Newtonian and Lagrangian/Hamiltonian mechanics: laws of motion, conservation laws, gravitation. Valid in the non-relativistic, non-quantum regime.
classical-mechanics
Newton's first law (inertia)
A body at rest remains at rest and a body in uniform motion remains in uniform motion unless acted on by a net external force.
Newton's second law (F=ma)
The net force on a body equals the time derivative of its momentum; for constant mass, F = ma.
Newton's third law (action-reaction)
Forces between two bodies are equal in magnitude and opposite in direction.
Conservation of energy
Total energy of an isolated system is invariant in time. Via Noether's theorem this follows from time-translation symmetry of the…
Conservation of linear momentum
Total momentum of an isolated system is invariant in time; follows from spatial-translation symmetry (Noether).
Conservation of angular momentum
Total angular momentum of an isolated system is invariant in time; follows from rotational symmetry (Noether).
Lagrangian mechanics (principle of stationary action)
Trajectories of a mechanical system extremize the action S = ∫L dt, with L = T − V. Yields the Euler-Lagrange equations.
Newton's law of universal gravitation
Every point mass attracts every other with a force along the line joining them, proportional to the product of masses and inversely…
Hamiltonian mechanics
Reformulation of classical mechanics on phase space (q, p) via H(q,p,t); canonical equations q̇ = ∂H/∂p, ṗ = −∂H/∂q. Natural setting for…
Poisson bracket
Bilinear antisymmetric operation {f,g} = Σ (∂f/∂q_i ∂g/∂p_i − ∂f/∂p_i ∂g/∂q_i) on phase-space functions. Encodes symplectic structure and…
Canonical transformation
Phase-space coordinate change (q,p) ↦ (Q,P) preserving Hamilton's equations (equivalently, preserving the symplectic 2-form dp∧dq).
Hamilton–Jacobi equation (classical mechanics)
Partial-differential equation for Hamilton's principal function S(q,t): ∂S/∂t + H(q, ∂S/∂q, t) = 0. Generates canonical transformation to…
Noether's theorem
Every continuous symmetry of a Lagrangian (or action) yields a conserved current. Time-translation → energy, spatial-translation →…
Rigid-body dynamics
Mechanics of rigid bodies: Euler's equations in the body frame, inertia tensor, principal axes, nutation/precession. Underlies gyroscopes…
Euler's equations (rigid body)
The three coupled ODEs governing angular velocity in the principal-axis body frame of a torque-free or torqued rigid body.
Small oscillations (normal modes)
Linearisation around equilibrium: the quadratic Lagrangian ½(M ẋ·ẋ − K x·x) diagonalises in normal modes x_k = a_k cos(ω_k t + φ_k).
Rotating reference frame
In a frame rotating with Ω, Newton's laws acquire centrifugal and Coriolis pseudoforces: a_inertial = a_rot + 2Ω×v + Ω×(Ω×r) + Ω̇×r.
d'Alembert's principle
Virtual work of constraint forces vanishes: Σ(F_i - m_i a_i)·δr_i = 0. Foundation for Lagrangian mechanics from Newton's laws.
Routhian reduction
Mixed Lagrangian/Hamiltonian for cyclic coordinates: R = Σ p_c q̇_c - L. Eliminates ignorable coords while keeping others in Lagrangian…
Maupertuis–Euler principle
δ∫p·dq = 0 at fixed energy (abbreviated action). Geodesic reformulation of trajectories on configuration manifold with Jacobi metric.
Arnold diffusion
In n≥3 DOF near-integrable systems, trajectories can wander along resonance web despite KAM tori. Exponentially slow but generic.
Nekhoroshev estimates
For analytic near-integrable Hamiltonians, actions drift is bounded by exp(-1/ε^a) over times exp(1/ε^b). Exponentially long stability.
Symplectic 2-form ω
Phase space carries closed non-degenerate 2-form ω = dp_i ∧ dq_i; Hamiltonian flow preserves ω (Liouville). Basis of symplectic geometry.
Lindstedt–Poincaré perturbation
Expand action-angle variables in ε; enforce removal of secular terms to obtain uniformly valid expansions of quasi-periodic motion.
Poincaré recurrence theorem
In measure-preserving dynamics on finite-measure space, almost every trajectory returns arbitrarily close to its initial state infinitely…
Euler rigid body equations
I_1 ω̇_1 = (I_2 - I_3) ω_2 ω_3 + τ_1 (cyc). Torque-free rotation of asymmetric top: tennis-racket theorem, intermediate-axis instability.
Precession and nutation of tops
Heavy symmetric top: steady precession Ω = Mgℓ/(I_3 ω_3); nutation superposed. Spin-stabilization of gyroscopes.
Foucault pendulum
Plane of oscillation rotates at ω sin(latitude) due to Coriolis force. Earth's rotation demonstration (1851).
Virial theorem (classical)
For bounded motion with U ~ r^n: 2⟨T⟩ = n⟨U⟩. Gravitational: 2⟨T⟩+⟨U⟩=0; applies to stellar clusters, galaxy clusters.
Restricted three-body problem
Test particle in potential of two circularly orbiting masses; Jacobi integral; Lagrange points L1–L5; Hill sphere; zero-velocity curves.
Lagrange points L1–L5
Five equilibrium points in rotating frame; L4/L5 stable if m_2/(m_1+m_2) < 1/26. Trojan asteroids, JWST at L2.
Tisserand parameter
T_J = a_J/a + 2√((1-e²) a/a_J) cos i — quasi-invariant after Jupiter flybys. Used to classify comets and asteroids.
Tidal locking and dissipation
Tidal torques synchronize rotation to orbital period (Moon, Pluto–Charon); Q-factor controls timescale; orbital evolution of satellites.
Sensitive dependence & Lyapunov exponents
|δz(t)| ~ |δz(0)| e^(λt); positive maximal λ signals deterministic chaos. Henon–Heiles, double pendulum.
Liouville–Arnold integrability
2n-DOF Hamiltonian with n Poisson-commuting integrals in involution admits action-angle variables; phase-space foliates into invariant tori.
Marsden–Weinstein symplectic reduction (classical-mechanics)
Classical-mechanical application of L0 Marsden–Weinstein symplectic reduction. A G-symmetry with equivariant moment map J reduces (T^*Q,…
Birkhoff normal form at an equilibrium (classical-mechanics)
Classical-mechanical application of L0 Williamson normal form. Near a non-resonant elliptic equilibrium of a smooth Hamiltonian,…