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Russian Mathematical Surveys, 2010, Volume 65, Issue 2, Pages 191–257
DOI: https://doi.org/10.1070/RM2010v065n02ABEH004671 (Mi rm9348) 		 
This article is cited in 18 scientific papers (total in 18 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
English version PDF (834 kB) Citations (18)
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DOI: https://doi.org/10.1070/RM2010v065n02ABEH004671
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.
Received: 25.02.2010
Russian version:
Uspekhi Matematicheskikh Nauk, 2010, Volume 65, Issue 2(392), Pages 3–70
DOI: https://doi.org/10.4213/rm9348
Bibliographic databases:
Document Type: Article
UDC: 531.01
MSC: 34D05, 37Jxx, 70H03
Language: English
Original paper language: Russian
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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    • This publication is cited in the following 18 articles:
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