Ivancevic_Applied-Diff-Geom

318 Applied Differential Geometry: A Modern Introduction[Udriste and Neagu (1999)]E p : Ω → R + , asE p (c) =∫ 10[g ij (c, ċ)ċ i ċ j ] p/2 dt =∫ 10[g(ċ, ċ)] p/2 dt =In particular, for p = 1 we get the length functionalL(c) =∫ 1and for p = 2 we get the energy functionalE(c) =0∫ 10‖ċ‖dt,‖ċ‖ 2 dt.∫ 10‖ċ‖ p dt.Also, for any naturally parametrized curve (i.e., ‖ċ‖ = const) we haveE p (c) = (L(c)) p = (E(c)) p/2 .Note that the p−energy of a curve is dependent of parametrization if p ≠ 1.For every curve c ∈ Ω, we define the tangent space T c Ω asT c Ω = {X : [0, 1] → T M | X is continuous, piecewise C ∞ , X(t) ∈ T c(t) M,for all t ∈ [0, 1], X(0) = X(1) = 0}.Let (c s ) s∈(−ɛ,ɛ) ⊂ Ω be a one–parameter variation of the curve c ∈ Ω. WedefineUsing the equalityX(t) = dc sds (0, t) ∈ T cΩ.( ) ( ∇ċs∇g∂s , ċ s = g∂t( ) ) ∂cs, ċ s ,∂swe can prove the following Theorem: The first variation of the p−energy is1pdE p (c s )(0) = − ∑ dst−∫ 10g(X, ∆ t (‖ċ‖ p−2 ċ))(‖ċ‖ p−4 g X, ‖ċ‖ 2 ∇ċ( ) ) ∇ċdt + (p − 2)g dt , ċ ċ dt,where ∆ t (‖ċ‖ p−2 ċ) = (‖ċ‖ p−2 ċ) t + − (‖ċ‖ p−2 ċ) t − represents the jump of‖ċ‖ p−2 ċ at the discontinuity point t ∈ (0, 1) [Udriste and Neagu (1999)].The curve c is a critical point of E p iff c is a geodesic.