Action Principle Ontology

flow "ActionPrincipleOntology" v1.0

# ============================================================
# ACTION PRINCIPLE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
ACTION_PRINCIPLE := {
    type: VariationalPhysicalPrinciple,
    domain: VariationalCalculus,
    purpose: "Represent physical evolution as a stationary path of an action functional defined by integrating a Lagrangian over the relevant domain."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    ACTION,
    ACTION_FUNCTIONAL,
    LAGRANGIAN,
    LAGRANGIAN_DENSITY,
    PATH,
    TRAJECTORY,
    VARIATION,
    STATIONARY_ACTION,
    EULER_LAGRANGE_EQUATION,
    GENERALIZED_COORDINATE,
    GENERALIZED_MOMENTUM,
    BOUNDARY_CONDITION,
    SYMMETRY,
    CONSERVED_QUANTITY,
    NOETHER_THEOREM
}

# ---------- ACTION FUNCTIONAL ----------
for generalized coordinates
    q_i(t)

ACTION
    := S[q]

S[q]
    =
integral_from_t1_to_t2(
    L(
        q_i,
        q_dot_i,
        t
    )
    dt
)

L
    -> LAGRANGIAN

ACTION
    -> scalar_functional_of(TRAJECTORY)

# ---------- PHYSICAL PATH ----------
TRAJECTORY
    := q_i(t)

PHYSICAL_TRAJECTORY
    -> selected_by(STATIONARY_ACTION)

ACTION_PRINCIPLE
    -> compares(NEARBY_ADMISSIBLE_TRAJECTORIES)

# ---------- VARIATION ----------
choose

eta_i(t)
    -> admissible_variation

define

q_i_epsilon(t)
    :=
q_i(t)
+
epsilon*eta_i(t)

for fixed endpoints:

eta_i(t1)
    = 0

eta_i(t2)
    = 0

VARIATION
    -> perturbs(PATH)

# ---------- STATIONARY ACTION ----------
PHYSICAL_PATH
    -> satisfies

delta_S
    = 0

for_all
    admissible_variations

STATIONARY_ACTION
    -> first_variation_vanishes

STATIONARY
    -/> necessarily(MINIMUM)

ACTION
    -> may_be(MINIMUM_MAXIMUM_OR_SADDLE_STATIONARY_VALUE)

# ---------- FIRST VARIATION ----------
delta_S
    =
integral_from_t1_to_t2(
    sum_i[
        partial_L/partial_q_i
        *
        eta_i
        +
        partial_L/partial_q_dot_i
        *
        eta_dot_i
    ]
    dt
)

FIRST_VARIATION
    -> contains(PATH_AND_VELOCITY_VARIATIONS)

# ---------- INTEGRATION BY PARTS ----------
integral(
    partial_L/partial_q_dot_i
    *
    eta_dot_i
    dt
)

=
[
    partial_L/partial_q_dot_i
    *
    eta_i
]_t1^t2

-
integral(
    d/dt(
        partial_L/partial_q_dot_i
    )
    *
    eta_i
    dt
)

FIXED_ENDPOINT_VARIATIONS
    -> eliminate(BOUNDARY_TERM)

# ---------- EULER-LAGRANGE EQUATIONS ----------
therefore

for each i:

d/dt(
    partial_L/partial_q_dot_i
)
-
partial_L/partial_q_i
=
0

STATIONARY_ACTION
    -> yields(EULER_LAGRANGE_EQUATIONS)

# ---------- CLASSICAL MECHANICS ----------
for many standard mechanical systems:

L
    :=
T - V

T
    -> KINETIC_ENERGY

V
    -> POTENTIAL_ENERGY

ACTION
    =
integral(
    (T-V)
    dt
)

LAGRANGIAN_MECHANICS
    -> reformulates(NEWTONIAN_DYNAMICS)

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

EULER_LAGRANGE:

d p_i/dt
    =
partial_L/partial_q_i

GENERALIZED_MOMENTUM
    -> conjugate_to(GENERALIZED_COORDINATE)

# ---------- CYCLIC COORDINATE ----------
if

partial_L/partial_q_i
    = 0

then

d p_i/dt
    = 0

therefore

p_i
    = constant

CYCLIC_COORDINATE
    -> CONSERVED_GENERALIZED_MOMENTUM

# ---------- ENERGY FUNCTION ----------
define

E
    :=
sum_i(
    q_dot_i*p_i
)
-
L

if
    partial_L/partial_t
    = 0

then

dE/dt
    = 0

under
    standard_regularities

TIME_TRANSLATION_SYMMETRY
    -> ENERGY_CONSERVATION

# ---------- HAMILTON PRINCIPLE ----------
HAMILTON_PRINCIPLE
    := stationary_action_for_mechanical_trajectories

delta
integral_from_t1_to_t2(
    L dt
)
=
0

HAMILTON_PRINCIPLE
    ~= ACTION_PRINCIPLE_IN_CLASSICAL_MECHANICS

# ---------- MAUPERTUIS RELATION ----------
MAUPERTUIS_PRINCIPLE
    -> alternative_variational_principle
       for(FIXED_ENERGY_MECHANICAL_PATHS)

