×

Two remarks about the connection of Jacobi and Neumann integrable systems. (English) Zbl 0815.58011

The paper can be considered as the addendum to the papers by H. Knörrer [J. Reine Angew. Math. 334, 69-78 (1982; Zbl 0478.58014)] and J. Moser [Proc. Math. 8, 233-290 (1980; Zbl 0468.58011)] and consists of two remarks about the connection of Jacobi and Neumann systems.
H. Knörrer [loc. cit.] discovered that the geodesics on the ellipsoid can be transformed into the trajectories of the motion of a point on the unit sphere under the influence of certain quadratic potential (Neumann system). The first remark says that Knörrer’s theorem can be extended to the case of the motion on the ellipsoid in the potential field with \(U(x) = {1\over 2} \varepsilon x^ 2\).
The second remark is an interpretation of the Lagrange multiplier for the geodesic flow on the ellipsoid in terms of the stability of the equilibrium point for a one-dimensional mechanical problem.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37C75 Stability theory for smooth dynamical systems
58E99 Variational problems in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [DNM] Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear equations of KdV type, finite-zone operators and abelian varieties. Russ. Math. Surv.31, 59–146 (1976) · Zbl 0346.35025 · doi:10.1070/RM1976v031n01ABEH001446
[2] [J] Jacobi, G.G.: Vorlesungen über Dynamik. (Gesammelte Werke, Supplement band) Berlin 1884
[3] [K] Knörrer, H.: Geodesics on quadrics and a mechanical problem of C. Neumann. J. Reine Angew. Math.334, 69–78 (1982) · Zbl 0478.58014 · doi:10.1515/crll.1982.334.69
[4] [M1] Moser, J.: Various aspects of integrable hamiltonian systems. (Prog. Math., vol. 8, pp. 233–289) Boston: Birkhäuser 1980
[5] [M2] Moser, J.: Integrable hamiltonian systems and spectral theory, Pisa: Lezione Fermiane 1981
[6] [N] Neumann, C.: De problemate quodam mechanico, quod ad primam integralium ultraellipticozum classem revocatur. J. Reine Angew. Math.56, 46–63 (1859) · doi:10.1515/crll.1859.56.46
[7] [V1] Veselov, A.P.: Finite-band potentials and an integrable system on the sphere with quadratic potential (in Russian). Funct. Anal. Appl.14 (no. 1), 48–50 (1980)
[8] [V2] Veselov, A.P.: Geometry of the integrable hamiltonian systems connected with nonlinear PDE’s (in Russian). Ph. Thesis, Moscow State University (1981)
[9] [V3] Veselov, A.P.: Change of time in integrable systems. Vestn. Mos. Univ., Ser. Mat. Mech. 25–29 (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.