Symplectic Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of symplectic geometry within the Kireth Research Semantic Library. Symplectic geometry studies even-dimensional smooth manifolds equipped with a closed nondegenerate differential 2-form. It provides the geometric language underlying Hamiltonian mechanics, canonical transformations, phase space, Poisson brackets, symplectomorphisms, Lagrangian submanifolds, Hamiltonian vector fields, moment maps and geometric quantisation. This ontology supports symplectic manifolds, symplectic forms, Darboux coordinates, Hamiltonian flows, canonical relations, cotangent bundles, symplectic reduction, contact boundaries and relationships with differential geometry, topology, mechanics and advanced geometry.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Symplectic Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Advanced Geometry / Symplectic Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/advanced-geometry/symplectic-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SymplecticGeometry
 SYMBOLIC_FORMS = [
  "(M,omega)",
  "d omega = 0",
  "omega nondegenerate",
  "i_XH omega = dH"
 ]
 ENTITY_CLASS = DifferentialGeometricStructure
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Symplectic geometry is the study of smooth even-dimensional manifolds equipped with a closed nondegenerate differential 2-form."
 SEMANTIC_ROLE = "Provides the geometric framework for Hamiltonian dynamics, canonical transformations, phase space, Poisson structures, Lagrangian geometry and symplectic reduction."
}

SCOPE {
 INCLUDE = [
  SymplecticGeometry,
  SymplecticManifold,
  SymplecticForm,
  ClosedTwoForm,
  NondegenerateTwoForm,
  DarbouxTheorem,
  DarbouxCoordinates,
  Symplectomorphism,
  HamiltonianFunction,
  HamiltonianVectorField,
  HamiltonianFlow,
  PhaseSpace,
  CanonicalTransformation,
  PoissonBracket,
  PoissonTensor,
  LagrangianSubmanifold,
  IsotropicSubmanifold,
  CoisotropicSubmanifold,
  SymplecticOrthogonal,
  CotangentBundle,
  CanonicalOneForm,
  CanonicalSymplecticForm,
  MomentMap,
  HamiltonianGroupAction,
  SymplecticReduction,
  MarsdenWeinsteinReduction,
  ExactSymplecticForm,
  LiouvilleVectorField,
  ContactTypeBoundary
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralRiemannianGeometryWithoutSymplecticStructure,
  PoissonGeometryWithoutNondegeneracyUnlessExplicit,
  ContactGeometryWithoutSymplectisationContext,
  ClassicalMechanicsWithoutGeometricFormulation
 ]

 DISAMBIGUATION_RULE =
  "When a smooth even-dimensional manifold carries a closed nondegenerate 2-form, or when the context concerns Hamiltonian vector fields, canonical transformations, Lagrangian submanifolds or symplectic reduction, interpret it as symplectic geometry."
}

CORE_CONCEPTS {

 SymplecticGeometry {
  definition = "The study of manifolds carrying symplectic forms and the transformations preserving them."
 }

 SymplecticManifold {
  definition = "A smooth manifold M equipped with a symplectic form omega."
 }

 SymplecticForm {
  symbol = "omega"
  definition = "A closed nondegenerate differential 2-form."
 }

 ClosedTwoForm {
  condition = "d omega = 0"
 }

 NondegenerateTwoForm {
  definition = "For every nonzero tangent vector v there exists w such that omega(v,w) != 0."
 }

 Symplectomorphism {
  definition = "A diffeomorphism phi satisfying phi^*omega_Y = omega_X."
 }

 HamiltonianFunction {
  symbol = "H"
  definition = "A smooth real-valued function generating a Hamiltonian vector field."
 }

 HamiltonianVectorField {
  symbol = "X_H"
  definition = "The unique vector field determined by i_XH omega = dH, up to adopted sign convention."
 }

 PoissonBracket {
  definition = "A bilinear bracket induced by the symplectic structure on smooth functions."
 }

 LagrangianSubmanifold {
  definition = "A half-dimensional submanifold on which the symplectic form restricts to zero."
 }

 IsotropicSubmanifold {
  definition = "A submanifold whose tangent spaces are contained in their symplectic orthogonals."
 }

