Research proof index

Explicit reformulation (by the chain rule): δL(t, q, dₜ q) = ∂F/∂t(t, q) + ⟨∇ᵣF(t, q), dₜ q⟩ or δL(t, q, dₜ q) = fderiv ℝ F (t, q) (1, dₜ q)

A velocity-only function that is a total time derivative must be linear in velocity. If δL depends only on velocity and equals d/dt F(t, q) for some F, then δL(dₜ q) = ⟨g, dₜ q⟩ for some constant vector g. This characterization comes from the requirement that: - d/dt F(t, q) = ∂F/∂t + ⟨∇F, dₜ q⟩ = ∂F/∂t + ⟨∇F, dₜ q⟩ - For the result to be independent of q...

If δL is a total time derivative (of a smooth function), then it is smooth

Corollary: If L and L' differ by a total time derivative, then the corresponding Euler-Lagrange operators coincide

Total time derivative has a variational derivative, which is zero

If two lagrangians, L and L', differ by a total time derivative, and L has a variational derivative grad, then so does L'. -/ lemma totalTimeDerivative_hasVarGradientAt_equivalence [CompleteSpace X] (L δL : Time → X → X → ℝ) (hδL : IsTotalTimeDerivative δL) (q : Time → X) (hq : ContDiff ℝ ∞ q) (grad : Time → X) (hgrad : HasVarGradientAt (fun q' t => L t (...