Research proof index

When γ = 0, the damped equation of motion is equivalent to the equation of motion for the corresponding undamped harmonic oscillator.

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...

The hamiltonian as a function of time, momentum and position. -/ noncomputable def hamiltonian (t : Time) (p : EuclideanSpace ℝ (Fin 1)) (x : EuclideanSpace ℝ (Fin 1)) : ℝ := ⟪p, (toCanonicalMomentum S t x).symm p⟫_ℝ - S.lagrangian t x ((toCanonicalMomentum S t x).symm p) /-! #### G.2.1. Equality for the Hamiltonian We prove a simple equality for the Hami...

The energy of the harmonic oscillator is the kinetic energy plus the potential energy. -/ noncomputable def energy (xₜ : Time → EuclideanSpace ℝ (Fin 1)) : Time → ℝ := fun t => kineticEnergy S xₜ t + potentialEnergy S (xₜ t) /-! ### C.2. Simple equalities for the energies -/ lemma kineticEnergy_eq (xₜ : Time → EuclideanSpace ℝ (Fin 1)) : kineticEnergy S x...

A vector-valued plane wave travelling in the direction of s with propagation speed c. -/ noncomputable def planeWave (f₀ : ℝ → EuclideanSpace ℝ (Fin d)) (c : ℝ) (s : Direction d) : Time → Space d → EuclideanSpace ℝ (Fin d) := fun t x => f₀ (⟪x, s.unit⟫ℝ - c * t) lemma planeWave_eq {d f₀ c x} {s : Direction d} : planeWave f₀ c s t x = f₀ (⟪x, s.unit⟫...