 CoisotropicSubmanifold {
  definition = "A submanifold whose symplectic orthogonal is contained in its tangent space."
 }

 MomentMap {
  definition = "A map encoding Hamiltonians associated with a Hamiltonian Lie-group action."
 }

 SymplecticReduction {
  definition = "A construction producing a lower-dimensional symplectic space from a level set modulo symmetry."
 }
}

RELATIONS {
 SymplecticGeometry --is_branch_of--> AdvancedGeometry
 SymplecticGeometry --is_branch_of--> DifferentialGeometry
 SymplecticManifold --has--> SymplecticForm
 SymplecticForm --is--> ClosedTwoForm
 SymplecticForm --is--> NondegenerateTwoForm
 SymplecticForm --forces--> EvenDimension
 Symplectomorphism --preserves--> SymplecticForm
 HamiltonianFunction --generates--> HamiltonianVectorField
 HamiltonianVectorField --generates--> HamiltonianFlow
 HamiltonianFlow --consists_of--> Symplectomorphisms
 SymplecticForm --induces--> PoissonBracket
 LagrangianSubmanifold --is_special_case_of--> IsotropicSubmanifold
 CotangentBundle --carries--> CanonicalSymplecticForm
 HamiltonianGroupAction --admits--> MomentMap
 MomentMap --supports--> SymplecticReduction
 ContactTypeBoundary --relates_to--> LiouvilleStructure
}

OPERATORS {
 IS_SYMPLECTIC_FORM(omega,M) :=
  d(omega)=0
  AND omega_is_nondegenerate

 IS_SYMPLECTIC_MANIFOLD(M,omega) :=
  M_is_smooth
  AND IS_SYMPLECTIC_FORM(omega,M)

 SYMPLECTIC_ORTHOGONAL(W,omega) :=
  {v : omega(v,w)=0 for_all_w_in_W}

 HAMILTONIAN_VECTOR_FIELD(H,omega) :=
  unique_X_H_such_that i_X_H(omega)=dH
  under_selected_sign_convention

 POISSON_BRACKET(f,g,omega) :=
  omega(X_f,X_g)
  with sign_matching_adopted convention

 IS_SYMPLECTOMORPHISM(phi,omega_X,omega_Y) :=
  phi_is_diffeomorphism
  AND pullback(phi,omega_Y)=omega_X

 IS_LAGRANGIAN(L,M,omega) :=
  dim(L)=0.5*dim(M)
  AND pullback(inclusion_L,omega)=0

 IS_ISOTROPIC(L,M,omega) :=
  tangent(L)_subset_symplectic_orthogonal(tangent(L))

 IS_COISOTROPIC(C,M,omega) :=
  symplectic_orthogonal(tangent(C))_subset_tangent(C)

 CANONICAL_SYMPLECTIC_FORM(T_star_Q) :=
  -d(theta_can)
  with sign_convention_declared

 REDUCED_SPACE(mu,alpha,G) :=
  mu^-1(alpha)/G_alpha
  under_regular_free_proper hypotheses
}

DEFINITIONS {
 D1 = "A symplectic form is closed and nondegenerate."
 D2 = "Every symplectic manifold has even dimension."
 D3 = "Symplectomorphisms preserve the symplectic form."
 D4 = "Darboux's theorem gives local canonical coordinates."
 D5 = "Symplectic geometry has no local curvature invariant analogous to Riemannian curvature."
 D6 = "Hamiltonian vector fields are determined by functions and the symplectic form."
 D7 = "Hamiltonian flows preserve the symplectic form."
 D8 = "Lagrangian submanifolds have half the ambient dimension."
 D9 = "Cotangent bundles carry a canonical symplectic structure."
 D10 = "Symplectic reduction lowers dimension by quotienting suitable conserved symmetries."
}

AXIOMS {
 A1_CLOSEDNESS:
  d omega = 0

 A2_NONDEGENERACY:
  omega(v,w)=0 for all w implies v=0

 A3_EVEN_DIMENSION:
  Nondegenerate skew-symmetric bilinear forms exist only in even dimension

 A4_DARBOUX:
  Every symplectic manifold is locally symplectomorphic to standard symplectic space

 A5_SYMPLECTOMORPHISM:
  phi is symplectic iff phi^*omega=omega

 A6_HAMILTONIAN:
  i_XH omega=dH determines X_H uniquely under fixed convention

 A7_FLOW_INVARIANCE:
  Hamiltonian flow preserves omega

 A8_LAGRANGIAN:
  L is Lagrangian iff it is maximal isotropic, equivalently isotropic of half dimension
}

REASONING {

 R1_SYMPLECTIC_FORM_TEST {
  INPUT = [M,omega]
  TEST = smoothness
  COMPUTE = d(omega)
  TEST = closedness
  TEST = nondegeneracy_at_each_point
  IF both_true
   RETURN SymplecticForm
  ELSE
   RETURN NotSymplectic
 }

 R2_DIMENSION_TEST {
  INPUT = manifold_dimension_n
  IF n_is_odd
   RETURN CannotCarryNondegenerateTwoForm
  ELSE
   RETURN SymplecticDimensionPossible
 }

 R3_DARBOUX_LOCAL_FORM {
  INPUT = symplectic_manifold_and_point
  APPLY = Darboux_theorem
  RETURN = local_coordinates(q_i,p_i)
  WITH omega=sum_i dq_i wedge dp_i
 }

 R4_HAMILTONIAN_VECTOR_FIELD {
  INPUT = H_and_omega
  COMPUTE = dH
  SOLVE = contraction_equation_for_X_H
  RETURN = HamiltonianVectorField
 }

 R5_HAMILTONIAN_FLOW {
  INPUT = X_H_and_initial_data
  SOLVE = flow_ODE
  VERIFY = pullback_of_omega_constant_in_time
  RETURN = symplectic_flow
 }

 R6_POISSON_BRACKET {
  INPUT = functions_f_g
  COMPUTE = X_f_and_X_g
  COMPUTE = omega(X_f,X_g)
  RETURN = PoissonBracket
 }

 R7_LAGRANGIAN_TEST {
  INPUT = submanifold_L
  TEST = dim(L)=half_dim(M)
  TEST = omega_restricts_to_zero_on_TL
  RETURN = LagrangianOrNot
 }

 R8_SYMPLECTOMORPHISM_TEST {
  INPUT = phi
  TEST = diffeomorphism
  COMPUTE = pullback(phi,omega_target)
  COMPARE = omega_source
  RETURN = SymplectomorphismOrNot
 }

 R9_REDUCTION {
  INPUT = Hamiltonian_G_action_and_moment_map_mu
  SELECT = regular_value_alpha
  VERIFY = suitable_group_action_hypotheses
  FORM = mu^-1(alpha)
  QUOTIENT = by_stabiliser_action
  RETURN = reduced_symplectic_space
 }
}

CONSTRAINTS {
 C1 = "Symplectic manifolds must be even-dimensional."
 C2 = "Closedness and nondegeneracy are independent requirements."
 C3 = "Hamiltonian sign conventions must be tracked."
 C4 = "Darboux coordinates are local, not generally global."
 C5 = "A symplectic form does not determine a Riemannian metric by itself."
 C6 = "Lagrangian dimension must be exactly half the ambient dimension."
 C7 = "Isotropic and coisotropic conditions must not be conflated."
 C8 = "Moment maps depend on Lie-group action and convention."
 C9 = "Symplectic reduction requires regularity and quotient hypotheses."
 C10 = "Exact symplectic structures depend on choice of primitive one-form."
}

SEMANTIC_EQUIVALENCE {
 "SymplecticGeometry" <=> "GeometryOfClosedNondegenerateTwoForms"
 "SymplecticManifold" <=> "SmoothManifoldWithSymplecticForm"
 "Symplectomorphism" <=> "SymplecticFormPreservingDiffeomorphism"
 "HamiltonianVectorField" <=> "VectorFieldGeneratedByHamiltonianFunction"
 "LagrangianSubmanifold" <=> "HalfDimensionalMaximalIsotropicSubmanifold"
 "DarbouxCoordinates" <=> "LocalCanonicalSymplecticCoordinates"
}

