Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 451together with the natural Poisson bracket (denoted by {, }) on the cotangentbundle. The geodesic system is given by˙q α = {q α , H} = g αµ p µ , ṗ α = {p α , H} = Γ µν α p µ p ν .The complete integrability of this system can be shown with the help ofa pair of matrices L and A with entries defined on the phase space (thecotangent bundle) and satisfying the Lax pair equation [Lax (1968)]˙L = {L, H} = [L, A]. (3.263)It follows from (3.263) that the quantities I k ≡ 1 k Lk are all constants of themotion. If in addition they commute with each other {I k , I j } = 0 (Liouvilleintegrability) then it is possible to integrate the system completely at leastin principle (see e.g., [Arnold (1989)]). The Lax representation (3.263) is notunique. In fact, the Lax pair equation is invariant under a transformationof the form˜L = ULU −1 , Ã = UAU −1 − ˙UU −1 . (3.264)We see that L transforms as a tensor while A transforms as a connection.As we will see, these statements acquire a more precise meaning in thegeometric formulation which we will now describe.Typically, the Lax matrices are linear in the momenta and in the geometricsetting they may also be assumed to be homogeneous. This motivatesthe introduction of two third rank geometrical objects, L α β γ and A α β γ ,such that the Lax matrices can be written in the form [Rosquist (1997)]L = (L α β) = (L α β µ p µ ) , A = (A α β) = (A α β µ p µ ).We will refer to L α β γ and A α β γ as the Lax tensor and the Lax connection,respectively. DefiningB = (B α β ) = (B α β µ p µ ) = A − Γ, where Γ = (Γ α β) = (Γ α β µ p µ )is the Levi–Civita connection with respect to g αβ , it then follows that theLax pair equation takes the covariant form (see [Rosquist (1997)] for details)L α β (γ;δ) = L α µ (γ B |µ| β δ) − B α µ (γ L |µ| β δ) ,where L α β γ and B α β γ are tensorial objects. Note that the right–hand sideof this equation is traceless, so that upon contracting over α and β we