Research proof index

For zero initial conditions, the trajectory is zero. -/ @[simp] lemma trajectory_zero : trajectory S 0 = fun _ => 0 := by simp [trajectory_eq] /-! ### B.3. Smoothness of the trajectories The trajectories for any initial conditions are smooth functions of time. -/ @[fun_prop] lemma trajectory_contDiff (S : HarmonicOscillator) (IC : InitialConditions) {n :...

For nonzero amplitude, the velocity is zero exactly at phase times φ + nπ.

The velocity vanishes exactly when the trajectory has norm equal to the amplitude.

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...

The oscillator mass metric is constant in the global tangent-bundle chart.

The mass metric unit ball is bounded in the model norm.