Calculus of Variations Ontology

flow "CalculusOfVariationsOntology" v1.0

# ============================================================
# CALCULUS OF VARIATIONS ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
CALCULUS_OF_VARIATIONS := {
    type: MathematicalOptimisationFramework,
    domain: VariationalCalculus,
    purpose: "Determine functions, curves, fields or trajectories that make a functional stationary or extremal subject to specified boundary and constraint conditions."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FUNCTIONAL,
    ADMISSIBLE_FUNCTION,
    VARIATION,
    FIRST_VARIATION,
    SECOND_VARIATION,
    EXTREMAL,
    STATIONARY_FUNCTION,
    EULER_LAGRANGE_EQUATION,
    LAGRANGIAN,
    FUNDAMENTAL_LEMMA,
    BOUNDARY_CONDITION,
    NATURAL_BOUNDARY_CONDITION,
    CONSTRAINT,
    LAGRANGE_MULTIPLIER,
    DIRECT_METHOD
}

# ---------- FUNCTIONAL ----------
FUNCTIONAL
    := J

J:
    FUNCTION_SPACE
    -> SCALAR_FIELD

example:

J[y]
    :=
integral_from_a_to_b(
    F(
        x,
        y(x),
        y_prime(x)
    )
    dx
)

FUNCTIONAL
    -> objective_of(VARIATIONAL_PROBLEM)

# ---------- ADMISSIBLE CLASS ----------
ADMISSIBLE_CLASS
    := {
        y :
        y satisfies(
            DOMAIN,
            REGULARITY,
            BOUNDARY_CONDITIONS,
            CONSTRAINTS
        )
    }

OPTIMISATION
    -> performed_over(ADMISSIBLE_CLASS)

# ---------- EXTREMUM ----------
y_star
    -> LOCAL_MINIMISER

if

J[y_star]
    <=
J[y]

for all admissible y
sufficiently_close_to(y_star)
under
    specified_function_space_topology

LOCAL_MAXIMISER
    -> analogous_with(REVERSED_INEQUALITY)

# ---------- VARIATION ----------
choose
    eta
        -> admissible_variation

define

y_epsilon
    :=
y + epsilon*eta

epsilon
    -> scalar_variation_parameter

VARIATION
    -> direction_in(FUNCTION_SPACE)

# ---------- FIRST VARIATION ----------
delta_J[y;eta]
    :=
d/d_epsilon(
    J[y+epsilon*eta]
)
evaluated_at
    epsilon=0

if
    y
        -> interior_local_extremum

then

delta_J[y;eta]
    = 0

for_all
    admissible_eta

FIRST_VARIATION
    -> stationarity_condition

# ---------- STATIONARY FUNCTION ----------
y
    -> STATIONARY

iff

delta_J[y;eta]
    = 0

for_all admissible eta

STATIONARY_FUNCTION
    -> candidate_for(EXTREMUM)

STATIONARY
    -/> automatically(MINIMUM_OR_MAXIMUM)

# ---------- BASIC FIRST VARIATION ----------
for

J[y]
    =
integral_from_a_to_b(
    F(x,y,y_prime)
    dx
)

delta_J
    =
integral_from_a_to_b(
    F_y*eta
    +
    F_y_prime*eta_prime
    dx
)

# ---------- INTEGRATION BY PARTS ----------
integral(
    F_y_prime*eta_prime
    dx
)

=
[
    F_y_prime*eta
]_a^b

-
integral(
    d/dx(F_y_prime)
    *
    eta
    dx
)

therefore

delta_J
    =
BOUNDARY_TERM
+
integral(
    [
        F_y
        -
        d/dx(F_y_prime)
    ]
    eta
    dx
)

# ---------- FIXED ENDPOINTS ----------
if

y(a)=A
and
y(b)=B

then

eta(a)
    = eta(b)
    = 0

BOUNDARY_TERM
    = 0

FIRST_VARIATION
    -> interior_integral_only

# ---------- FUNDAMENTAL LEMMA ----------
if

integral_from_a_to_b(
    g(x)*eta(x)
    dx
)
=
0

for_all admissible smooth eta
vanishing_at_endpoints

then

g(x)
    = 0

under
    standard_regularities

FUNDAMENTAL_LEMMA
    -> converts(INTEGRAL_IDENTITY_TO_POINTWISE_EQUATION)

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

F_y
-
d/dx(
    F_y_prime
)
=
0

EULER_LAGRANGE_EQUATION
    -> necessary_condition_for(SMOOTH_STATIONARY_FUNCTION)

# ---------- MULTIPLE DEPENDENT VARIABLES ----------
for
    y_1,...,y_m

functional:

J[y_1,...,y_m]
    =
integral(
    F(
        x,
        y_i,
        y_i_prime
    )
    dx
)

for each i:

partial_F/partial_y_i
-
d/dx(
    partial_F/partial_y_i_prime
)
=
0

MULTIVARIABLE_FUNCTIONAL
    -> coupled_Euler_Lagrange_system

# ---------- HIGHER ORDER ----------
if

F
    =
F(
    x,
    y,
    y_prime,
    ...,
    y^(n)
)

then

sum_from_k_0_to_n(
    (-1)^k
    *
    d^k/dx^k(
        partial_F/partial_y^(k)
    )
)
=
0

HIGHER_ORDER_VARIATIONAL_PROBLEM
    -> HIGHER_ORDER_EULER_LAGRANGE_EQUATION

# ---------- FIELD FORM ----------
for field
    u: Omega -> R

J[u]
    =
integral_Omega(
    F(
        x,
        u,
        grad(u)
    )
    dx
)

Euler-Lagrange PDE:

partial_F/partial_u
-
div(
    partial_F/partial_grad_u
)
=
0

CALCULUS_OF_VARIATIONS
    -> generates(PDEs)

# ---------- SECOND VARIATION ----------
delta_squared_J[y;eta]
    :=
d^2/d_epsilon^2(
    J[y+epsilon*eta]
)
evaluated_at
    epsilon=0

SECOND_VARIATION
    -> quadratic_form_in(eta)

POSITIVE_SECOND_VARIATION
    -> evidence_for(LOCAL_MINIMUM)
       under(SUFFICIENT_CONDITIONS)

NEGATIVE_SECOND_VARIATION
    -> evidence_for(LOCAL_MAXIMUM)

# ---------- LEGENDRE CONDITION ----------
for local minimum:

F_y_prime_y_prime
    >= 0

along extremal
under
    standard_smoothness_conditions

LEGENDRE_CONDITION
    -> necessary_second_order_condition

STRICT_POSITIVITY
    -> strengthened_condition

# ---------- JACOBI CONDITION ----------
SECOND_VARIATION
    -> related_to(JACOBI_DIFFERENTIAL_EQUATION)

CONJUGATE_POINTS
    -> affect(MINIMALITY)

JACOBI_CONDITION
    -> further_second_order_test_for(MINIMUM)

# ---------- WEIERSTRASS CONDITION ----------
WEIERSTRASS_EXCESS_FUNCTION
    -> compares(LAGRANGIAN_WITH_TANGENT_LINEARISATION_IN_DERIVATIVE_VARIABLE)

WEIERSTRASS_NECESSARY_CONDITION
    -> stronger_local_minimum_condition
       for(STRONG_VARIATIONS)

# ---------- NATURAL BOUNDARY ----------
if
    endpoint_value
        -> FREE

then

eta(endpoint)
    -> arbitrary

boundary term:

F_y_prime
*
eta

requires

F_y_prime
    = 0

at free endpoint
under
    standard_vertical_endpoint_context

NATURAL_BOUNDARY_CONDITION
    -> generated_by(BOUNDARY_VARIATION)

# ---------- TRANSVERSALITY ----------
if
    endpoint
        -> allowed_to_move_on(CONSTRAINT_CURVE_OR_SURFACE)

then

BOUNDARY_TERM
    -> produces(TRANSVERSALITY_CONDITION)

TRANSVERSALITY
    -> general_boundary_stationarity_condition

# ---------- ISOPERIMETRIC PROBLEM ----------
optimise

J[y]

subject_to

K[y]
    = c

introduce scalar
    lambda

AUGMENTED_FUNCTIONAL
    :=
J[y]
-
lambda*K[y]

EULER_LAGRANGE
    -> applied_to(AUGMENTED_INTEGRAND)

ISOPERIMETRIC_CONSTRAINT
    -> handled_by(LAGRANGE_MULTIPLIER)

# ---------- BELTRAMI IDENTITY ----------
if
    F
        -> independent_of(x)

then

F
-
y_prime*F_y_prime
    =
constant

along extremal

BELTRAMI_IDENTITY
    -> first_integral

# ---------- CYCLIC VARIABLE ----------
if
    F
        -> independent_of(y)

then

F_y_prime
    =
constant

CYCLIC_COORDINATE
    -> CONSERVED_QUANTITY

# ---------- PARAMETER-INVARIANT PROBLEMS ----------
CURVE_FUNCTIONAL
    -> may_be(INVARIANT_UNDER_REPARAMETERISATION)

REPARAMETERISATION_INVARIANCE
    -> can_produce(DEGENERATE_EULER_LAGRANGE_STRUCTURE)

GEOMETRIC_FUNCTIONALS
    -> require(PARAMETERISATION_AWARE_REASONING)

# ---------- GEODESICS ----------
LENGTH_FUNCTIONAL
    :=
integral(
    sqrt(
        g_ij
        *
        x_dot^i
        *
        x_dot^j
    )
    dt
)

STATIONARY_CURVES
    -> GEODESICS

CALCULUS_OF_VARIATIONS
    -> derives(GEODESIC_EQUATIONS)

# ---------- ACTION ----------
ACTION_FUNCTIONAL
    :=
integral_from_t1_to_t2(
    L(q,q_dot,t)
    dt
)

delta_ACTION
    = 0

-> EULER_LAGRANGE_EQUATIONS

CLASSICAL_MECHANICS
    -> variational_formulation_by(STATIONARY_ACTION)

# ---------- DIRECT METHOD ----------
to establish existence of minimiser:

choose
    MINIMISING_SEQUENCE

seek:
    COERCIVITY
    COMPACTNESS
    LOWER_SEMICONTINUITY
    CLOSEDNESS_OF_ADMISSIBLE_CLASS

then

limit_of_minimising_sequence
    -> candidate_minimiser

DIRECT_METHOD
    -> existence_theory

# ---------- WEAK LOWER SEMICONTINUITY ----------
if
    y_n
        -> weakly_converges_to(y)

then desirable property:

J[y]
    <=
liminf(
    J[y_n]
)

WEAK_LOWER_SEMICONTINUITY
    -> preserves(MINIMISATION_UNDER_WEAK_LIMITS)

# ---------- COERCIVITY ----------
J[y]
    -> infinity

as
    ||y||
    -> infinity

in relevant admissible directions

COERCIVITY
    -> prevents(MINIMISING_SEQUENCE_ESCAPE)

# ---------- APPLICATIONS ----------
CALCULUS_OF_VARIATIONS
    -> applied_to(MECHANICS)
    -> applied_to(GEOMETRY)
    -> applied_to(PDEs)
    -> applied_to(OPTIMAL_CONTROL)
    -> applied_to(ELASTICITY)
    -> applied_to(FLUID_MECHANICS)
    -> applied_to(FINITE_ELEMENT_METHODS)
    -> applied_to(PHYSICS)

# ---------- RELATION GRAPH ----------
FUNCTIONAL
    -> acts_on(ADMISSIBLE_FUNCTION)

VARIATION
    -> perturbs(ADMISSIBLE_FUNCTION)

FIRST_VARIATION
    -> determines(STATIONARITY)

FUNDAMENTAL_LEMMA
    -> yields(EULER_LAGRANGE_EQUATION)

SECOND_VARIATION
    -> investigates(MINIMALITY_OR_MAXIMALITY)

BOUNDARY_TERM
    -> yields(NATURAL_BOUNDARY_CONDITIONS)

CONSTRAINT
    -> introduces(LAGRANGE_MULTIPLIER)

DIRECT_METHOD
    -> addresses(EXISTENCE_OF_MINIMISERS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "calculus of variations" ~= "optimisation of functionals over spaces of functions",
    "functional" ~= "scalar-valued mapping whose input is a function or field",
    "first variation" ~= "directional derivative of a functional",
    "stationary function" ~= "function for which the first variation vanishes in every admissible direction",
    "Euler-Lagrange equation" ~= "differential stationarity equation derived from a variational functional",
    "direct method" ~= "existence method using minimising sequences, compactness and lower semicontinuity"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    Euler_Lagrange_equation_is_necessary_not_generally_sufficient: true,
    admissible_variations_must_respect_boundary_and_constraint_conditions: true,
    fixed_and_free_endpoint_problems_have_different_boundary_terms: true,
    second_variation_tests_require_additional_regularities_and_conditions: true,
    constrained_problems_require_multiplier_or_equivalent_constraint_handling: true,
    existence_of_formal_extremal_does_not_guarantee_global_minimiser: true,
    direct_method_requires_topology_compactness_and_lower_semicontinuity_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    first_variation_relation: 0.99,
    Euler_Lagrange_relation: 0.99,
    second_variation_relation: 0.99,
    boundary_constraint_relation: 0.99,
    direct_method_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(functional_and_admissible_class)
    -> identify(boundary_constraints_and_regularities)
    -> construct(admissible_variation)
    -> compute(first_variation)
    -> integrate_by_parts(if_required)
    -> apply(fundamental_lemma)
    -> derive(Euler_Lagrange_equations)
    -> derive(boundary_or_transversality_conditions_if_required)
    -> analyse(second_variation_and_minimality_conditions)
    -> analyse(existence_via_direct_method_if_requested)
    -> verify(all_hypotheses_and_admissibility)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed calculus-of-variations semantic graph",
    primary_entity: CALCULUS_OF_VARIATIONS,
    preserve: [
        functional,
        admissible_class,
        Lagrangian,
        boundary_conditions,
        constraints,
        regularity,
        topology,
        candidate_extremal
    ],
    infer_through: [
        variations,
        first_variations,
        Euler_Lagrange_equations,
        second_variations,
        natural_boundary_conditions,
        transversality,
        Lagrange_multipliers,
        direct_methods
    ],
    avoid: [
        treating_stationary_function_as guaranteed minimiser,
        ignoring_boundary_terms,
        using_arbitrary_variations_that_violate constraints,
        applying_second_order_conditions_without hypotheses,
        confusing_formal_solution_with existence_or_uniqueness,
        omitting_function_space_topology_in direct method arguments
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + CALCULUS_OF_VARIATIONS
    + FUNCTIONAL
    + VARIATION
    + FIRST_VARIATION
    + SECOND_VARIATION
    + EULER_LAGRANGE_EQUATION
    + NATURAL_BOUNDARY_CONDITION
    + LAGRANGE_MULTIPLIER
    + DIRECT_METHOD
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> CALCULUS_OF_VARIATIONS
    -> FUNCTIONAL_ADMISSIBLE_CLASS_IDENTIFICATION
    -> BOUNDARY_CONSTRAINT_ANALYSIS
    -> VARIATION_CONSTRUCTION
    -> FIRST_VARIATION_ANALYSIS
    -> INTEGRATION_BY_PARTS
    -> FUNDAMENTAL_LEMMA_APPLICATION
    -> EULER_LAGRANGE_DERIVATION
    -> BOUNDARY_TRANSVERSALITY_ANALYSIS
    -> SECOND_VARIATION_MINIMALITY_ANALYSIS
    -> DIRECT_METHOD_EXISTENCE_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW