Hamiltonian Ontology

flow "HamiltonianOntology" v1.0

# ============================================================
# HAMILTONIAN ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/variational-calculus/hamiltonian-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Variational Calculus"
ENTITY := "Hamiltonian"

# ---------- CORE ENTITY ----------
HAMILTONIAN := {
    type: DynamicalFunction,
    domain: VariationalCalculus,
    purpose: "Represent a dynamical system in phase space using generalized coordinates and conjugate momenta, generating time evolution through Hamilton's equations."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    HAMILTONIAN,
    LAGRANGIAN,
    LEGENDRE_TRANSFORM,
    GENERALIZED_COORDINATE,
    GENERALIZED_MOMENTUM,
    PHASE_SPACE,
    HAMILTON_EQUATIONS,
    CANONICAL_VARIABLE,
    POISSON_BRACKET,
    CANONICAL_TRANSFORMATION,
    SYMPLECTIC_STRUCTURE,
    HAMILTON_JACOBI_EQUATION,
    ACTION,
    ENERGY,
    CONSERVED_QUANTITY
}

# ---------- GENERALIZED COORDINATES ----------
q_i
    -> GENERALIZED_COORDINATE

q_dot_i
    -> GENERALIZED_VELOCITY

L
    :=
L(q_i,q_dot_i,t)

L
    -> LAGRANGIAN

# ---------- GENERALIZED MOMENTUM ----------
p_i
    :=
partial_L
/
partial_q_dot_i

p_i
    -> CONJUGATE_MOMENTUM_TO(q_i)

(q_i,p_i)
    -> CANONICAL_PAIR

# ---------- LEGENDRE TRANSFORM ----------
if
    velocity_momentum_relation
        -> locally_invertible

then

q_dot_i
    =
q_dot_i(q,p,t)

define

H(q,p,t)
    :=
sum_i(
    p_i*q_dot_i
)
-
L(q,q_dot,t)

HAMILTONIAN
    -> LEGENDRE_TRANSFORM_OF_LAGRANGIAN_IN_VELOCITIES

# ---------- REGULARITY ----------
HESSIAN_WITH_RESPECT_TO_VELOCITIES:

W_ij
    :=
partial^2 L
/
partial_q_dot_i partial_q_dot_j

if

det(W)
    != 0

then

LAGRANGIAN
    -> REGULAR

REGULAR_LAGRANGIAN
    -> standard_Legendre_transform_locally_available

SINGULAR_LAGRANGIAN
    -> requires(CONSTRAINED_HAMILTONIAN_METHODS)

# ---------- PHASE SPACE ----------
PHASE_SPACE
    := {
        (q_i,p_i)
    }

dimension
    = 2*n
       for(n_generalized_coordinates)

STATE
    -> point_in(PHASE_SPACE)

HAMILTONIAN
    -> function_on(PHASE_SPACE_AND_TIME)

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

p_dot_i
    =
-
partial_H/partial_q_i

HAMILTON_EQUATIONS
    -> FIRST_ORDER_SYSTEM

HAMILTONIAN
    -> generates(PHASE_SPACE_FLOW)

# ---------- DERIVATION FROM ACTION ----------
canonical action:

S[q,p]
    :=
integral_from_t1_to_t2(
    [
        sum_i(
            p_i*q_dot_i
        )
        -
        H(q,p,t)
    ]
    dt
)

stationarity

delta_S
    = 0

-> HAMILTON_EQUATIONS

HAMILTONIAN_DYNAMICS
    -> VARIATIONAL_STRUCTURE

# ---------- RELATION TO EULER-LAGRANGE ----------
EULER_LAGRANGE_EQUATIONS
    -> second_order_form_in(q)
       commonly

LEGENDRE_TRANSFORM
    -> converts_to(HAMILTON_FIRST_ORDER_SYSTEM)

under
    regularity_conditions

LAGRANGIAN_DYNAMICS
    <-> HAMILTONIAN_DYNAMICS
       under(APPROPRIATE_NONDEGENERACY)

