — they predict identical motion — but each takes a different starting point and excels at different problems.

Aspect	Newtonian	Lagrangian	Hamiltonian
Central quantity	Force $\vec{F}$	Lagrangian $L = T - V$	Hamiltonian $H = T + V$
Variables	Positions, accelerations	Generalized coordinates $q_i, \dot{q}_i$	Coordinates and momenta $q_i, p_i$
Core equation	$\vec{F} = m\vec{a}$	$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0$	$\dot{q}_i = \frac{\partial H}{\partial p_i},\ \dot{p}_i = -\frac{\partial H}{\partial q_i}$
Handles constraints	Awkwardly (constraint forces)	Naturally (pick smart coordinates)	Naturally
Best for	Direct force problems, intuition	Complex/constrained systems, symmetries	Phase-space geometry, chaos, the bridge to QM
Mathematical home	Vectors in space	Configuration space (tangent bundle)	Phase space (cotangent bundle)

Which one should you reach for? Use Newton when forces are simple and you want physical intuition. Switch to Lagrange the moment constraints appear (a bead on a wire, a double pendulum) — choosing the right generalized coordinates makes constraint forces vanish. Move to Hamilton when you care about the structure of all possible motions, conserved quantities, statistical mechanics, or the route to quantum theory.

Key Takeaways

Three formulations, one physics. Newtonian, Lagrangian, and Hamiltonian mechanics are equivalent, but each makes different problems easy and reveals different structure.
Conservation laws come from symmetry. Noether’s theorem ties time-translation to energy, space-translation to momentum, and rotation to angular momentum.
Phase space is the natural arena. Hamiltonian dynamics lives in $(q,p)$ phase space, where the symplectic structure is preserved by the flow.
Action is fundamental. The principle of least action underlies all of physics and is the bridge to quantum mechanics via the path integral.
Determinism is not predictability. Chaotic systems obey exact laws yet diverge exponentially, limiting long-term prediction (the butterfly effect).
It is a limiting case. Classical mechanics emerges from quantum mechanics ($\hbar \to 0$) and relativity ($v \ll c$); know where it breaks down.