Tamtam Proceedings - lamsin

66 Benzekri et al.ẏ = ∂H∂x = ∂V∂xĖ = − ∂H∂τ ,˙τ = ∂H∂EThe operator {H 0 } is given by := ɛ cos(x + B sin τ) sin y,= 1. (12)since :ω ={H 0 } = ω(A) ·∂∂θ , (13)( ∂H0∂x , ∂H )0= (0, 1) . (14)∂EIn order to calculate the operator Γ, we rewrite V by expanding sin(B sin τ) and cos(B sin τ)as a series of Bessel functions of first kind :cos(B sin τ) = J 0 (B) + 2 ∑ n≥1J 2n (B) cos 2nτ,sin(B sin τ) = 2 ∑ n≥0J 2n+1 (B) sin(2n + 1)τ, (15)so :⎛⎞V (x, y, τ) = ɛ sin y sin x ⎝J 0 (B) + 2 ∑ J 2n (B) cos 2nτ⎠n≥1⎛⎞+2ɛ sin y cos x ⎝ ∑ J 2n+1 (B) sin(2n + 1)τ⎠ . (16)n≥1By applying the operator Γ to V we have :⎛⎞ΓV (x, y, τ) = 2ɛ sin y sin x ⎝ ∑ 12n J 2n(B) sin 2nτ⎠n≥1⎛⎞−2ɛ sin y cos x ⎝ ∑ 1(2n + 1) J 2n+1(B) cos(2n + 1)τ⎠ . (17)n≥0TAMTAM –Tunis– 2005