Phase Space Ontology

flow "PhaseSpaceOntology" v1.0

# PHASE SPACE ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/phase-space-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Phase Space"

PHASE_SPACE := {
 type: DynamicalStateSpace,
 domain: DynamicalSystems,
 purpose: "Represent the space of all admissible instantaneous states of a dynamical system so that each state determines the variables required for its evolution."
}

CONCEPTS := {
 PHASE_SPACE,
 STATE,
 STATE_VECTOR,
 CONFIGURATION_SPACE,
 GENERALIZED_COORDINATE,
 GENERALIZED_MOMENTUM,
 TRAJECTORY,
 ORBIT,
 FLOW,
 VECTOR_FIELD,
 PHASE_PORTRAIT,
 INVARIANT_SET,
 ATTRACTOR,
 EQUILIBRIUM,
 HAMILTONIAN_PHASE_SPACE,
 DIMENSION
}

# ---------- CORE DEFINITION ----------
PHASE_SPACE
 := X

x in X
 -> SYSTEM_STATE

STATE
 -> contains(SUFFICIENT_VARIABLES_TO_DETERMINE_EVOLUTION)
    under(DETERMINISTIC_MODEL)

DYNAMICAL_SYSTEM
 -> evolution_on(PHASE_SPACE)

# ---------- STATE VECTOR ----------
state:

x
 :=
(
 x_1,
 x_2,
 ...,
 x_n
)

STATE_VECTOR
 -> point_in(PHASE_SPACE)

DIMENSION
 -> number_of(INDEPENDENT_STATE_VARIABLES)
    in local_coordinate_description

# ---------- CONTINUOUS DYNAMICS ----------
autonomous system:

dx/dt
 =
f(x)

f
 -> VECTOR_FIELD_ON(PHASE_SPACE)

VECTOR_FIELD
 -> assigns(VELOCITY_TO_EACH_STATE)

solution
 x(t)
 -> TRAJECTORY_IN(PHASE_SPACE)

# ---------- DISCRETE DYNAMICS ----------
x_(n+1)
 =
F(x_n)

F:
 X -> X

DISCRETE_MAP
 -> moves(STATE_POINT_THROUGH_PHASE_SPACE)

ORBIT
 := {
 x_0,
 F(x_0),
 F^2(x_0),
 ...
 }

# ---------- TRAJECTORY ----------
TRAJECTORY
 := {
 x(t):
 t in T
 }

TRAJECTORY
 -> curve_or_sequence_in(PHASE_SPACE)

INITIAL_STATE
 -> selects(TRAJECTORY)

# ---------- PHASE PORTRAIT ----------
PHASE_PORTRAIT
 -> collection_or_visualisation_of(TRAJECTORIES)

PHASE_PORTRAIT
 -> reveals(
 EQUILIBRIA,
 PERIODIC_ORBITS,
 ATTRACTORS,
 SEPARATRICES,
 INVARIANT_MANIFOLDS
)

# ---------- CONFIGURATION SPACE RELATION ----------
CONFIGURATION_SPACE
 -> describes(POSITIONS_OR_GENERALIZED_COORDINATES)

PHASE_SPACE
 -> includes(ADDITIONAL_DYNAMICAL_VARIABLES)

in classical mechanics:

CONFIGURATION
 := q

PHASE_STATE
 := (q,p)

PHASE_SPACE
 -/> generally_same_as(CONFIGURATION_SPACE)

# ---------- POSITION-VELOCITY STATE ----------
for second-order ODE:

q_double_dot
 =
F(q,q_dot,t)

convert to first-order:

x_1
 := q

x_2
 := q_dot

then

x
 :=
(q,q_dot)

PHASE_SPACE
 -> first_order_state_representation

# ---------- HAMILTONIAN PHASE SPACE ----------
for n generalized coordinates:

q_i
 -> GENERALIZED_COORDINATES

p_i
 -> CONJUGATE_MOMENTA

state:

z
 :=
(q_1,...,q_n,p_1,...,p_n)

