Ivancevic_Applied-Diff-Geom

454 Applied Differential Geometry: A Modern Introduction(1997)]The resulting Lax pair matrices are¯q 1 = q 1 + ln p 1 , ¯p 1 = p 1 ,¯q 2 = q 2 , ¯p 2 = p 2 ,¯q 3 = q 3 − ln p 3 , ¯p 3 = p 3 .⎛⎞ ⎛⎞p 1 a 1 p 1 00 a 1 p 1 0L = ⎝ a 1 p 1 p 2 a 2 p 3⎠ , A = ⎝ −a 1 p 1 0 a 2 p 3⎠ ,0 a 2 p 3 p 3 0 −a 2 p 3 0where a 1 = exp(q 1 − q 2 ), a 2 = exp(q 2 − q 3 ).The Hamiltonian is now purely kineticH = 1 2 L2 = 1 2Using (3.262) we identify a metric[(1 + 2a12 ) p 1 2 + p 2 2 + ( 1 + 2a 22 ) p 32 ] .ds 2 = g 11 (dq 1 ) 2 + (dq 2 ) 2 + g 33 (dq 3 ) 2 , whereg 11 = ( 21 + 2a ) −11 , g33 = ( 21 + 2a ) −12 .The non–zero Levi–Civita connection coefficients, Γ α βγ = Γα (βγ), of this metricareΓ 1 11 = −2a 2 1 g 11 , Γ 2 33 = 2a 2 2 (g 33 ) 2 ,Γ 1 12 = 2a 2 1 g 11 , Γ 3 23 = −2a 2 2 g 33 ,Γ 2 11 = −2a 2 1 (g 11 ) 2 , Γ 3 33 = 2a 2 2 g 33 .Following the arguments above, the homogeneous Lax matrix should correspondto a tensor with mixed indices L α β. It is a reasonable assumptionthat the covariant Lax formulation inherits the symmetries of the standardformulation we started with. We therefore expect L αβ and B αβ to have thesymmetriesL (αβ) = L αβ and B [αβ] = B αβ .Note that the symmetry properties are not imposed on the Lax matrices,L α β and Bα β, themselves. In fact, the required symmetries are not consistentwith the representation (3.266). We can however perform a similaritytransformation (3.264) of the Lax matrix, L → ˜L in such a way that ˜L αβ