IndisputableMonolith.Action.EulerLagrange
The EulerLagrange module derives the Euler-Lagrange equation for the cost-rate action S[γ] = ∫ J(γ(t)) dt. Researchers modeling variational principles from the J-functional in Recognition Science would cite these results when analyzing critical paths. The derivation exploits the velocity independence of L(q, q̇) = J(q) to reduce the equation to J'(q) = 0, supported by convexity lemmas from upstream modules.
claimFor the action functional $S[γ] = ∫ J(γ(t)) dt$, the Euler-Lagrange equation simplifies to $J'(q) = 0$ because the Lagrangian $L(q, q̇) = J(q)$ does not depend on velocity.
background
PathSpace defines admissible paths as continuous strictly positive functions on an interval and sets up the action as the integral of J(γ(t)) dt for the variational principle of least action from the d'Alembert cost functional J. Cost.Convexity establishes that Jcost(x) = ½(x + x^{-1}) - 1 is strictly convex on the positive reals, foundational for T5 J-uniqueness. FunctionalConvexity proves convexity of the full action functional S[γ] and discharges the h_min witness that previously made the variational principle conditional.
proof idea
The module organizes its argument by specializing the Euler-Lagrange operator to a Lagrangian without velocity dependence, which directly yields the condition that the derivative of J vanishes. Supporting lemmas verify the equation for the cost rate, establish equivalence to constant rate one, and connect to geodesic equations via the Hessian metric and Christoffel symbols.
why it matters in Recognition Science
This module supplies the EL reduction that feeds directly into the Hamiltonian module, which derives Hamilton's equations from the J-action via the Legendre transform in the small-strain limit. It completes the variational step linking the single functional equation to classical mechanics, consistent with the forcing chain landmarks T5 J-uniqueness, T6 phi fixed point, and T8 D = 3.
scope and limits
- Does not treat Lagrangians that depend explicitly on velocity.
- Does not incorporate external forces or non-conservative terms.
- Does not solve the resulting differential equation J'(q)=0 explicitly.
- Does not address multi-particle or field-theoretic extensions.
used by (1)
depends on (4)
declarations in this module (12)
-
def
costRateELHolds -
theorem
costRateEL_const_one -
theorem
costRateEL_implies_const_one -
theorem
costRateEL_iff_const_one -
def
hessianMetric -
lemma
hessianMetric_eq -
def
christoffel -
def
geodesicEquationHolds -
theorem
geodesic_iff_hessianEnergy_EL -
theorem
const_one_is_geodesic -
theorem
ground_state_is_unique_critical_point -
def
eulerLagrange_status