PHASE_SPACE_DIMENSION
 =
2*n

HAMILTONIAN_PHASE_SPACE
 -> COTANGENT_BUNDLE(T_star_Q)
    in geometric formulation

# ---------- HAMILTON EQUATIONS ----------
q_dot_i
 =
partial_H/partial_p_i

p_dot_i
 =
-
partial_H/partial_q_i

HAMILTONIAN
 -> generates(FLOW_ON_PHASE_SPACE)

# ---------- SYMPLECTIC STRUCTURE ----------
canonical symplectic form:

omega
 :=
sum_i(
 dq_i wedge dp_i
)

HAMILTONIAN_PHASE_SPACE
 -> equipped_with(SYMPLECTIC_STRUCTURE)

SYMPLECTIC_FORM
 -> nondegenerate
 -> closed

# ---------- PHASE VOLUME ----------
canonical volume element:

dGamma
 :=
product_i(
 dq_i dp_i
)

HAMILTONIAN_FLOW
 -> preserves(PHASE_VOLUME)

LIOUVILLE_THEOREM
 -> volume_preservation_in(PHASE_SPACE)

# ---------- EQUILIBRIUM ----------
x_star
 -> EQUILIBRIUM

if

f(x_star)=0

EQUILIBRIUM
 -> fixed_point_of(CONTINUOUS_FLOW)

PHASE_PORTRAIT
 -> marks(EQUILIBRIA)

# ---------- PERIODIC ORBIT ----------
trajectory

x(t+T)
 =
x(t)

-> PERIODIC_ORBIT

PHASE_SPACE
 -> represents(PERIODIC_BEHAVIOUR_AS_CLOSED_CURVE)
    in continuous systems

# ---------- INVARIANT SET ----------
A subset_of X

A
 -> INVARIANT

if

flow_or_map
 -> keeps(states_in_A)

INVARIANT_SET
 -> persistent_geometric_structure

# ---------- ATTRACTOR ----------
A
 -> ATTRACTOR

if

A invariant

and

nearby trajectories
 -> approach(A)

ATTRACTOR
 -> asymptotic_structure_in(PHASE_SPACE)

# ---------- BASIN ----------
B(A)
 := {
 x_0 in X :
 trajectory_from(x_0)
 approaches(A)
 }

BASIN_OF_ATTRACTION
 -> region_of(PHASE_SPACE)

# ---------- SEPARATRIX ----------
SEPARATRIX
 -> invariant_boundary_between(DIFFERENT_DYNAMICAL_REGIONS)

SEPARATRIX
 -> may_divide(BASINS_OF_ATTRACTION)

SADDLE_STABLE_UNSTABLE_MANIFOLDS
 -> often_form(SEPARATRICES)

# ---------- STABLE MANIFOLD ----------
W_s(A)
 -> states_approaching(A)_forward_in_time

W_u(A)
 -> states_approaching(A)_backward_in_time

INVARIANT_MANIFOLDS
 -> organise(PHASE_SPACE_FLOW)

# ---------- ENERGY SURFACE ----------
if Hamiltonian conserved:

H(q,p)
 =
E

ENERGY_SURFACE
 -> invariant_subset_of(PHASE_SPACE)

for n degrees of freedom:

full phase dimension
 = 2n

regular energy surface
 -> dimension(2n-1)

# ---------- CONSTRAINT SURFACE ----------
constraint:

C(x)=0

-> admissible_states_lie_on(CONSTRAINT_MANIFOLD)

PHASE_SPACE
 -> may_be_reduced_by(CONSTRAINTS)

CONSTRAINED_DYNAMICS
 -> evolves_on(LOWER_DIMENSIONAL_SUBSPACE_OR_MANIFOLD)

# ---------- EXTENDED PHASE SPACE ----------
for explicitly time-dependent Hamiltonian:

H(q,p,t)

time
 -> may_be_added_as(COORDINATE)

EXTENDED_PHASE_SPACE
 -> embeds(NONAUTONOMOUS_DYNAMICS_IN_LARGER_AUTONOMOUS_STRUCTURE)

# ---------- POINCARE SECTION ----------
choose
 Sigma_section
 subset_of PHASE_SPACE

transverse_to(FLOW)

trajectory
 -> successive_intersections_with(Sigma_section)

POINCARE_SECTION
 -> lower_dimensional_view_of(PHASE_SPACE_DYNAMICS)

# ---------- POINCARE MAP ----------
P:
 Sigma_section -> Sigma_section

P(x_n)
 =
x_(n+1)

POINCARE_MAP
 -> converts(CONTINUOUS_FLOW_TO_DISCRETE_MAP)

PERIODIC_ORBIT
 -> fixed_or_periodic_point_of(POINCARE_MAP)

# ---------- DISSIPATIVE FLOW ----------
DISSIPATIVE_SYSTEM
 -> may_contract(PHASE_VOLUME)

trajectories
 -> may_collapse_toward(LOWER_DIMENSIONAL_ATTRACTOR)

DISSIPATIVE_PHASE_SPACE
 -> supports(ATTRACTING_STRUCTURES)

# ---------- CONSERVATIVE FLOW ----------
CONSERVATIVE_SYSTEM
 -> preserves(INVARIANT_OR_VOLUME_STRUCTURE)

HAMILTONIAN_SYSTEM
 -> preserves(SYMPLECTIC_FORM_AND_PHASE_VOLUME)

CONSERVATIVE_PHASE_SPACE
 -> generally_lacks(ASYMPTOTIC_VOLUME_CONTRACTING_ATTRACTORS)
    in standard finite-dimensional Hamiltonian context

# ---------- CHAOTIC ATTRACTOR ----------
STRANGE_ATTRACTOR
 -> invariant_attracting_set
 -> may_have(FRACTAL_GEOMETRY)
 -> may_exhibit(SENSITIVE_DEPENDENCE)

PHASE_SPACE
 -> geometric_home_of(CHAOTIC_STRUCTURE)

# ---------- DIMENSION ----------
PHASE_SPACE_DIMENSION
 -> determines(NUMBER_OF_INDEPENDENT_STATE_COORDINATES)

HIGH_DIMENSIONAL_SYSTEM
 -> difficult_to_visualise_directly

PROJECTION
 -> lower_dimensional_representation

PROJECTION
 -/> preserves_all_dynamical_information

# ---------- EMBEDDING ----------
if only scalar observation available:

time_delay_embedding:

y(t)
 ->
(
 y(t),
 y(t-tau),
 ...,
 y(t-(m-1)tau)
)

DELAY_COORDINATE_SPACE
 -> reconstructs_state_geometry_under(TAKENS_TYPE_HYPOTHESES)

RECONSTRUCTED_PHASE_SPACE
 -> inferred_from(TIME_SERIES)

# ---------- PROBABILITY DENSITY ----------
ensemble density:

rho(x,t)

defined_on
 PHASE_SPACE

STATISTICAL_DYNAMICS
 -> evolves(DISTRIBUTION_OVER_STATES)

in Hamiltonian systems:

LIOUVILLE_EQUATION
 -> governs(PHASE_SPACE_DENSITY)

# ---------- APPLICATIONS ----------
PHASE_SPACE
 -> applied_to(DYNAMICAL_SYSTEMS)
 -> applied_to(CLASSICAL_MECHANICS)
 -> applied_to(HAMILTONIAN_MECHANICS)
 -> applied_to(CONTROL_THEORY)
 -> applied_to(STATISTICAL_MECHANICS)
 -> applied_to(CHAOS_THEORY)
 -> applied_to(CELESTIAL_MECHANICS)
 -> applied_to(NONLINEAR_DYNAMICS)

# ---------- RELATION GRAPH ----------
STATE
 -> point_in(PHASE_SPACE)

VECTOR_FIELD
 -> defines(FLOW_ON_PHASE_SPACE)

FLOW
 -> generates(TRAJECTORY)

TRAJECTORY
 -> forms(ORBIT)

