IndisputableMonolith.Action.EulerLagrange
The module reduces the Euler-Lagrange equation for the J-cost action functional to the condition J prime equals zero. Researchers deriving equations of motion from the Recognition cost would cite the reduction. The argument follows at once from the standard EL formula once the Lagrangian is seen to lack velocity dependence.
claimFor the action $S[\gamma]=\int_a^b J(\gamma(t))\,dt$ with Lagrangian $L(q,\dot q)=J(q)$, the Euler-Lagrange equation reduces to $J'(\gamma(t))=0$.
background
PathSpace defines admissible paths on intervals and the J-action functional $S[\gamma]=\int J(\gamma(t)),dt$ together with fixed-endpoint boundary conditions. FunctionalConvexity supplies the convexity needed to remove conditional hypotheses from the variational principle. Cost.Convexity establishes that both $J_{\log}(t)=\cosh t-1$ and $J_{\rm cost}(x)=\frac12(x+x^{-1})-1$ are strictly convex, the foundation for T5 uniqueness.
proof idea
The module assembles direct reductions of the EL operator for velocity-independent Lagrangians. It proves equivalences between the cost-rate satisfying EL, constancy of one, and geodesic conditions via the Hessian metric. Sibling lemmas connect these statements to the ground-state uniqueness result.
why it matters in Recognition Science
The module supplies the critical-point condition $J'(q)=0$ that the downstream Hamiltonian module uses to obtain conjugate momentum and Hamilton's equations in the small-strain limit of the J-action. It therefore bridges the T5 J-uniqueness step of the forcing chain to classical mechanics.
scope and limits
- Does not derive the EL equation for velocity-dependent Lagrangians.
- Does not solve the reduced equation $J'(q)=0$ explicitly.
- Does not incorporate time-dependent potentials or external forces.
- Does not address boundary variations beyond fixed endpoints.
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