VALIDATION {
 V1 = "Ambient dimension is even."
 V2 = "Symplectic form is closed."
 V3 = "Symplectic form is nondegenerate."
 V4 = "Symplectomorphism test uses pullback of target form."
 V5 = "Hamiltonian convention is internally consistent."
 V6 = "Lagrangian dimension is half ambient dimension."
 V7 = "Restriction of omega to Lagrangian tangent spaces vanishes."
 V8 = "Reduction hypotheses are stated and checked."
 V9 = "Local Darboux form is not mistaken for global canonical coordinates."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMPLECTIC_GEOMETRY_DEFINITION = 0.99
 SYMPLECTIC_FORM_REASONING = 0.99
 HAMILTONIAN_REASONING = 0.98
 POISSON_REASONING = 0.98
 LAGRANGIAN_REASONING = 0.99
 SYMPLECTOMORPHISM_REASONING = 0.99
 REDUCTION_REASONING = 0.97

 POLICY =
  "Reduce confidence when sign convention, manifold dimension, regularity, group action, moment-map convention, quotient hypotheses or whether structures are local versus global are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SymplecticGeometryReasoning,
  DifferentialGeometryReasoning,
  HamiltonianMechanicsReasoning,
  SymplecticFormReasoning,
  HamiltonianVectorFieldReasoning,
  PoissonBracketReasoning,
  LagrangianSubmanifoldReasoning,
  SymplectomorphismReasoning,
  MomentMapReasoning,
  SymplecticReductionReasoning,
  CotangentBundleReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "symplectic geometry" -> SymplecticGeometry
  "symplectic manifold" -> SymplecticManifold
  "symplectic form" -> SymplecticForm
  "symplectomorphism" -> Symplectomorphism
  "Hamiltonian vector field" -> HamiltonianVectorField
  "Hamiltonian flow" -> HamiltonianFlow
  "Poisson bracket" -> PoissonBracket
  "Lagrangian submanifold" -> LagrangianSubmanifold
  "moment map" -> MomentMap
  "symplectic reduction" -> SymplecticReduction
 }

 INTERPRETATION_ORDER = [
  detect_symplectic_geometry_context,
  identify smooth_manifold_dimension_and_two_form,
  verify closedness_and_nondegeneracy,
  identify Hamiltonian_Lagrangian_symplectomorphism_moment_map_or_reduction goal,
  apply symplectic_specific reasoning,
  track sign_conventions_local_global_and_group_action assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("symplectic geometry")
  OR closed_nondegenerate_two_form_context
  THEN activate(SymplecticGeometryOntology)

 E2:
  IF manifold_dimension_is_odd
  THEN reject ordinary_symplectic_structure

 E3:
  IF candidate_symplectic_form
  THEN verify d_omega=0 and nondegeneracy

 E4:
  IF Hamiltonian_function_given
  THEN solve contraction_equation for X_H

 E5:
  IF symplectomorphism_requested
  THEN verify diffeomorphism_and_pullback_form equality

 E6:
  IF Lagrangian_submanifold_requested
  THEN verify isotropy_and_half_dimension

 E7:
  IF cotangent_bundle_context
  THEN use canonical_one_form_and_symplectic_form

 E8:
  IF moment_map_and_group_action_present
  THEN permit symplectic_reduction reasoning

 E9:
  IF Darboux_coordinates_requested
  THEN state locality explicitly
}

RECURSION {
 ENTRY = SymplecticGeometry

 EXPAND(SymplecticGeometry) -> [
  SymplecticManifold,
  SymplecticForm,
  ClosedTwoForm,
  NondegenerateTwoForm,
  DarbouxTheorem,
  DarbouxCoordinates,
  Symplectomorphism,
  HamiltonianFunction,
  HamiltonianVectorField,
  HamiltonianFlow,
  PoissonBracket,
  LagrangianSubmanifold,
  IsotropicSubmanifold,
  CoisotropicSubmanifold,
  CotangentBundle,
  CanonicalOneForm,
  MomentMap,
  HamiltonianGroupAction,
  SymplecticReduction,
  LiouvilleVectorField,
  ContactTypeBoundary
 ]

 EXPAND(HamiltonianStructure) -> [
  HamiltonianFunction,
  HamiltonianVectorField,
  HamiltonianFlow,
  ConservedQuantity,
  PoissonBracket
 ]