ACTION_PRINCIPLES
    -> exist_in(MULTIPLE_RELATED_FORMS)

CONTEXT
    -> determines(ACTION_FUNCTIONAL_AND_CONSTRAINTS)

# ---------- FIELD THEORY ----------
for fields
    phi_a(x)

ACTION
    :=
S[phi]

S[phi]
    =
integral(
    L_density(
        phi_a,
        partial_mu(phi_a),
        x
    )
    d^n x
)

L_density
    -> LAGRANGIAN_DENSITY

# ---------- FIELD EULER-LAGRANGE ----------
stationarity:

delta_S
    = 0

yields

partial_L_density/partial_phi_a
-
partial_mu(
    partial_L_density
    /
    partial(partial_mu phi_a)
)
=
0

ACTION_PRINCIPLE
    -> generates(FIELD_EQUATIONS)

# ---------- EXAMPLE: SCALAR FIELD ----------
L_density
    =
1/2
*
partial_mu(phi)
*
partial^mu(phi)
-
V(phi)

Euler-Lagrange
    ->

BOX(phi)
+
dV/dphi
=
0

under
    chosen_metric_and_sign_convention

FIELD_ACTION
    -> determines(DYNAMICS)

# ---------- ELECTROMAGNETIC ACTION ----------
ELECTROMAGNETIC_FIELD
    -> describable_by(ACTION_FUNCTIONAL)

variation_with_respect_to
    gauge_potential

-> MAXWELL_EQUATIONS
   under(STANDARD_ELECTROMAGNETIC_ACTION)

ACTION_PRINCIPLE
    -> unifies(FIELD_DYNAMICS)

# ---------- GEODESIC ACTION ----------
for particle in curved spacetime:

ACTION
    -> proportional_to(PROPER_TIME_OR_PATH_LENGTH)

variation
    -> GEODESIC_EQUATION

GEODESIC
    -> stationary_worldline_of(ACTION)

# ---------- CONSTRAINTS ----------
if coordinates satisfy

C_a(
    q,
    q_dot,
    t
)
=
0

then

CONSTRAINED_ACTION
    -> may_require(LAGRANGE_MULTIPLIERS)

AUGMENTED_LAGRANGIAN
    :=
L
+
sum_a(
    lambda_a*C_a
)

VARIATION
    -> performed_with(CONSTRAINT_STRUCTURE)

# ---------- BOUNDARY TERMS ----------
ACTION_VARIATION
    -> generates(BOUNDARY_TERMS)

BOUNDARY_CONDITIONS
    -> determine(WHICH_TERMS_VANISH)

FREE_BOUNDARY
    -> may_require(NATURAL_BOUNDARY_CONDITIONS)

FIELD_THEORY
    -> may_require(BOUNDARY_ACTION_TERMS)
       for(WELL_POSED_VARIATIONAL_PRINCIPLE)

# ---------- SYMMETRY ----------
CONTINUOUS_TRANSFORMATION
    -> symmetry_of(ACTION)

if
    delta_S
        -> invariant_up_to_allowed_boundary_term

then

NOETHER_THEOREM
    -> yields(CONSERVED_CURRENT_OR_QUANTITY)

SYMMETRY
    -> central_to(ACTION_BASED_PHYSICS)

# ---------- NOETHER THEOREM ----------
continuous_symmetry
    <-> conservation_law
       under(NOETHER_HYPOTHESES)

examples:

TIME_TRANSLATION
    -> ENERGY

SPACE_TRANSLATION
    -> LINEAR_MOMENTUM

ROTATION
    -> ANGULAR_MOMENTUM

ACTION_PRINCIPLE
    -> exposes(SYMMETRY_CONSERVATION_STRUCTURE)

# ---------- CANONICAL STRUCTURE ----------
p_i
    :=
partial_L/partial_q_dot_i

Hamiltonian:

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

LEGENDRE_TRANSFORM
    -> connects(LAGRANGIAN_AND_HAMILTONIAN_FORMULATIONS)

ACTION_PRINCIPLE
    -> compatible_with(CANONICAL_MECHANICS)

# ---------- HAMILTONIAN ACTION ----------
canonical action:

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

variation
    -> HAMILTON_EQUATIONS

ACTION_FORMALISM
    -> supports(MULTIPLE_DYNAMICAL_REPRESENTATIONS)

# ---------- QUANTUM RELATION ----------
in path integral formulation:

QUANTUM_AMPLITUDE
    -> weighted_sum_over(PATHS)

weight
    :=
exp(
    i*S[path]/hbar
)

CLASSICAL_LIMIT
    -> dominated_by(STATIONARY_PHASE_PATHS)

STATIONARY_ACTION
    -> emerges_as(CLASSICAL_LIMIT_STRUCTURE)

# ---------- DIMENSION ----------
ACTION
    -> physical_dimension_of(
        ENERGY * TIME
    )

PLANCK_CONSTANT
    -> same_physical_dimension_as(ACTION)

S/hbar
    -> DIMENSIONLESS

DIMENSIONAL_CONSISTENCY
    -> required_in(PHYSICAL_ACTION)

# ---------- APPLICATIONS ----------
ACTION_PRINCIPLE
    -> applied_to(CLASSICAL_MECHANICS)
    -> applied_to(FIELD_THEORY)
    -> applied_to(GENERAL_RELATIVITY)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(OPTIMAL_CONTROL)
    -> applied_to(GEOMETRIC_MECHANICS)

# ---------- RELATION GRAPH ----------
LAGRANGIAN
    -> integrated_to(ACTION)

TRAJECTORY
    -> input_to(ACTION_FUNCTIONAL)

VARIATION
    -> changes(TRAJECTORY)

STATIONARY_ACTION
    -> yields(EULER_LAGRANGE_EQUATIONS)

SYMMETRY_OF_ACTION
    -> yields(CONSERVED_QUANTITY)

GENERALIZED_COORDINATE
    -> paired_with(GENERALIZED_MOMENTUM)

LEGENDRE_TRANSFORM
    -> connects(LAGRANGIAN, HAMILTONIAN)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "action principle" ~= "variational principle selecting stationary-action physical histories",
    "principle of stationary action" ~= "requirement that the first variation of the action vanish",
    "Hamilton's principle" ~= "stationary-action principle for classical mechanical trajectories",
    "action functional" ~= "integral of a Lagrangian over a trajectory or field configuration",
    "generalized momentum" ~= "derivative of the Lagrangian with respect to generalized velocity",
    "Noether symmetry" ~= "continuous action symmetry associated with a conservation law under suitable conditions"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    stationary_action_does_not_mean_action_is_always_minimum: true,
    admissible_variations_must_respect_boundary_and_constraint_conditions: true,
    Lagrangian_is_not_unique_under_total_derivative_changes_with_fixed_boundary_context: true,
    field_action_variation_requires_boundary_term_handling: true,
    conservation_laws_require_actual_action_symmetry_and_Noether_hypotheses: true,
    sign_and_metric_conventions_must_be_preserved_in_field_theory: true,
    physical_action_must_be_dimensionally_consistent: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    Euler_Lagrange_relation: 0.99,
    mechanics_relation: 0.99,
    field_theory_relation: 0.99,
    symmetry_Noether_relation: 0.99,
    Hamiltonian_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(system_coordinates_fields_and_domain)
    -> identify(Lagrangian_or_Lagrangian_density)
    -> construct(action_functional)
    -> identify(boundary_and_constraint_conditions)
    -> construct(admissible_variation)
    -> compute(first_variation)
    -> integrate_by_parts(if_required)
    -> analyse(boundary_terms)
    -> derive(Euler_Lagrange_equations)
    -> identify(symmetries_and_conserved_quantities_if_requested)
    -> transform_to(Hamiltonian_form_if_relevant)
    -> validate(sign_units_boundary_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed action-principle semantic graph",
    primary_entity: ACTION_PRINCIPLE,
    preserve: [
        system,
        coordinates_or_fields,
        Lagrangian,
        action,
        domain,
        boundary_conditions,
        constraints,
        sign_conventions,
        units
    ],
    infer_through: [
        variations,
        stationary_action,
        Euler_Lagrange_equations,
        generalized_momenta,
        conserved_quantities,
        Noether_theorem,
        Hamiltonian_formulation,
        field_equations
    ],
    avoid: [
        equating_stationary_action_with minimum action universally,
        discarding_boundary_terms_without justification,
        varying_fields_or_paths_outside admissible class,
        claiming_conservation_without symmetry conditions,
        mixing_metric_or_sign conventions,
        treating_Lagrangian_as unique under total derivative equivalence
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + ACTION_PRINCIPLE
    + ACTION
    + LAGRANGIAN
    + VARIATION
    + STATIONARY_ACTION
    + EULER_LAGRANGE_EQUATION
    + GENERALIZED_MOMENTUM
    + SYMMETRY
    + NOETHER_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> ACTION_PRINCIPLE
    -> SYSTEM_FIELD_IDENTIFICATION
    -> LAGRANGIAN_IDENTIFICATION
    -> ACTION_CONSTRUCTION
    -> BOUNDARY_CONSTRAINT_ANALYSIS
    -> VARIATION_CONSTRUCTION
    -> FIRST_VARIATION_ANALYSIS
    -> BOUNDARY_TERM_ANALYSIS
    -> EULER_LAGRANGE_DERIVATION
    -> SYMMETRY_CONSERVATION_ANALYSIS
    -> HAMILTONIAN_RELATION_ANALYSIS
    -> CONVENTION_UNIT_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW