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 Uspekhi Mat. Nauk, 2010, Volume 65, Issue 2(392), Pages 3–70 (Mi umn9348)

Hill's formula

S. V. Bolotinab, D. V. Treschevca

a Steklov Mathematical Institute, Russian Academy of Sciences
c M. V. Lomonosov Moscow State University

Abstract: In his study of periodic orbits of the three-body problem, Hill obtained a formula connecting the characteristic polynomial of the monodromy matrix of a periodic orbit with the infinite determinant of the Hessian of the action functional. A mathematically rigorous definition of the Hill determinant and a proof of Hill's formula were obtained later by Poincaré. Here two multidimensional generalizations of Hill's formula are given: for discrete Lagrangian systems (symplectic twist maps) and for continuous Lagrangian systems. Additional aspects appearing in the presence of symmetries or reversibility are discussed. Also studied is the change of the Morse index of a periodic trajectory upon reduction of order in a system with symmetries. Applications are given to the problem of stability of periodic orbits.
Bibliography: 34 titles.

Keywords: periodic solution, stability, Lagrangian system, multipliers, billiard system.

DOI: https://doi.org/10.4213/rm9348

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English version:
Russian Mathematical Surveys, 2010, 65:2, 191–257

Bibliographic databases:

Document Type: Article
UDC: 531.01
MSC: 34D05, 37Jxx, 70H03

Citation: S. V. Bolotin, D. V. Treschev, “Hill's formula”, Uspekhi Mat. Nauk, 65:2(392) (2010), 3–70; Russian Math. Surveys, 65:2 (2010), 191–257

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9348
• https://doi.org/10.4213/rm9348
• http://mi.mathnet.ru/eng/umn/v65/i2/p3

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This publication is cited in the following articles:
1. Hu Xijun, Wang Penghui, “Conditional Fredholm determinant for the $S$-periodic orbits in Hamiltonian systems”, J. Funct. Anal., 261:11 (2011), 3247–3278
2. Bolotin S.V., Negrini P., “Variational approach to second species periodic solutions of Poincaré of the 3 body problem”, Discret. Contin. Dyn. Syst., 33:3 (2013), 1009–1032
3. E. Petrisor, “Twist number and order properties of periodic orbits”, Phys. D, 263 (2013), 57–73
4. M. N. Davletshin, “Hill’s formula for $g$-periodic trajectories of Lagrangian systems”, Trans. Moscow Math. Soc., 74 (2013), 65–96
5. Xijun Hu, Yuwei Ou, Penghui Wang, “Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions”, Arch. Ration. Mech. Anal., 216:1 (2015), 313–357
6. Hu X., Wang P., “Hill-Type Formula and Krein-Type Trace Formula For S-Periodic Solutions in ODEs”, 36, no. 2, SI, 2016, 763–784
7. Hu X., Wang P., “Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions”, Math. Z., 283:1-2 (2016), 339–348
8. Buono P.-L., Offin D.C., “Instability Criterion For Periodic Solutions With Spatio-Temporal Symmetries in Hamiltonian Systems”, J. Geom. Mech., 9:4 (2017), 439–457
9. Antonio J. Ureña, “The Spectrum of Reversible Minimizers”, Regul. Chaotic Dyn., 23:3 (2018), 248–256
10. A. A. Agrachev, “Spectrum of the Second Variation”, Proc. Steklov Inst. Math., 304 (2019), 26–41
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