 EXPAND(LagrangianGeometry) -> [
  IsotropicSubmanifold,
  CoisotropicSubmanifold,
  LagrangianSubmanifold,
  SymplecticOrthogonal
 ]

 EXPAND(Reduction) -> [
  LieGroupAction,
  MomentMap,
  LevelSet,
  Stabiliser,
  QuotientSpace,
  MarsdenWeinsteinReduction
 ]

 STOP_CONDITION =
  "Stop when the symplectic form, Hamiltonian dynamics, Lagrangian structure, symplectomorphism, moment map, cotangent structure or reduced space is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Differential Geometry",
  "Manifold Ontology",
  "Topology Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Differential Geometry",
  "Manifold Ontology",
  "Metric Tensor Ontology",
  "Topology Ontology",
  "Vector Geometry Ontology",
  "Calculus & Analysis Ontologies",
  "Differential Forms Ontologies",
  "Lie Group Ontologies",
  "Classical Mechanics Ontologies",
  "Poisson Geometry Ontologies",
  "Contact Geometry Ontologies"
 ]

 ROUTE_RULES = [
  "Symplectic Geometry -> Advanced Geometry for parent structural semantics",
  "Symplectic Geometry -> Differential Geometry for smooth manifolds and differential forms",
  "Symplectic Geometry -> Topology for global manifold structure",
  "Symplectic Geometry -> Lie Groups for Hamiltonian group actions",
  "Symplectic Geometry -> Classical Mechanics for phase-space dynamics",
  "Symplectic Geometry -> Poisson Geometry for bracket structure",
  "Symplectic Geometry -> Contact Geometry for contact-type boundaries and symplectisation",
  "Symplectic Geometry -> Calculus for differential and flow equations"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible symplectic structures, constructions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve closed-nondegenerate two-form semantics.",
  "Preserve even-dimensionality.",
  "Preserve symplectomorphism pullback semantics.",
  "Preserve Hamiltonian sign-convention awareness.",
  "Preserve Lagrangian half-dimensional maximal-isotropic semantics.",
  "Preserve Darboux locality.",
  "Preserve reduction hypotheses.",
  "Preserve machine-readable symplectic-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Symplectic geometry studies smooth even-dimensional manifolds equipped with a closed nondegenerate differential 2-form."

 FORM_SUMMARY =
  "A symplectic form omega satisfies d omega=0 and is nondegenerate."

 DIMENSION_SUMMARY =
  "Every symplectic manifold has even dimension."

 DARBOUX_SUMMARY =
  "Locally, every symplectic form can be written as sum_i dq_i wedge dp_i."

 HAMILTONIAN_SUMMARY =
  "A Hamiltonian function H defines X_H through i_XH omega=dH, up to sign convention."

 FLOW_SUMMARY =
  "Hamiltonian flows preserve the symplectic form."

 LAGRANGIAN_SUMMARY =
  "A Lagrangian submanifold is isotropic and has half the dimension of the ambient symplectic manifold."

 COTANGENT_SUMMARY =
  "Cotangent bundles carry a canonical symplectic structure."

 REDUCTION_SUMMARY =
  "Symplectic reduction constructs lower-dimensional symplectic spaces from moment-map level sets modulo symmetry."
}

FLOW {
 INPUT
 -> detect_symplectic_geometry_context
 -> identify manifold_dimension_smooth_structure_two_form_group_action_and_Hamiltonian data
 -> verify closedness_nondegeneracy_and_even_dimension
 -> identify Darboux_Hamiltonian_Poisson_Lagrangian_symplectomorphism_cotangent_or_reduction goal
 -> apply symplectic_specific differential_geometric reasoning
 -> verify sign_conventions_local_global_regular_value_group_action_and_dimension constraints
 -> route to advanced_geometry differential_geometry topology differential_forms Lie_groups mechanics Poisson_or_contact geometry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Symplectic Geometry Ontology defines smooth even-dimensional manifolds equipped with closed nondegenerate 2-forms. It supports Darboux coordinates, symplectomorphisms, Hamiltonian vector fields and flows, Poisson brackets, isotropic and Lagrangian submanifolds, cotangent bundles, Hamiltonian group actions, moment maps and symplectic reduction."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-SYMPLECTIC-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW