Euler–Lagrange Equation Ontology

flow "EulerLagrangeEquationOntology" v1.0

# ============================================================
# EULER-LAGRANGE EQUATION ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/variational-calculus/euler-lagrange-equation-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Variational Calculus"
ENTITY := "Euler-Lagrange Equation"

# ---------- CORE ENTITY ----------
EULER_LAGRANGE_EQUATION := {
    type: VariationalDifferentialEquation,
    domain: CalculusOfVariations,
    purpose: "Provide the differential stationarity condition satisfied by sufficiently smooth extremals of variational functionals."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FUNCTIONAL,
    LAGRANGIAN,
    VARIATION,
    FIRST_VARIATION,
    EXTREMAL,
    STATIONARY_FUNCTION,
    EULER_LAGRANGE_EQUATION,
    FUNDAMENTAL_LEMMA,
    INTEGRATION_BY_PARTS,
    BOUNDARY_CONDITION,
    NATURAL_BOUNDARY_CONDITION,
    GENERALIZED_COORDINATE,
    ACTION,
    FIELD_EQUATION,
    CONSERVED_QUANTITY
}

# ---------- BASIC FUNCTIONAL ----------
let

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

F
    -> LAGRANGIAN_OR_INTEGRAND

y
    -> ADMISSIBLE_FUNCTION

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

eta(x)
    -> ADMISSIBLE_VARIATION

define

y_epsilon(x)
    :=
y(x)
+
epsilon*eta(x)

VARIATION
    -> perturbs(CANDIDATE_FUNCTION)

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

STATIONARITY
    -> requires(
        delta_J[y;eta]=0
        for_all_admissible_eta
    )

# ---------- DIFFERENTIATION OF FUNCTIONAL ----------
delta_J
    =
integral_from_a_to_b(
    F_y
    *
    eta
    +
    F_y_prime
    *
    eta_prime
    dx
)

FIRST_VARIATION
    -> contains(ETA_AND_ETA_PRIME_TERMS)

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

=
[
    F_y_prime
    *
    eta
]_a^b

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

therefore

delta_J
    =
[
    F_y_prime*eta
]_a^b

+
integral_from_a_to_b(
    [
        F_y
        -
        d/dx(F_y_prime)
    ]
    *
    eta
    dx
)

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

y(a)=A

and

y(b)=B

then

eta(a)=0

eta(b)=0

therefore

BOUNDARY_TERM
    = 0

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

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

for_all
    sufficiently_smooth_eta
    vanishing_at_endpoints

then

g(x)
    = 0

under
    standard_regularities

FUNDAMENTAL_LEMMA
    -> converts(INTEGRAL_STATIONARITY_TO_POINTWISE_EQUATION)

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

partial_F/partial_y
-
d/dx(
    partial_F/partial_y_prime
)
=
0

EULER_LAGRANGE_EQUATION
    -> necessary_condition_for(SMOOTH_EXTREMAL)

# ---------- ALTERNATIVE FORM ----------
d/dx(
    partial_F/partial_y_prime
)
-
partial_F/partial_y
=
0

SIGN_CONVENTION
    -> algebraically_equivalent

# ---------- MULTIPLE DEPENDENT VARIABLES ----------
for

q_1,...,q_n

and

J[q]
    =
integral(
    F(
        x,
        q_i,
        q_i_prime
    )
    dx
)

for each i:

d/dx(
    partial_F/partial_q_i_prime
)
-
partial_F/partial_q_i
=
0

MULTIPLE_COORDINATES
    -> SYSTEM_OF_EULER_LAGRANGE_EQUATIONS

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

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

then

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

HIGHER_ORDER_EULER_LAGRANGE
    -> requires(REPEATED_INTEGRATION_BY_PARTS)

# ---------- FIELD THEORY ----------
for

u:
    Omega subset_of R^n
    -> R

functional

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

Euler-Lagrange PDE:

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

FIELD_EULER_LAGRANGE
    -> variational_PDE

# ---------- MULTIPLE FIELDS ----------
for fields

phi_a

Lagrangian_density

L(
    x,
    phi_a,
    partial_mu phi_a
)

equations:

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

FIELD_THEORY
    -> SYSTEM_OF_EULER_LAGRANGE_PDEs

# ---------- NATURAL BOUNDARY CONDITION ----------
if endpoint
    -> FREE

then

eta(endpoint)
    -> arbitrary

boundary stationarity requires

partial_F/partial_y_prime
    = 0

at free endpoint
under
    standard_vertical_endpoint_variation

NATURAL_BOUNDARY_CONDITION
    -> arises_from(BOUNDARY_TERM)

# ---------- TRANSVERSALITY ----------
if endpoint
    -> moves_on(CONSTRAINT_CURVE)

then

BOUNDARY_TERM
    -> combined_with(ENDPOINT_GEOMETRY)

TRANSVERSALITY_CONDITION
    -> generalises(NATURAL_BOUNDARY_CONDITION)

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

partial_F/partial_y
    = 0

then

d/dx(
    partial_F/partial_y_prime
)
=
0

therefore

partial_F/partial_y_prime
    =
constant

CYCLIC_COORDINATE
    -> CONSERVED_GENERALIZED_MOMENTUM

# ---------- BELTRAMI IDENTITY ----------
if

partial_F/partial_x
    = 0

then along extremal:

F
-
y_prime
*
partial_F/partial_y_prime
=
constant

BELTRAMI_IDENTITY
    -> first_integral_of(EULER_LAGRANGE_EQUATION)

# ---------- MECHANICS ----------
ACTION
    :=
S[q]

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

STATIONARY_ACTION
    -> yields

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

EULER_LAGRANGE_EQUATIONS
    -> equations_of_motion_in(LAGRANGIAN_MECHANICS)

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

EULER_LAGRANGE
    ->

d p_i/dt
    =
partial_L/partial_q_i

if
    q_i cyclic

then

p_i
    -> CONSERVED

# ---------- EXAMPLE: FREE PARTICLE ----------
L
    =
1/2*m*q_dot^2

partial_L/partial_q
    = 0

partial_L/partial_q_dot
    =
m*q_dot

Euler-Lagrange:

d/dt(
    m*q_dot
)
=
0

therefore

m*q_double_dot
    = 0

FREE_PARTICLE
    -> constant_momentum

# ---------- EXAMPLE: POTENTIAL ----------
L
    =
1/2*m*q_dot^2
-
V(q)

Euler-Lagrange:

m*q_double_dot
+
dV/dq
=
0

equivalently

m*q_double_dot
=
-
dV/dq

LAGRANGIAN_MECHANICS
    -> reproduces(NEWTONIAN_FORCE_EQUATION)

# ---------- GEODESIC ----------
for metric
    g_ij

energy Lagrangian:

L
    =
1/2
*
g_ij(x)
*
x_dot^i
*
x_dot^j

Euler-Lagrange
    -> GEODESIC_EQUATION

x_double_dot^k
+
Gamma^k_ij
*
x_dot^i
*
x_dot^j
=
0

GEODESIC
    -> variational_extremal

# ---------- CONSTRAINED PROBLEM ----------
if

J[y]
    -> extremised

subject_to

K[y]
    = c

then introduce

lambda

and use

F_tilde
    :=
F
+
lambda*G

EULER_LAGRANGE
    -> applied_to(AUGMENTED_LAGRANGIAN)

# ---------- SECOND VARIATION RELATION ----------
EULER_LAGRANGE
    -> identifies(STATIONARY_CANDIDATE)

SECOND_VARIATION
    -> needed_to_investigate(MINIMUM_OR_MAXIMUM)

EULER_LAGRANGE_SATISFIED
    -/> guarantees(MINIMUM)

# ---------- WEAK FORM ----------
Euler-Lagrange PDE

-> multiply_by(TEST_FUNCTION)
-> integrate
-> integrate_by_parts

WEAK_FORM
    -> equivalent_under(APPROPRIATE_REGULARITY)

VARIATIONAL_FORMULATION
    -> basis_of(FINITE_ELEMENT_METHODS)

# ---------- NOETHER RELATION ----------
CONTINUOUS_SYMMETRY_OF_ACTION
    -> CONSERVED_QUANTITY

NOETHER_THEOREM
    -> connects(SYMMETRY, EULER_LAGRANGE_DYNAMICS)

EULER_LAGRANGE_STRUCTURE
    -> central_to(VARIATIONAL_PHYSICS)

# ---------- APPLICATIONS ----------
EULER_LAGRANGE_EQUATION
    -> applied_to(CALCULUS_OF_VARIATIONS)
    -> applied_to(CLASSICAL_MECHANICS)
    -> applied_to(GEOMETRY)
    -> applied_to(PDEs)
    -> applied_to(OPTIMAL_CONTROL)
    -> applied_to(FIELD_THEORY)
    -> applied_to(ELASTICITY)
    -> applied_to(FINITE_ELEMENT_ANALYSIS)

# ---------- RELATION GRAPH ----------
FUNCTIONAL
    -> varied_by(VARIATION)

FIRST_VARIATION
    -> integrated_by_parts

FUNDAMENTAL_LEMMA
    -> yields(EULER_LAGRANGE_EQUATION)

EULER_LAGRANGE_EQUATION
    -> characterises(STATIONARY_EXTREMALS)

BOUNDARY_TERM
    -> yields(NATURAL_BOUNDARY_CONDITION)

SYMMETRY
    -> may_yield(CONSERVED_QUANTITY)

SECOND_VARIATION
    -> classifies(STATIONARY_CANDIDATE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Euler-Lagrange equation" ~= "differential stationarity condition for a variational functional",
    "Euler equation of the calculus of variations" ~= "Euler-Lagrange equation",
    "Lagrangian derivative equation" ~= "equation balancing direct and derivative dependence of a variational integrand",
    "stationary action equation" ~= "Euler-Lagrange equation applied to an action functional",
    "cyclic coordinate" ~= "coordinate absent explicitly from the Lagrangian",
    "natural boundary condition" ~= "boundary stationarity condition arising from a free endpoint variation"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    Euler_Lagrange_equation_is_generally_necessary_not_sufficient_for_extremum: true,
    admissible_variations_must_obey_boundary_and_constraint_conditions: true,
    integration_by_parts_boundary_terms_must_not_be_discarded_for_free_endpoints: true,
    higher_order_functionals_require_generalised_equations: true,
    field_functionals_require_divergence_or_partial_derivative_form: true,
    constrained_problems_require_augmented_or_equivalent_formulation: true,
    regularity_assumptions_are_required_for_classical_derivation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    first_variation_derivation: 0.99,
    fundamental_lemma_relation: 0.99,
    boundary_relation: 0.99,
    mechanics_field_relation: 0.99,
    conservation_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(functional_and_Lagrangian)
    -> identify(dependent_independent_variables)
    -> identify(boundary_and_constraint_conditions)
    -> construct(admissible_variation)
    -> compute(first_variation)
    -> integrate_by_parts
    -> retain_or_eliminate(boundary_terms_as_conditions_require)
    -> apply(fundamental_lemma)
    -> derive(Euler_Lagrange_equation)
    -> simplify_using(symmetry_or_first_integrals_if_available)
    -> analyse(second_variation_if_extremum_type_requested)
    -> validate(regularity_and_admissibility)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Euler-Lagrange semantic graph",
    primary_entity: EULER_LAGRANGE_EQUATION,
    preserve: [
        functional,
        Lagrangian,
        independent_variable,
        dependent_variables,
        derivatives,
        boundary_conditions,
        constraints,
        regularity
    ],
    infer_through: [
        variations,
        first_variations,
        integration_by_parts,
        fundamental_lemma,
        natural_boundary_conditions,
        conserved_momenta,
        Beltrami_identity,
        field_equations
    ],
    avoid: [
        treating_Euler_Lagrange_solution_as guaranteed minimum,
        ignoring_boundary_terms,
        applying_fixed_endpoint variation_to free_endpoint problem,
        omitting_constraints,
        using_ordinary_equation_for higher_derivative functional,
        ignoring_regularities_needed_for classical derivatives
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + EULER_LAGRANGE_EQUATION
    + FUNCTIONAL
    + LAGRANGIAN
    + FIRST_VARIATION
    + FUNDAMENTAL_LEMMA
    + BOUNDARY_CONDITION
    + NATURAL_BOUNDARY_CONDITION
    + ACTION
    + CONSERVED_QUANTITY
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> EULER_LAGRANGE_EQUATION
    -> FUNCTIONAL_LAGRANGIAN_IDENTIFICATION
    -> VARIABLE_ANALYSIS
    -> BOUNDARY_CONSTRAINT_ANALYSIS
    -> VARIATION_CONSTRUCTION
    -> FIRST_VARIATION_COMPUTATION
    -> INTEGRATION_BY_PARTS
    -> BOUNDARY_TERM_ANALYSIS
    -> FUNDAMENTAL_LEMMA_APPLICATION
    -> EQUATION_DERIVATION
    -> SYMMETRY_FIRST_INTEGRAL_ANALYSIS
    -> SECOND_VARIATION_ANALYSIS
    -> REGULARITY_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW