By Michael Spivak
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Extra resources for A Comprehensive Introduction To Differential Geometry Volume 5, Second Edition
Then we want to write L, [X, Y ] = [C12 , L1 ], X ⊗ Y , however, this formula n , L]⊗E ∗n and does not make sense as it stands because [C12 , L1 ] = α [Eij ij n ∗ Eij ∈ G while L ∈ G and the commutator is not deﬁned. To overcome this problem we embed G and its dual G ∗ into the full loop algebra G˜ n , n ∈ Z. 28) Note that in this sum, G and G ∗ do not commute. Let us compute C12 , assuming |λ| < |µ|: ∞ C12 (λ, µ) = C12 n=0 λn C12 , =− n+1 µ λ−µ Eij ⊗ Eji C12 = i,j 46 3 Synopsis of integrable systems We can now write L(λ)[X(λ), Y (λ)] = [C12 (λ, µ), L(λ) ⊗ 1] , X(λ) ⊗ Y (µ) .
Then the phase space is of dimension 2, and the system is integrable with conserved quantity H. Note that the trajectories are immediately obtained as the intersection of the sphere J12 + J22 + J32 = J 2 and the ellipsoid J12 /I1 + J22 /I2 + J32 /I3 = 2H. e. J˙1 = (I3−1 − I2−1 )J2 J3 , yields an equation of the form J˙1 = α + βJ12 + γJ14 , so that J1 is an elliptic function of t. 9 The Lagrange top When the top is in a gravitational ﬁeld its weight has to be taken into account and the problem is more complicated.
K) We have seen in these three examples that the singular parts, A− , of the matrix A(k) (λ) are independent of the dynamical variables. We will show in the next section that, in this case, eq. 14) admits an important interpretation as a coadjoint orbit. 3 Coadjoint orbits and Hamiltonian formalism In this section we show that the Zakharov–Shabat construction, when the (k) matrices A− are non-dynamical, can be interpreted as coadjoint orbits. This introduces a natural symplectic structure in the problem and gives a Hamiltonian interpretation to the Lax equation.