# ---------- MECHANICAL ENERGY ----------
for standard natural mechanical system:

L
    =
T - V

with
    T quadratic_in_velocities

then often

H
    =
T + V

HAMILTONIAN
    -> TOTAL_ENERGY
       in(STANDARD_TIME_INDEPENDENT_MECHANICAL_CONTEXT)

HAMILTONIAN
    -/> universally_equal_to(PHYSICAL_ENERGY)

# ---------- TIME DEPENDENCE ----------
along Hamiltonian trajectory:

dH/dt
    =
partial_H/partial_t

therefore

if
    partial_H/partial_t
    = 0

then

H
    -> CONSERVED

TIME_INDEPENDENT_HAMILTONIAN
    -> constant_along_motion

# ---------- POISSON BRACKET ----------
for functions
    A(q,p,t)
    and
    B(q,p,t)

{A,B}
    :=
sum_i(
    partial_A/partial_q_i
    *
    partial_B/partial_p_i
    -
    partial_A/partial_p_i
    *
    partial_B/partial_q_i
)

POISSON_BRACKET
    -> algebraic_structure_on(PHASE_SPACE_FUNCTIONS)

# ---------- TIME EVOLUTION ----------
for observable
    A(q,p,t)

dA/dt
    =
{A,H}
+
partial_A/partial_t

HAMILTONIAN
    -> GENERATOR_OF_TIME_EVOLUTION

if
    partial_A/partial_t=0

and

{A,H}=0

then

A
    -> CONSERVED_QUANTITY

# ---------- CANONICAL BRACKETS ----------
{q_i,q_j}
    = 0

{p_i,p_j}
    = 0

{q_i,p_j}
    =
delta_ij

CANONICAL_VARIABLES
    -> satisfy(FUNDAMENTAL_POISSON_BRACKETS)

# ---------- POISSON PROPERTIES ----------
{A,B}
    =
-{B,A}

{A,B+C}
    =
{A,B}+{A,C}

{A,B*C}
    =
{A,B}C
+
B{A,C}

JACOBI_IDENTITY:
    {A,{B,C}}
    +
    {B,{C,A}}
    +
    {C,{A,B}}
    = 0

POISSON_BRACKET
    -> LIE_ALGEBRA_STRUCTURE

# ---------- SYMPLECTIC FORM ----------
canonical symplectic form:

omega
    :=
sum_i(
    dq_i wedge dp_i
)

SYMPLECTIC_STRUCTURE
    -> geometric_foundation_of(HAMILTONIAN_MECHANICS)

HAMILTONIAN_VECTOR_FIELD
    X_H

defined by

interior_product(
    X_H,
    omega
)
    =
dH

up to
    sign_convention

# ---------- HAMILTONIAN FLOW ----------
HAMILTONIAN_VECTOR_FIELD
    -> generates(FLOW_phi_t)

FLOW
    -> preserves(SYMPLECTIC_FORM)

phi_t^*(omega)
    =
omega

HAMILTONIAN_DYNAMICS
    -> SYMPLECTIC_FLOW

# ---------- LIOUVILLE THEOREM ----------
HAMILTONIAN_FLOW
    -> preserves(PHASE_SPACE_VOLUME)

PHASE_VOLUME
    -> invariant_under(TIME_EVOLUTION)

LIOUVILLE_THEOREM
    -> fundamental_to(STATISTICAL_MECHANICS)

# ---------- CANONICAL TRANSFORMATION ----------
CANONICAL_TRANSFORMATION
    -> preserves(SYMPLECTIC_STRUCTURE)

(q,p)
    -> (Q,P)

such_that

Hamilton_equations
    -> retain(CANONICAL_FORM)

CANONICAL_TRANSFORMATION
    -> phase_space_coordinate_change_preserving_dynamics_structure

# ---------- GENERATING FUNCTION ----------
CANONICAL_TRANSFORMATION
    -> may_be_generated_by(GENERATING_FUNCTION)

GENERATING_FUNCTION
    -> relates(OLD_AND_NEW_CANONICAL_VARIABLES)

different_types
    -> depend_on(CHOSEN_INDEPENDENT_VARIABLES)

