Weinstein, Alan On the hypotheses of Rabinowitz’ periodic orbit theorems. (English) Zbl 0388.58020 J. Differ. Equations 33, 353-358 (1979). Summary: In recent work of Rabinowitz, Ekeland, Clarke, and the author, the existence of periodic orbits with prescribed energy for Hamiltonian systems has been proven under the assumption of convexity or some kind of starshapedness of the energy surface. The ideas described in the present paper arose in an attempt to extend these existence theorems to more general systems by canonical transformations of the Hamiltonian. It is shown that the presence of a contact structure is a common feature of all the theorems, and it is conjectured that the contact structure is sufficient to imply the existence of a periodic orbit. An example is given of an embedded sphere which cannot be canonically transformed to become starshaped. Reviewer: Alan Weinstein (Houston) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 9 ReviewsCited in 76 Documents MSC: 53D35 Global theory of symplectic and contact manifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Keywords:periodic orbit; Hamiltonian and Lagrangian systems × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Birkhoff, G. D., Dynamical Systems, (Colloquium Publications, Vol. IX (1927), Amer. Math. Soc: Amer. Math. Soc Providence) · Zbl 0171.05402 [2] Klingenberg, W., Closed Geodesics, (Grundlehren der Mathematischen Wissenschaften, Vol. 230 (1978), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0223.53037 [3] Lusternik, L. A.; Fet, A. I., Variational problems on closed manifolds, Dokl. Akad. Nauk SSSR, 81, 17-18 (1951), (Russian) · Zbl 0045.20903 [4] Rabinowitz, P., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31, 157-184 (1978) · Zbl 0358.70014 [5] P. Rabinowitzin; P. Rabinowitzin · Zbl 0486.35009 [6] Rabinowitz, P., Periodic solutions of a Hamiltonian system on a prescribed energy surface, J. Differential Equations, 33, 336-352 (1979) · Zbl 0424.34043 [7] Schweitzer, P. A., Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math., 100, 386-400 (1974) · Zbl 0295.57010 [8] Seifert, H., Periodische bewegungen mechanischer Systeme, Math. Z., 51, 197-216 (1948) · Zbl 0030.22103 [9] Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6, 329-346 (1971) · Zbl 0213.48203 [10] Weinstein, A., Periodic orbits for convex hamiltonian systems, Ann. of Math., 108, 507-518 (1978) · Zbl 0403.58001 [11] Wilson, F. W., On the minimal sets of non-singular vector fields, Ann. of Math., 84, 529-536 (1966) · Zbl 0156.43803 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.