Research proof index

In the underdamped regime, the selected trajectory satisfies the damped equation of motion.

The selected trajectory has initial velocity IC.v₀.

The Euler Lagrange operator, for a trajectory q : Time → X, and a lagrangian Time → X → X → ℝ, the Euler-Lagrange operator is ∂L/∂q - dₜ(∂L/∂(dₜ q)). -/ noncomputable def eulerLagrangeOp (L : Time → X → X → ℝ) (q : Time → X) : Time → X := fun t => gradient (L t · (∂ₜ q t)) (q t) - ∂ₜ (fun t' => gradient (L t' (q t') ·) (∂ₜ q t')) t lemma eulerLagran...

fromInitialConditions is a right inverse of toInitialConditions: converting initial conditions to amplitude–phase form and back recovers them exactly.

The operator on the momentum-position phase-space whose vanishing is equivalent to the hamilton's equations between the momentum and position. -/ noncomputable def hamiltonEqOp (p : Time → EuclideanSpace ℝ (Fin 1)) (q : Time → EuclideanSpace ℝ (Fin 1)) := ClassicalMechanics.hamiltonEqOp (hamiltonian S) p q /-! ### G.5. Equation of motion if and only if Ha...

The operator on the momentum-position phase-space whose vanishing is equivalent to the hamilton's equations between the momentum and position. -/ noncomputable def hamiltonEqOp (p : Time → EuclideanSpace ℝ (Fin 1)) (q : Time → EuclideanSpace ℝ (Fin 1)) := ClassicalMechanics.hamiltonEqOp (hamiltonian S) p q /-! ### G.5. Equation of motion if and only if Ha...