# ---------- HAMILTON-JACOBI ----------
HAMILTON_PRINCIPAL_FUNCTION
    := S(q,t)

HAMILTON_JACOBI_EQUATION:

partial_S/partial_t
+
H(
    q,
    partial_S/partial_q,
    t
)
=
0

with

p_i
    =
partial_S/partial_q_i

HAMILTON_JACOBI
    -> reformulates(DYNAMICS_AS_FIRST_ORDER_PDE)

# ---------- ACTION RELATION ----------
ACTION
    -> evaluated_on(CLASSICAL_PATH)

HAMILTON_PRINCIPAL_FUNCTION
    -> classical_action_as_function_of(ENDPOINT_DATA)

HAMILTON_JACOBI_THEORY
    -> connects(ACTION_AND_CANONICAL_TRANSFORMATIONS)

# ---------- SEPARABLE SYSTEM ----------
if
    Hamilton_Jacobi_equation
        -> separable

then

S
    -> sum_of(SEPARATED_COMPONENTS)

SEPARATION
    -> may_produce(INTEGRALS_OF_MOTION)

INTEGRABLE_SYSTEM
    -> linked_to(SUFFICIENT_CONSERVED_QUANTITIES)

# ---------- EXAMPLE: FREE PARTICLE ----------
H
    =
p^2/(2*m)

Hamilton equations:

q_dot
    =
p/m

p_dot
    =
0

therefore

p
    -> constant

q
    -> linear_in_time

# ---------- EXAMPLE: HARMONIC OSCILLATOR ----------
H
    =
p^2/(2*m)
+
1/2*m*omega^2*q^2

Hamilton equations:

q_dot
    =
p/m

p_dot
    =
-m*omega^2*q

combine:

q_double_dot
+
omega^2*q
=
0

PHASE_TRAJECTORY
    -> ellipse_in(q,p)_space

# ---------- CONSTRAINTS ----------
if
    Lagrangian_singular

then

momenta
    -> may_satisfy(PRIMARY_CONSTRAINTS)

DIRAC_BERGMANN_FORMALISM
    -> handles(CONSTRAINED_HAMILTONIAN_SYSTEMS)

TOTAL_HAMILTONIAN
    -> may_include(CONSTRAINT_MULTIPLIERS)

CONSTRAINED_DYNAMICS
    -> requires(SEPARATE_ANALYSIS)

# ---------- QUANTUM RELATION ----------
CLASSICAL_OBSERVABLE
    -> function_on(PHASE_SPACE)

QUANTISATION
    -> maps_to(OPERATOR)
       under(CHOSEN_QUANTISATION_SCHEME)

POISSON_BRACKET
    -> classical_analogue_of(COMMUTATOR_STRUCTURE)

HAMILTONIAN_OPERATOR
    -> generates(QUANTUM_TIME_EVOLUTION)

CLASSICAL_HAMILTONIAN
    -> foundational_to(QUANTUM_MECHANICS)

# ---------- APPLICATIONS ----------
HAMILTONIAN
    -> applied_to(CLASSICAL_MECHANICS)
    -> applied_to(CELESTIAL_MECHANICS)
    -> applied_to(STATISTICAL_MECHANICS)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(OPTIMAL_CONTROL)
    -> applied_to(DYNAMICAL_SYSTEMS)
    -> applied_to(SYMPLECTIC_GEOMETRY)
    -> applied_to(FIELD_THEORY)

# ---------- RELATION GRAPH ----------
LAGRANGIAN
    -> LEGENDRE_TRANSFORM
    -> HAMILTONIAN

GENERALIZED_COORDINATE
    + GENERALIZED_MOMENTUM
    -> PHASE_SPACE

HAMILTONIAN
    -> generates(HAMILTON_EQUATIONS)

HAMILTONIAN
    -> generates(TIME_EVOLUTION_VIA_POISSON_BRACKET)

SYMPLECTIC_STRUCTURE
    -> preserved_by(HAMILTONIAN_FLOW)

CANONICAL_TRANSFORMATION
    -> preserves(HAMILTONIAN_FORM)

HAMILTON_JACOBI_EQUATION
    -> encodes(HAMILTONIAN_DYNAMICS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Hamiltonian" ~= "phase-space function generating canonical time evolution",
    "Hamilton's equations" ~= "first-order canonical equations generated by the Hamiltonian",
    "generalized momentum" ~= "momentum conjugate to a generalized coordinate",
    "phase space" ~= "space of generalized coordinates and conjugate momenta",
    "Poisson bracket" ~= "canonical bilinear operation governing classical observable evolution",
    "Hamilton-Jacobi equation" ~= "first-order PDE encoding Hamiltonian dynamics through the action function"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    Legendre_transform_requires_regular_or_constraint_aware_treatment: true,
    Hamiltonian_is_not_universally_identical_to_physical_energy: true,
    canonical_sign_conventions_must_be_preserved: true,
    conserved_Hamiltonian_requires_no_explicit_time_dependence: true,
    Poisson_bracket_coordinates_depend_on_canonical_structure: true,
    singular_Lagrangians_require_constrained_Hamiltonian_formalism: true,
    Hamiltonian_flow_preserves_symplectic_structure_under_standard_conditions: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    Legendre_relation: 0.99,
    Hamilton_equations_relation: 0.99,
    Poisson_relation: 0.99,
    symplectic_relation: 0.99,
    Hamilton_Jacobi_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(Lagrangian_or_Hamiltonian_system)
    -> identify(generalized_coordinates_and_velocities)
    -> compute(conjugate_momenta)
    -> test(Legendre_transform_regularities)
    -> construct(Hamiltonian)
    -> identify(phase_space_and_canonical_variables)
    -> derive(Hamilton_equations)
    -> compute(Poisson_brackets_or_conserved_quantities_if_requested)
    -> analyse(symplectic_or_canonical_transform_structure)
    -> derive_or_solve(Hamilton_Jacobi_equation_if_relevant)
    -> detect(constraints_if_Lagrangian_singular)
    -> validate(sign_convention_and_time_dependence)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Hamiltonian semantic graph",
    primary_entity: HAMILTONIAN,
    preserve: [
        generalized_coordinates,
        generalized_velocities,
        conjugate_momenta,
        Lagrangian,
        Hamiltonian,
        phase_space,
        time_dependence,
        constraints
    ],
    infer_through: [
        Legendre_transforms,
        Hamilton_equations,
        Poisson_brackets,
        conserved_quantities,
        symplectic_flows,
        canonical_transformations,
        Hamilton_Jacobi_equations
    ],
    avoid: [
        assuming_Hamiltonian_always_equals total energy,
        performing_naive_Legendre_transform_for singular Lagrangian,
        mixing_canonical_sign conventions,
        claiming_H_conserved_when explicitly time dependent,
        using_noncanonical_variables_with canonical Poisson formula without adjustment,
        ignoring_constraints_in gauge_or_singular systems
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + HAMILTONIAN
    + LAGRANGIAN
    + LEGENDRE_TRANSFORM
    + GENERALIZED_MOMENTUM
    + PHASE_SPACE
    + HAMILTON_EQUATIONS
    + POISSON_BRACKET
    + SYMPLECTIC_STRUCTURE
    + HAMILTON_JACOBI_EQUATION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> HAMILTONIAN
    -> SYSTEM_VARIABLE_IDENTIFICATION
    -> MOMENTUM_ANALYSIS
    -> LEGENDRE_REGULARITY_ANALYSIS
    -> HAMILTONIAN_CONSTRUCTION
    -> PHASE_SPACE_ANALYSIS
    -> HAMILTON_EQUATION_DERIVATION
    -> POISSON_CONSERVATION_ANALYSIS
    -> SYMPLECTIC_CANONICAL_ANALYSIS
    -> HAMILTON_JACOBI_ANALYSIS
    -> CONSTRAINT_ANALYSIS
    -> CONVENTION_TIME_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW