POISSON BRACKETS AND COMPLEX TORI

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16 A. P. VESELOV AND S. P. NOVIKOVExample 4. The Hamiltonian structure generated by the so-called latent isomorphismof Moser-Trubowitz (see [28]) and [13]) between the KdV dynamicson the space of finite-zone potentials and the classical Neumann systems describingthe dynamics on the sphere S g in R 2g+1 ,S g = { x: |x 2 | = 1, x ∈ R g+1} ,under the action of the quadratic potentialU(x) = 1 2g∑a i x 2 i .i=0The coefficients a i , representing the set of end points of the zones λ i , here playthe role of the annihilator. Using [28] and [13], we obtain (see also [27])Q = √ −R(λ)g∏(λ − λ 2j ) −1 dλ, A = {λ 0 , λ 2 , . . . , λ 2g } .j=0Example 5. The integrable case of Goryachev and Chaplygin in the dynamicsof a rigid body with a fixed point [30]. HereQ(Γ, λ) = arcsin 1 ( λ2µ 2 − 1 2 H − 2G ),λwhere H is the energy of the gyroscope, G is the Goryachev-Chaplygin integral,and µ is a parameter of the problem. The curve Γ is given by the equationy 2 = 4µ 2 λ 2 − (λ 3 − Hλ − 4G) 2 .Example 6. In the well-known Kovalevskaya case the action variables werenot computed previously. Let I 1 , I 2 and I 3 be, respectively, the energy, area,and Kovalevskaya integrals, and let γ i = s i be the Kovalevskaya variables (see[30]). Calculations lead to the resultQ(Γ, λ) = 1 [ √−λ2 √ −λ ln ( )(λ − I1 ) 2 − I32 ν 2−2 √ −λ (λ − 2I2 2) + √ ]−R(λ) ,R(λ) = ( λ ( ((λ − I 1 ) 2 + ν 2 − I3) 2 − 2ν 2 I2) 2 (λ − I1 ) 2 − I3) 2 ,while the curve Γ is given by the equation y 2 = R(λ). IntegratingQ = Q(Γ, λ) dλ over the “real” cycles a j on which the γ i lie, we obtain the
POISSON BRACKETS AND COMPLEX TORI 17action variables J j . In correspondence with the result of Kovalevskaya theAbel transformation linearizes the dynamics (we note that the replacement ofthe time and Hamiltonian structure proposed by G. V. Kolosov (see [29]) leadsfor this system to non-Abelian tori).The integrable Steklov case for a solid body in a fluid is also of majorinterest.Bibliography[1] S. P. Novikov (editor), Theory of solitons, “Nauka”, Moscow, 1980; Englishtransl., Plenum Press, New York, 1984.[2] S. P. Novikov, A periodic problem for the Korteweg-de Vries equation.I, Funktsional. Anal. i Prilozhen. 8 (1974), no. 3, 54–66; English transl.in Functional Anal. Appl. 8 (1974).[3] B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodicanalogs of the many-soliton solutions of the Korteweg-de Vries equation,Zh. Èksper. Teoret. Fiz. 67 (1974), 2131–2144; English transl. in SovietPhys. JETP 40 (1974).[4] A. R. Its and V. B. Matveev, Schrödinger operators with finite-zero spectrumand N-soliton solutions of the Korteweg-de Vries equation, Teoret.i Mat. Fiz. 23 (1975), 51–68; English transl. in Theoret. Math.Phys. 23 (1975).[5] Peter D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl.Math 28 (1975), 141–188.[6] H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent.Math. 30 (1975), 217–274.[7] V. A. Dubrovin, V. B. Matveev and S. P. Novikov, Nonlinear equations ofKorteweg-de Vries type, finite-zone linear operators, and Abelian varieties,Uspekhi Mat. Nauk 31 (1976), no. 1 (187), 55–136; English transl. inRussian Math. Surveys 31 (1976).[8] I. M. Krichever, Methods of algebraic geometry in the theory of nonlinearequations, Uspekhi Mat. Nauk. 32 (1977), no. 6 (198), 183–208; Englishtransl. in Russian Math. Surveys 32 (1977).
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