Gradient hamiltonian position eq
ClassicalMechanics.HarmonicOscillator.gradient_hamiltonian_position_eq
Project documentation
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...
Exact Lean statement
lemma gradient_hamiltonian_position_eq (t : Time) (x : EuclideanSpace ℝ (Fin 1))
(p : EuclideanSpace ℝ (Fin 1)) :
gradient (hamiltonian S t p) x = S.k • xFormal artifact
Lean source
lemma gradient_hamiltonian_position_eq (t : Time) (x : EuclideanSpace ℝ (Fin 1)) (p : EuclideanSpace ℝ (Fin 1)) : gradient (hamiltonian S t p) x = S.k • x := by have h_eq : (fun y : EuclideanSpace ℝ (Fin 1) => hamiltonian S t p y) = fun y => ((1 / (2 : ℝ)) * S.k) * ⟪y, y⟫_ℝ + ((1 / (2 : ℝ)) * (1 / S.m) * ⟪p, p⟫_ℝ) := by funext y simp only [hamiltonian_eq] ring change gradient (fun y : EuclideanSpace ℝ (Fin 1) => hamiltonian S t p y) x = S.k • x rw [h_eq, gradient_add_const', gradient_const_mul_inner_self] module- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean:672-683
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