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

This article is cited in 10 scientific papers (total in 10 papers)

Hill's formula

S. V. Bolotinab, D. V. Treschevca

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Wisconsin-Madison, USA
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

Full text: PDF file (1127 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
UDC: 531.01
MSC: 34D05, 37Jxx, 70H03
Received: 25.02.2010

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

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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    3. E. Petrisor, “Twist number and order properties of periodic orbits”, Phys. D, 263 (2013), 57–73  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. M. N. Davletshin, “HillТs formula for $g$-periodic trajectories of Lagrangian systems”, Trans. Moscow Math. Soc., 74 (2013), 65–96  mathnet  crossref  mathscinet  zmath  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi
    7. Hu X., Wang P., “Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions”, Math. Z., 283:1-2 (2016), 339–348  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi
    9. Antonio J. Ureña, “The Spectrum of Reversible Minimizers”, Regul. Chaotic Dyn., 23:3 (2018), 248–256  mathnet  crossref  mathscinet  adsnasa
    10. A. A. Agrachev, “Spectrum of the Second Variation”, Proc. Steklov Inst. Math., 304 (2019), 26–41  mathnet  crossref  crossref  elib
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