IndisputableMonolith.Action.EulerLagrange
The EulerLagrange module derives the Euler-Lagrange equation for the cost-rate action S[γ] = ∫ J(γ(t)) dt, reducing it to J'(q) = 0 because the Lagrangian depends only on position. Researchers applying the variational principle from Recognition Science to derive equations of motion would cite these results when analyzing critical paths of the J-functional. The derivations follow by direct substitution into the standard EL operator, using convexity established upstream to confirm uniqueness of solutions.
claimFor the action functional $S[γ] = ∫_a^b J(γ(t)) dt$ with Lagrangian $L(q, q̇) = J(q)$, the Euler-Lagrange equation reduces to $J'(γ(t)) = 0$.
background
This module sits inside the Action domain and builds directly on the J-action functional introduced in PathSpace, where admissible paths are continuous strictly positive functions on a closed interval and actionJ γ denotes the integral of J along the path. Upstream Cost.Convexity proves Jcost(x) = ½(x + x⁻¹) - 1 is strictly convex on the positive reals, while FunctionalConvexity lifts this to convexity of the full integral functional S[γ]. The local setting is the variational principle for the d'Alembert cost J, with the Lagrangian independent of velocity.
proof idea
The module organizes lemmas that apply the standard Euler-Lagrange operator to the velocity-independent Lagrangian L(q) = J(q). The time-derivative term vanishes identically, leaving the stationarity condition J'(q) = 0. Supporting results establish equivalence to constant paths and uniqueness of the ground state critical point, drawing on the convexity properties imported from FunctionalConvexity and Cost.Convexity.
why it matters in Recognition Science
The module supplies the explicit Euler-Lagrange reduction that the downstream Hamiltonian module uses to pass from the J-action to the Legendre-transformed Hamiltonian formulation. It completes the step from the variational setup in PathSpace to the equations of motion, aligning with the J-uniqueness result T5 in the forcing chain. The results also underpin later claims that constant paths are geodesics and that the ground state is the unique critical point.
scope and limits
- Does not treat Lagrangians that depend explicitly on velocity.
- Does not derive the full geodesic equation without the hessianMetric companion lemma.
- Does not address time-dependent or non-autonomous extensions of the action.
- Does not connect the EL equation to the eight-tick octave or spatial dimension D = 3.
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