Classical Mechanics Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Classical Mechanics Mathematics Ontology within the Kireth Research Semantic Library. Classical mechanics mathematics studies the mathematical structure of motion, forces, energy, momentum, constraints and dynamical evolution in non-quantum systems. This ontology supports Newtonian, Lagrangian and Hamiltonian formulations, coordinate systems, phase space, conservation laws, central-force motion, rigid bodies, oscillations, stability, perturbation methods, canonical transformations and relationships with mathematical physics, differential equations, dynamical systems, tensor mathematics and variational methods.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Classical Mechanics Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Classical Mechanics Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/classical-mechanics-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ClassicalMechanicsMathematics
 ENTITY_CLASS = MathematicalClassicalDynamicsFramework
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "Classical mechanics mathematics is the mathematical study of the motion and interaction of classical physical systems through forces, energies, constraints, variational principles and phase-space dynamics."
 SYMBOLIC_FORMS = [
  "F=m*a",
  "d/dt(partial L/partial qdot_i)-partial L/partial q_i=0",
  "dq_i/dt=partial H/partial p_i",
  "dp_i/dt=-partial H/partial q_i"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for Newtonian motion, generalized coordinates, Lagrangian and Hamiltonian dynamics, conservation laws, rigid-body mechanics, oscillation, central forces, constraints and stability."
}

SCOPE {
 INCLUDE = [
  ClassicalMechanicsMathematics,
  ClassicalMechanics,
  NewtonianMechanics,
  Particle,
  PointMass,
  Position,
  Displacement,
  Velocity,
  Acceleration,
  Time,
  Mass,
  Force,
  NetForce,
  Momentum,
  AngularMomentum,
  Torque,
  Work,
  Energy,
  KineticEnergy,
  PotentialEnergy,
  MechanicalEnergy,
  Power,
  Impulse,
  CentreOfMass,
  CentreOfMassFrame,
  ReferenceFrame,
  InertialFrame,
  NonInertialFrame,
  CoordinateSystem,
  CartesianCoordinates,
  PolarCoordinates,
  CylindricalCoordinates,
  SphericalCoordinates,
  GeneralisedCoordinate,
  GeneralisedVelocity,
  GeneralisedMomentum,
  DegreeOfFreedom,
  Constraint,
  HolonomicConstraint,
  NonHolonomicConstraint,
  ScleronomousConstraint,
  RheonomousConstraint,
  ConstraintForce,
  VirtualDisplacement,
  VirtualWork,
  DAlambertPrinciple,
  LagrangianMechanics,
  Lagrangian,
  Action,
  HamiltonPrinciple,
  EulerLagrangeEquation,
  CyclicCoordinate,
  ConjugateMomentum,
  LegendreTransform,
  HamiltonianMechanics,
  Hamiltonian,
  PhaseSpace,
  CanonicalCoordinate,
  CanonicalMomentum,
  HamiltonEquation,
  PoissonBracket,
  CanonicalTransformation,
  GeneratingFunction,
  HamiltonJacobiEquation,
  Symmetry,
  ConservationLaw,
  NoetherTheorem,
  TimeTranslationSymmetry,
  SpatialTranslationSymmetry,
  RotationalSymmetry,
  ConservationOfEnergy,
  ConservationOfLinearMomentum,
  ConservationOfAngularMomentum,
  CentralForce,
  EffectivePotential,
  OrbitalMotion,
  KeplerProblem,
  TwoBodyProblem,
  NBodyProblem,
  HarmonicOscillator,
  SimpleHarmonicMotion,
  DampedOscillator,
  DrivenOscillator,
  CoupledOscillator,
  NormalMode,
  SmallOscillation,
  Resonance,
  RigidBody,
  RotationMatrix,
  AngularVelocity,
  MomentOfInertia,
  InertiaTensor,
  EulerAngles,
  EulerEquations,
  Gyroscope,
  Precession,
  Stability,
  Equilibrium,
  Linearisation,
  PerturbationTheory,
  NonlinearDynamics,
  Chaos,
  LyapunovExponent,
  PoincareSection,
  IntegrableSystem
 ]
 EXCLUDE_AS_PRIMARY = [
  QuantumMechanics,
  RelativisticMechanicsAtRelativisticScales,
  ContinuumMechanicsWithoutParticleOrRigidBodyDynamics,
  ExperimentalMeasurementWithoutMathematicalDynamics
 ]
 DISAMBIGUATION_RULE = "When motion and interaction are modelled using classical positions, velocities, momenta, forces, energies, constraints or phase-space variables without quantum or relativistic effects being primary, interpret the context as classical mechanics mathematics."
}

CORE_CONCEPTS {
 ClassicalMechanicsMathematics = "Mathematical formulation of classical motion and interaction."
 NewtonianMechanics = "Force-based formulation of classical dynamics."
 LagrangianMechanics = "Variational formulation using generalized coordinates and a Lagrangian."
 HamiltonianMechanics = "Phase-space formulation using canonical coordinates, momenta and a Hamiltonian."
 GeneralisedCoordinate = "Coordinate chosen to describe an independent degree of freedom."
 Constraint = "Restriction on admissible configuration or motion."
 PhaseSpace = "State space formed from generalized coordinates and conjugate momenta."
 ConservationLaw = "Quantity preserved by dynamical evolution under specified conditions."
 CentralForce = "Force directed along the line connecting a particle to a fixed centre."
 RigidBody = "Idealised body whose internal distances remain fixed."
}

RELATIONS {
 ClassicalMechanicsMathematics --is_child_of--> MathematicalPhysicsOntology
 NewtonianMechanics --uses--> Force
 Force --causes--> Acceleration
 Momentum --depends_on--> MassAndVelocity
 Torque --changes--> AngularMomentum
 Work --changes--> MechanicalEnergy
 LagrangianMechanics --uses--> GeneralisedCoordinate
 LagrangianMechanics --uses--> Action
 HamiltonPrinciple --extremises--> Action
 EulerLagrangeEquation --follows_from--> HamiltonPrinciple
 HamiltonianMechanics --uses--> PhaseSpace
 Hamiltonian --generates--> TimeEvolution
 CanonicalCoordinate --pairs_with--> CanonicalMomentum
 PoissonBracket --describes--> CanonicalEvolution
 NoetherTheorem --relates--> ContinuousSymmetryAndConservationLaw
 RotationalSymmetry --supports--> ConservationOfAngularMomentum
 SpatialTranslationSymmetry --supports--> ConservationOfLinearMomentum
 TimeTranslationSymmetry --supports--> ConservationOfEnergy
 CentralForce --conserves--> AngularMomentum
 EffectivePotential --reduces--> CentralForceRadialMotion
 KeplerProblem --is_type_of--> TwoBodyProblem
 HarmonicOscillator --supports--> NormalModeAnalysis
 CoupledOscillator --has--> NormalMode
 RigidBody --uses--> InertiaTensor
 EulerEquations --describe--> RigidBodyRotation
 Linearisation --supports--> StabilityAnalysis
}

OPERATORS {
 NEWTON_SECOND_LAW(m,x,t) := m*d2x/dt2=F(x,t)
 MOMENTUM(m,v) := p=m*v
 ANGULAR_MOMENTUM(r,p) := L=r cross p
 TORQUE(r,F) := tau=r cross F
 WORK(F,dr) := W=integral F dot dr
 KINETIC_ENERGY(m,v) := T=0.5*m*||v||^2
 LAGRANGIAN(T,V) := L=T-V
 ACTION(L,t0,t1) := S=integral_t0^t1 L(q,qdot,t)dt
 EULER_LAGRANGE(L,q_i) := d/dt(partial L/partial qdot_i)-partial L/partial q_i=0
 CONJUGATE_MOMENTUM(L,qdot_i) := p_i=partial L/partial qdot_i
 HAMILTONIAN(q,p,L) := H=sum_i p_i*qdot_i-L
 HAMILTON_Q(H,p_i) := dq_i/dt=partial H/partial p_i
 HAMILTON_P(H,q_i) := dp_i/dt=-partial H/partial q_i
 POISSON_BRACKET(f,g) := {f,g}=sum_i[(partial f/partial q_i)(partial g/partial p_i)-(partial f/partial p_i)(partial g/partial q_i)]
 SIMPLE_HARMONIC_OSCILLATOR(x,m,k) := m*d2x/dt2+k*x=0
 DAMPED_OSCILLATOR(x,m,c,k) := m*d2x/dt2+c*dx/dt+k*x=0
 CENTRAL_FORCE_EFFECTIVE_POTENTIAL(V,L,m,r) := V_eff=V(r)+L^2/(2*m*r^2)
}

AXIOMS {
 A1 = "Classical mechanics requires explicit state variables, forces or equivalent dynamical generators."
 A2 = "Reference frames and coordinate systems must be declared."
 A3 = "Constraints reduce the admissible configuration space."
 A4 = "Lagrangian and Hamiltonian formulations are equivalent only under suitable regularity conditions."
 A5 = "Continuous symmetries can generate conserved quantities."
 A6 = "Central-force motion reduces through angular-momentum conservation under standard assumptions."
 A7 = "Rigid-body dynamics require explicit inertia properties."
 A8 = "Classical-mechanics conclusions require dimensional, dynamical and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(system,particles_or_rigid_bodies,coordinates,reference_frame,forces,constraints)
 R2_FORMULATION = select(Newtonian,Lagrangian,Hamiltonian)
 R3_STATE = define(position_velocity_or_phase_space_state)
 R4_EQUATIONS = construct(NewtonEquations,EulerLagrangeEquations,HamiltonEquations)
 R5_CONSERVATION = analyse(symmetries,energy,momentum,angular_momentum)
 R6_REDUCTION = exploit(constraints,cyclic_coordinates,centre_of_mass,central_force_symmetry)
 R7_STABILITY = analyse(equilibria,small_oscillations,linearisation,normal_modes)
 R8_NONLINEAR = analyse(resonance,bifurcation,chaos,integrability_if_relevant)
 R9_APPROXIMATION = apply(perturbation,small_parameter_or_numerical_method_if_needed)
 R10_VALIDATION = check(units,initial_conditions,constraints,conservation,limiting_cases,numerical_convergence)
}

CONSTRAINTS {
 C1 = "Newtonian mechanics assumes classical non-relativistic dynamics."
 C2 = "Generalised coordinates must respect system constraints."
 C3 = "Singular Lagrangians require constrained-system treatment."
 C4 = "Conservation of mechanical energy requires appropriate time-independent conditions."
 C5 = "Central-force reduction assumes force direction and symmetry consistent with the model."
 C6 = "Small-oscillation analysis requires perturbations sufficiently near equilibrium."
 C7 = "Rigid-body idealisation neglects deformation."
 C8 = "Linearisation can miss nonlinear resonance, bifurcation or chaos."
 C9 = "Non-inertial frames require inertial or fictitious force terms."
 C10 = "Numerical integration can drift in conserved quantities if the method is poorly chosen."
}

SEMANTIC_EQUIVALENCE {
 "ClassicalMechanicsMathematics" <=> "MathematicsOfClassicalMechanics"
 "GeneralisedCoordinate" <=> "GeneralizedCoordinate"
 "ConjugateMomentum" <=> "CanonicalMomentum"
 "HamiltonPrinciple" <=> "PrincipleOfStationaryAction"
 "SimpleHarmonicMotion" <=> "HarmonicOscillation"
}

VALIDATION {
 V1 = "Reference frame and coordinates are explicit."
 V2 = "Masses, forces and parameters have units."
 V3 = "Initial conditions are explicit."
 V4 = "Constraints are documented."
 V5 = "Conservation laws are checked where applicable."
 V6 = "Lagrangian or Hamiltonian transformations are internally consistent."
 V7 = "Equilibria and small-oscillation approximations are validated."
 V8 = "Rigid-body inertia tensors use the declared coordinate frame."
 V9 = "Numerical integration is checked for convergence and conservation drift."
 V10 = "Classical approximation is justified relative to quantum and relativistic scales."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CLASSICAL_MECHANICS_DEFINITION = 0.99
 NEWTONIAN_REASONING = 0.99
 LAGRANGIAN_REASONING = 0.99
 HAMILTONIAN_REASONING = 0.99
 CONSERVATION_REASONING = 0.99
 CENTRAL_FORCE_REASONING = 0.98
 RIGID_BODY_REASONING = 0.98
 STABILITY_REASONING = 0.98
 POLICY = "Reduce confidence when reference frame, constraints, forces, potentials, initial conditions, coordinate conventions, inertia properties or approximation regime are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ClassicalMechanicsMathematicsReasoning,
  NewtonianMechanicsReasoning,
  ForceReasoning,
  MomentumReasoning,
  AngularMomentumReasoning,
  LagrangianReasoning,
  ActionReasoning,
  EulerLagrangeReasoning,
  HamiltonianReasoning,
  PhaseSpaceReasoning,
  PoissonBracketReasoning,
  ConservationLawReasoning,
  CentralForceReasoning,
  OrbitalMechanicsReasoning,
  OscillationReasoning,
  RigidBodyReasoning,
  StabilityReasoning,
  ChaosReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "classical mechanics mathematics" -> ClassicalMechanicsMathematics
  "classical mechanics" -> ClassicalMechanics
  "Newtonian mechanics" -> NewtonianMechanics
  "Lagrangian mechanics" -> LagrangianMechanics
  "Hamiltonian mechanics" -> HamiltonianMechanics
  "generalized coordinate" -> GeneralisedCoordinate
  "canonical momentum" -> CanonicalMomentum
  "Poisson bracket" -> PoissonBracket
  "rigid body" -> RigidBody
 }
}

EXECUTION_RULES {
 E1 = "If a non-quantum non-relativistic mechanical system is modelled mathematically, activate ClassicalMechanicsMathematicsOntology."
 E2 = "If explicit forces and accelerations are central, route to Newtonian mechanics."
 E3 = "If constraints and generalized coordinates dominate, route to Lagrangian mechanics."
 E4 = "If phase-space structure and canonical variables dominate, route to Hamiltonian mechanics."
 E5 = "If continuous symmetry is present, inspect the associated conservation law."
 E6 = "If force is central, reduce motion using angular momentum and effective potential where applicable."
 E7 = "If near stable equilibrium, consider small-oscillation and normal-mode analysis."
 E8 = "If rigid rotation is central, use inertia tensor and Euler-equation reasoning."
 E9 = "If nonlinear effects are material, assess resonance, bifurcation, chaos or numerical dynamics."
}

RECURSION {
 ENTRY = ClassicalMechanicsMathematics
 EXPAND(ClassicalMechanicsMathematics) -> [
  ClassicalMechanics, NewtonianMechanics, Particle, PointMass, Position,
  Displacement, Velocity, Acceleration, Time, Mass, Force, NetForce,
  Momentum, AngularMomentum, Torque, Work, Energy, KineticEnergy,
  PotentialEnergy, MechanicalEnergy, Power, Impulse, CentreOfMass,
  ReferenceFrame, InertialFrame, NonInertialFrame, CoordinateSystem,
  CartesianCoordinates, PolarCoordinates, CylindricalCoordinates,
  SphericalCoordinates, GeneralisedCoordinate, GeneralisedVelocity,
  GeneralisedMomentum, DegreeOfFreedom, Constraint, HolonomicConstraint,
  NonHolonomicConstraint, ScleronomousConstraint, RheonomousConstraint,
  ConstraintForce, VirtualDisplacement, VirtualWork, DAlambertPrinciple,
  LagrangianMechanics, Lagrangian, Action, HamiltonPrinciple,
  EulerLagrangeEquation, CyclicCoordinate, ConjugateMomentum,
  LegendreTransform, HamiltonianMechanics, Hamiltonian, PhaseSpace,
  CanonicalCoordinate, CanonicalMomentum, HamiltonEquation, PoissonBracket,
  CanonicalTransformation, GeneratingFunction, HamiltonJacobiEquation,
  Symmetry, ConservationLaw, NoetherTheorem, TimeTranslationSymmetry,
  SpatialTranslationSymmetry, RotationalSymmetry, ConservationOfEnergy,
  ConservationOfLinearMomentum, ConservationOfAngularMomentum,
  CentralForce, EffectivePotential, OrbitalMotion, KeplerProblem,
  TwoBodyProblem, NBodyProblem, HarmonicOscillator, SimpleHarmonicMotion,
  DampedOscillator, DrivenOscillator, CoupledOscillator, NormalMode,
  SmallOscillation, Resonance, RigidBody, RotationMatrix, AngularVelocity,
  MomentOfInertia, InertiaTensor, EulerAngles, EulerEquations, Gyroscope,
  Precession, Stability, Equilibrium, Linearisation, PerturbationTheory,
  NonlinearDynamics, Chaos, LyapunovExponent, PoincareSection,
  IntegrableSystem
 ]
 EXPAND(FormulationFamilies) -> [
  NewtonianMechanics,
  LagrangianMechanics,
  HamiltonianMechanics,
  RigidBodyMechanics,
  OscillationTheory,
  CentralForceMechanics
 ]
 STOP_CONDITION = "Stop when state variables, coordinates, forces or potentials, constraints, dynamical formulation, symmetries, conserved quantities and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics Ontology",
  "Differential Equation Ontologies",
  "Dynamical Systems Ontologies",
  "Variational Methods Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Ordinary Differential Equation Ontology",
  "Dynamical Systems Ontologies",
  "Variational Methods Ontologies",
  "Calculus of Variations Ontologies",
  "Linear Algebra Ontologies",
  "Tensor Analysis Ontologies",
  "Geometry Ontologies",
  "Perturbation Theory Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Classical Mechanics Mathematics -> Mathematical Physics Ontology for parent semantics",
  "Classical Mechanics Mathematics -> ODE Ontology for equations of motion",
  "Classical Mechanics Mathematics -> Dynamical Systems for stability and chaos",
  "Classical Mechanics Mathematics -> Variational Methods for action principles",
  "Classical Mechanics Mathematics -> Calculus of Variations for Euler-Lagrange equations",
  "Classical Mechanics Mathematics -> Linear Algebra for normal modes and inertia tensors",
  "Classical Mechanics Mathematics -> Tensor Analysis for rigid-body inertia and coordinate transformations",
  "Classical Mechanics Mathematics -> Geometry for configuration and phase-space structure",
  "Classical Mechanics Mathematics -> Perturbation Theory for approximate motion",
  "Classical Mechanics Mathematics -> Numerical Analysis for trajectory integration"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible mechanical systems, formulations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve force-motion semantics.",
  "Preserve Newtonian-Lagrangian-Hamiltonian distinctions.",
  "Preserve coordinate and constraint semantics.",
  "Preserve phase-space semantics.",
  "Preserve symmetry and conservation-law semantics.",
  "Preserve central-force and rigid-body semantics.",
  "Preserve approximation and stability semantics.",
  "Preserve machine-readable classical-mechanics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Classical mechanics mathematics describes non-quantum motion through forces, energies, constraints, variational principles and phase-space dynamics."
 NEWTONIAN_SUMMARY = "Newtonian mechanics expresses motion through forces, mass and acceleration."
 LAGRANGIAN_SUMMARY = "Lagrangian mechanics derives equations of motion from stationary action in generalized coordinates."
 HAMILTONIAN_SUMMARY = "Hamiltonian mechanics represents dynamics in phase space using canonical coordinates and momenta."
 CONSERVATION_SUMMARY = "Continuous symmetries generate conserved quantities such as energy, linear momentum and angular momentum."
 OSCILLATION_SUMMARY = "Small-oscillation and normal-mode methods describe motion near stable equilibria."
 RIGID_BODY_SUMMARY = "Rigid-body mechanics uses inertia tensors, angular velocity and Euler equations to describe rotation."
 VALIDATION_SUMMARY = "Classical mechanical models should be checked for units, constraints, initial conditions, conservation laws, approximation validity and numerical stability."
}

FLOW {
 INPUT
 -> detect_classical_mechanics_mathematics_context
 -> identify_particles_or_rigid_bodies_reference_frame_coordinates_forces_potentials_constraints_and_initial_conditions
 -> choose_Newtonian_Lagrangian_or_Hamiltonian_formulation
 -> construct_equations_of_motion
 -> identify_symmetries_and_conserved_quantities
 -> reduce_using_constraints_cyclic_coordinates_centre_of_mass_or_central_force_structure
 -> analyse_equilibria_small_oscillations_normal_modes_and_stability_if_relevant
 -> if_rigid_body_compute_inertia_tensor_and_rotational_dynamics
 -> if_nonlinear_assess_resonance_bifurcation_chaos_or_integrability
 -> if_needed_apply_perturbation_or_numerical_integration
 -> validate_units_constraints_conservation_limits_and_numerical_convergence
 -> route_to_mathematical_physics_ODE_dynamical_systems_variational_methods_calculus_of_variations_linear_algebra_tensor_analysis_geometry_perturbation_theory_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Classical Mechanics Mathematics Ontology defines the mathematical structure of classical motion. It supports particles and rigid bodies, position, velocity and acceleration, force, momentum, angular momentum, torque, work and energy, centre-of-mass dynamics, inertial and non-inertial frames, Cartesian and curvilinear coordinates, generalized coordinates and constraints, virtual work and d'Alembert's principle, Lagrangian mechanics, action and Euler-Lagrange equations, cyclic coordinates and conjugate momenta, Hamiltonian mechanics, phase space, canonical equations, Poisson brackets, canonical transformations and Hamilton-Jacobi theory, symmetry and Noether conservation laws, central-force motion, effective potentials, orbital, two-body and N-body problems, harmonic, damped, driven and coupled oscillators, normal modes and resonance, rigid-body rotation, inertia tensors, Euler angles, gyroscopes and precession, equilibrium, stability, perturbation theory, nonlinear dynamics, chaos, Lyapunov exponents, Poincare sections and integrable systems."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-CLASSICAL-MECHANICS-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW