Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics. (English) Zbl 1513.70068
Summary: We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert’s conjecture about the multiplicity of brake orbits to Hamiltonian functions of the classical type.
MSC:
| 70H12 | Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics |
| 70G75 | Variational methods for problems in mechanics |
| 70H05 | Hamilton’s equations |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
References:
| [1] | H. Seifert, Periodische bewegungen mechanischer systeme, Math. Z. 51 (1948), 197-216. · Zbl 0030.22103 |
| [2] | A. Ambrosetti, V. Benci, and Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), no. 9, 643-649. · Zbl 0811.70015 |
| [3] | X. Hu, L. Wu, and R. Yang, Morse index theorem of Lagrangian systems and stability of brake orbit, J. Dynam. Differ. Equ. 32 (2020), no. 1, 61-84. · Zbl 1437.37071 |
| [4] | C. Liu, Y. Long, and D. Zhang, Index iteration theory for brake orbit type solutions and applications, Anal. Theory Appl. 37 (2021), no. 2, 129-156. · Zbl 1488.37044 |
| [5] | C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math. 67 (2014), 1563-1604. · Zbl 1336.37043 |
| [6] | Y. Long, D. Zhang, and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math. 203 (2006), no. 2, 568-635. · Zbl 1118.58006 |
| [7] | P. H. Rabinowitz, Critical point theory and applications to differential equations: a survey, In: M. Matzeu and A. Vignoli, editors, Topological Nonlinear Analysis, volume 15, chapter Critical Point Theory and Applications to Differential Equations: A Survey, Birkhäuser Boston, Boston, 1995, pp. 464-513. · Zbl 0823.58009 |
| [8] | A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), no. 2, 241-255. · Zbl 0642.58030 |
| [9] | F. Wang and D. Zhang, Multiple brake orbits of even Hamiltonian systems on torus, Sci. China Math. 63 (2020), no. 7, 1429-1440. · Zbl 1448.58010 |
| [10] | D. Zhang and C. Liu, Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R2n, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 531-554. · Zbl 1300.52006 |
| [11] | R. Giambò, F. Giannoni, P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert’s conjecture on brake orbits, 2021, http://arxiv.org/abs/2002.09687v2. |
| [12] | R. Giambò, F. Giannoni, and P. Piccione, On the multiplicity of orthogonal geodesies in riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud. 9 (2009), no. 4, 763-782. · Zbl 1213.37032 |
| [13] | R. Giambò, F. Giannoni, and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk, Nonlinear Anal. 72 (2010), no. 2, 290-337. · Zbl 1193.37027 |
| [14] | R. Giambò, F. Giannoni, and P. Piccione, Multiple brake orbits in m-dimensional disks, Calc. Var. Partial Differ. Equ. 54 (2015), 2553-2580. · Zbl 1343.37051 |
| [15] | R. Giambò, F. Giannoni, and P. Piccione, On the normal exponential map in singular conformal metrics, Nonlinear Anal. 127 (2015), 35-44. · Zbl 1329.53049 |
| [16] | R. Giambò, F. Giannoni, and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differ. Equ. 57 (2018), no. 117. · Zbl 1401.37039 |
| [17] | R. Giambò, F. Giannoni, and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in riemannian manifolds, Adv. Differ. Equ. 10 (2004), 1-24. |
| [18] | O. R. Ruiz, Existence of brake orbits in Finsler mechanical systems, In: J. Palis and M. doCarmo, editors, Geometry and Topology, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1977, pp. 542-567. · Zbl 0357.70019 |
| [19] | A. Weinstein, Periodic orbits for convex hamiltonian systems, Ann. of Math. 108 (1978), no. 3, 507-518. · Zbl 0403.58001 |
| [20] | H. Rund, The Differential Geometry of Finsler Spaces, Springer-Verlag, Berlin, 1959. · Zbl 0087.36604 |
| [21] | Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, Singapore, 2001. · Zbl 0974.53002 |
| [22] | R. Bartolo, E. Caponio, A. V. Germinario, and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calculus Variat. 40 (2011), 335-356. · Zbl 1216.53064 |
| [23] | D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl. 22 (2020), no. 60. · Zbl 1473.70031 |
| [24] | D. Corona, A multiplicity result for orthogonal geodesic chords in Finsler disks, Discrete Contin. Dyn. Syst. 41 (2021), no. 11, 5329-5357. · Zbl 1498.70024 |
| [25] | D. Corona and F. Giannoni, A new approach for Euler-Lagrange orbits on compact manifolds with boundary, Symmetry 12 (2020), no. 11, 1917. |
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