Hill’s formula. (English. Russian original) Zbl 1369.37085

Russ. Math. Surv. 65, No. 2, 191-257 (2010); translation from Usp. Mat. Nauk 65, No. 2, 3-70 (2010).
Summary: In his study of periodic orbits of the three-body problem, G. W. Hill [Acta Math. 8, 1–36 (1886; JFM 18.1106.01)] 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.


37N05 Dynamical systems in classical and celestial mechanics
34C14 Symmetries, invariants of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
70F10 \(n\)-body problems
70H03 Lagrange’s equations


JFM 18.1106.01
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