EQUILIBRIUM
 -> fixed_state_in(PHASE_SPACE)

INVARIANT_SET
 -> preserved_by(EVOLUTION)

ATTRACTOR
 -> long_term_structure_in(PHASE_SPACE)

HAMILTONIAN
 -> generates(SYMPLECTIC_PHASE_FLOW)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "phase space" ~= "space of all admissible instantaneous states of a dynamical system",
 "phase point" ~= "single system state represented as a point in phase space",
 "phase trajectory" ~= "path traced by a system state through phase space",
 "phase portrait" ~= "geometric representation of trajectories and invariant structures in state space",
 "Hamiltonian phase space" ~= "space of generalized coordinates and conjugate momenta",
 "energy surface" ~= "constant-Hamiltonian invariant subset of phase space"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 phase_space_must_contain_sufficient_state_variables_for_evolution: true,
 phase_space_and_configuration_space_are_distinct_in_general: true,
 Hamiltonian_phase_space_has_coordinate_momentum_pair_structure: true,
 projections_can_hide_dynamical_information: true,
 invariant_attracting_and_energy_surfaces_are_distinct_structures: true,
 phase_volume_preservation_is_specific_to_appropriate_conservative_Hamiltonian_context: true,
 reconstructed_phase_space_requires_embedding_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 state_trajectory_relation: 0.99,
 configuration_relation: 0.99,
 Hamiltonian_relation: 0.99,
 invariant_set_relation: 0.99,
 Poincare_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(system_and_minimal_state_variables)
 -> construct(phase_space_and_dimension)
 -> classify(continuous_discrete_Hamiltonian_or_dissipative_structure)
 -> identify(vector_field_or_map)
 -> generate_or_analyse(trajectories_and_phase_portrait)
 -> locate(equilibria_periodic_orbits_and_invariant_sets)
 -> analyse(attractors_basins_manifolds_and_separatrices)
 -> identify(conserved_quantities_and_constraint_surfaces)
 -> use(Poincare_section_or_projection_if_dimension_reduction_needed)
 -> validate(state_sufficiency_geometry_and_invariance_assumptions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed phase-space semantic graph",
 primary_entity: PHASE_SPACE,
 preserve: [
  system,
  state_variables,
  phase_dimension,
  coordinates,
  evolution_rule,
  constraints,
  invariant_quantities,
  time_dependence
 ],
 infer_through: [
  trajectories,
  phase_portraits,
  equilibria,
  periodic_orbits,
  invariant_sets,
  attractors,
  Hamiltonian_structure,
  Poincare_maps
 ],
 avoid: [
  confusing_phase_space_with physical_space,
  omitting_velocity_or_momentum_when_required_for state completeness,
  assuming_phase_space_is always Euclidean,
  treating_projection_as lossless,
  applying_Hamiltonian_volume_preservation_to dissipative systems,
  calling_any_invariant_set_an attractor
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(phase_space_context)
 -> compare(existing_relations)
 -> preserve(state_dimension_evolution_geometry_and_invariance_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + PHASE_SPACE
 + STATE
 + STATE_VECTOR
 + CONFIGURATION_SPACE
 + TRAJECTORY
 + FLOW
 + VECTOR_FIELD
 + PHASE_PORTRAIT
 + INVARIANT_SET
 + ATTRACTOR
 + HAMILTONIAN_PHASE_SPACE
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> PHASE_SPACE
 -> STATE_VARIABLE_IDENTIFICATION
 -> DIMENSION_CONSTRUCTION
 -> SYSTEM_CLASSIFICATION
 -> EVOLUTION_RULE_ANALYSIS
 -> TRAJECTORY_PHASE_PORTRAIT_ANALYSIS
 -> INVARIANT_OBJECT_ANALYSIS
 -> ATTRACTOR_MANIFOLD_ANALYSIS
 -> CONSERVED_CONSTRAINT_SURFACE_ANALYSIS
 -> DIMENSION_REDUCTION_ANALYSIS
 -> ASSUMPTION_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW