api: YAML JSON TOON Database
version: 1.0.0
format: toon
dataset:
  id: 2744
  slug: classical-mechanics
  title: "Classical Mechanics: Laws of Motion and Energy"
  description: "Foundations of classical mechanics: Newton's laws, conservation laws (energy, momentum, angular momentum), Lagrangian and Hamiltonian mechanics, and oscillations."
  summary: "The foundational principles of classical mechanics: Newton's three laws, the conservation of energy, momentum and angular momentum, and the Lagrangian and Hamiltonian formulations, each with a statement, formula and significance. Extra entries add universal gravitation, the work-energy theorem, Hooke's law and Kepler's third law."
  use_when[2]: You need a statement and formula for a mechanics law with its significance.,You are choosing between a force-based and an energy-based approach to a problem.
  caveats[2]: "Classical mechanics is accurate for everyday speeds and sizes, but needs relativity near the speed of light and quantum mechanics at atomic scales.",Formulas are given in their usual vector or scalar form; check the symbol definitions before using them.
  related[4]: waves-and-sound,thermodynamics-laws,electromagnetism-basics,quantum-mechanics-basics
  entry_count: 13
  fields[4]{section,field,type,example}:
    principles,name,string,Newton's First Law (Inertia)
    principles,statement,string,"An object at rest stays at rest, and an object …"
    principles,formula,string,ΣF = 0 ⟹ dv/dt = 0
    principles,significance,string,Defines inertial reference frames; introduces c…
  category: Physics
  category_slug: physics
  tags: "physics,classical-mechanics,newton,conservation-laws,lagrangian,hamiltonian,oscillations"
  view_count: 52
  created_at: 1778695229
  updated_at: 1778695229
data:
  principles[13]{name,statement,formula,significance}:
    Newton's First Law (Inertia),"An object at rest stays at rest, and an object in motion stays in uniform motion, unless acted upon by a net external force",ΣF = 0 ⟹ dv/dt = 0,Defines inertial reference frames; introduces concept of force as cause of acceleration
    Newton's Second Law,The rate of change of momentum equals the applied force,F = ma = dp/dt,"Quantitative relationship between force, mass, and acceleration; foundation of dynamics"
    Newton's Third Law,"For every action, there is an equal and opposite reaction",F_AB = -F_BA,"Forces always occur in pairs; explains propulsion, collisions, and contact forces"
    Conservation of Energy,"Total energy in an isolated system is constant; energy transforms between kinetic, potential, thermal, etc.",E_total = KE + PE = constant (conservative systems),Universal principle; simplifies problem-solving by avoiding force analysis
    Conservation of Momentum,Total momentum of an isolated system is constant,p_total = Σm_i v_i = constant,"Applies even when energy is not conserved (inelastic collisions); fundamental symmetry (Noether: spatial translation)"
    Conservation of Angular Momentum,Total angular momentum of an isolated system is constant,L = r × p = constant (when τ = 0),"Explains planetary orbits, spinning ice skaters, gyroscopic stability; Noether: rotational symmetry"
    Lagrangian Mechanics,"Equations of motion derived from principle of least action: δ∫L dt = 0",L = T - V; d/dt(∂L/∂q̇) - ∂L/∂q = 0 (Euler-Lagrange equation),Coordinate-independent formulation; foundation for quantum mechanics and field theory
    Hamiltonian Mechanics,"Reformulation using generalized coordinates and momenta; energy function H(p,q)","H = Σp_i q̇_i - L; dq/dt = ∂H/∂p, dp/dt = -∂H/∂q",Phase space formulation; bridge to quantum mechanics (canonical quantization)
    Simple Harmonic Motion,"Restoring force proportional to displacement: F = -kx",x(t) = A cos(ωt + φ); ω = √(k/m); T = 2π√(m/k),"Universal approximation near stable equilibrium; models springs, pendulums, molecular vibrations"
    Law of Universal Gravitation,Every mass attracts every other with a force proportional to the product of the masses and inversely proportional to the square of the distance,F = G m1 m2 / r^2,Explains planetary orbits and everyday weight; superseded by general relativity in strong fields
    Work-Energy Theorem,The net work done on an object equals the change in its kinetic energy,W_net = delta KE = (1/2) m v2^2 - (1/2) m v1^2,Connects forces acting over distance to changes in speed without solving the equation of motion
    Hooke's Law,For small deformations the restoring force of a spring is proportional to its displacement,F = -k x,"Basis of simple harmonic motion and elastic materials, valid within the elastic limit"
    Kepler's Third Law,The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit,T^2 proportional to a^3,Lets orbital period and distance be derived from each other for bodies orbiting the same mass