Metrics of curves in shape optimization and analysis - Andrea Carlo ...

The proof is by direct computation.For example, from (4.5) we easily obtain (2.7), that is∫D c,h avg c (c) = h + (c − avg c (c))(D s h · D s c) ds .cIf C(t, θ) is a homotopy, we can obtain a different interpretation of all previousequalities substituting formally C for O, ∂ t for D c,h and eventually ∂ t C for h.The above proposition helps in formalizing and speeding up calculus ongeometric quantities, as in this example.Example 4.6 We can Gâteaux-derive the “second arc parameter derivativeoperator D ss ” by repeated applications of (4.3), as followsD c,h (D ss g) = −(D s h · D s c)D ss g + D s [D c,h (D s g)] == −(D s h · D s c)D ss g + D s [−(D s h · D s c)(D s g) + D s (D c,h g)] == D ss (D c,h g) − 2(D s h · D s c)D ss g − [D s (D s h · D s c)](D s g)and hence obtain the Gâteaux differential of the curvatureD c,h D ss c = D ss h − 2(D s h · D s c)D ss c − [D s (D s h · D s c)]D s cand eventually obtain the Gâteaux differential of the elastica energy bysubstituting this last equality in (4.4)∫∫D c,h |D ss c| 2 ds = 2D ss c · (D c,h D ss c) + |D ss c| 2 . (D s h · D s c) ds =cc∫= 2D ss c · D ss h − 3(D s h · D s c)|D ss c| 2 dssince (D ss c · D s c) = 0.cRemark 4.7 Most objects we deal with share an important property: they arereparameterization invariant. In the case of operators, in all cases in whichO(c) is a curve or a vector field, then this means that for any diffeomorphismϕ ∈ Diff + (S 1 ) there holdsO(c ◦ φ)(θ) = O(c)(φ(θ)) . (4.6)(whereas for ϕ ∈ Diff − (S 1 ) there is a possible sign choice as explained in Definition1.5).By considering an infinitesimal perturbation a : S 1 → lR of the identity inDiff(S 1 ), we obtain the formulaD c,h O(c)(θ) = a(θ) . ∂ θ O(c)(θ) for any h(θ) = a(θ).∂ θ c(θ) (4.7)and this last may be rewritten in arc parameterD c,h O(c)(s) = a(s) . D s O(c)(s) for any h(s) = a(s).D s c(s) . (4.8)Similarly if E : M → lR is a reparameterization invariant energy, thenD c,h E(c) = 0 for any h = a.D s c . (4.9)36