Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

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December 2017, 9(4): 439-457. doi: 10.3934/jgm.2017017

Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

Pietro-Luciano Buono ^1,, and Daniel C. Offin ^2,

1.	Faculty of Science, University of Ontario Institute of Technology, Oshawa, ONT L1H 7K4, Canada
2.	Department of Mathematics and Statistics, Queen's University, Kingston, ONT K7L 3N6, Canada

* Corresponding author: Pietro-Luciano Buono

Received June 2010 Revised April 2017 Published October 2017

We consider the question of linear stability of a periodic solution $z(t)$ with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that $z(t)$ is unstable if a subspace $W$ associated with the boundary conditions of the minimizing problem is a Lagrangian subspace with no focal points on the time interval defined by the boundary conditions and the second variation restricted to the subspace $W$ at the minimizer has positive directions. We show that the conditions of our theorem are always met for a class of minimizing periodic orbits with the standard mechanical reversing symmetry. Comparison theorems for Lagrangian subspaces and the use of time-reversing symmetries are essential tools in constructing stable and unstable subspaces for $z(t)$. In particular, our results are complementary to the recent paper of Hu and Sun Commun. Math. Phys. 290, (2009).

Keywords: Periodic solutions, symmetry, Hamiltonian, instability, Lagrangian subspaces, reversibility.

Mathematics Subject Classification: Primary: 37J25, 37J45; Secondary: 37G40.

Citation: Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017

References:

[1]	V. I. Arnol'd, The Sturm theorems and symplectic geometry, Funktsional. Anal. i Prilozhen., 19 (1985), 1-10, 95. Google Scholar
[2]	S. V. Bolotin and D. V. Treschev, Hill's formula, Russian Math. Surveys, 65 (2010), 191-257. doi: 10.1070/RM2010v065n02ABEH004671. Google Scholar
[3]	R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206. doi: 10.1002/cpa.3160090204. Google Scholar
[4]	S. Cappell, R. Lee and E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202. Google Scholar
[5]	K.-C. Chen, Action minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 158 (2001), 293-318. doi: 10.1007/s002050100146. Google Scholar
[6]	K.-C. Chen, Binary decompositions for the planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2. Google Scholar
[7]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357. Google Scholar
[8]	A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du Probléme newtonien de 4 corps de masses égales dans $\mathbb{R}^{3}$ : orbites hip-hop, Celestial Mechanics and Dynamical Astronomy, 77 (2000), 139-152. doi: 10.1023/A:1008381001328. Google Scholar
[9]	G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems, 19 (1999), 901-952. doi: 10.1017/S014338579913387X. Google Scholar
[10]	J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math., 21 (1976), 173-195. doi: 10.1016/0001-8708(76)90074-8. Google Scholar
[11]	D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7. Google Scholar
[12]	M. Golubitsky, I. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. Ⅱ, Applied Mathematical Sciences, 69, Springer-Verlag, New-York, 1988. doi: 10.1007/978-1-4612-4574-2. Google Scholar
[13]	P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. Google Scholar
[14]	X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y. Google Scholar
[15]	M. Lewis, D. Offin, P.-L. Buono and M. Kovacic, Instability of the periodic Hip-Hop orbit in the 2N-body problem with equal masses, Discrete and Continuous Dynamical Systems -A, 33 (2013), 1137-1155. Google Scholar
[16]	J. E. Marsden, Lectures on Mechanics, LMS Lecture Note Series, 174, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624001. Google Scholar
[17]	M. Morse, The Calculus of Variations in the Large, American Mathematical Society Colloquium Publications, 18 American Mathematical Society, Providence, 1996. Google Scholar
[18]	D. Offin, A spectral theorem for reversible second order equations with periodic coefficients, Differential and Integral Equations, 5 (1992), 615-629. Google Scholar
[19]	D. Offin, Hyperbolic minimizing geodesics, Trans. AMS, 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1. Google Scholar
[20]	D. Offin and H. Cabral, Hyperbolic symmetric periodic orbits in the isosceles three-body problem, Disc. Cont. Dyn. Syst. Ser. S, 2 (2009), 379-392. doi: 10.3934/dcdss.2009.2.379. Google Scholar
[21]	G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284. Google Scholar

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References:

[1]	V. I. Arnol'd, The Sturm theorems and symplectic geometry, Funktsional. Anal. i Prilozhen., 19 (1985), 1-10, 95. Google Scholar
[2]	S. V. Bolotin and D. V. Treschev, Hill's formula, Russian Math. Surveys, 65 (2010), 191-257. doi: 10.1070/RM2010v065n02ABEH004671. Google Scholar
[3]	R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206. doi: 10.1002/cpa.3160090204. Google Scholar
[4]	S. Cappell, R. Lee and E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202. Google Scholar
[5]	K.-C. Chen, Action minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 158 (2001), 293-318. doi: 10.1007/s002050100146. Google Scholar
[6]	K.-C. Chen, Binary decompositions for the planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2. Google Scholar
[7]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901. doi: 10.2307/2661357. Google Scholar
[8]	A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du Probléme newtonien de 4 corps de masses égales dans $\mathbb{R}^{3}$ : orbites hip-hop, Celestial Mechanics and Dynamical Astronomy, 77 (2000), 139-152. doi: 10.1023/A:1008381001328. Google Scholar
[9]	G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems, 19 (1999), 901-952. doi: 10.1017/S014338579913387X. Google Scholar
[10]	J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math., 21 (1976), 173-195. doi: 10.1016/0001-8708(76)90074-8. Google Scholar
[11]	D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7. Google Scholar
[12]	M. Golubitsky, I. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. Ⅱ, Applied Mathematical Sciences, 69, Springer-Verlag, New-York, 1988. doi: 10.1007/978-1-4612-4574-2. Google Scholar
[13]	P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. Google Scholar
[14]	X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y. Google Scholar
[15]	M. Lewis, D. Offin, P.-L. Buono and M. Kovacic, Instability of the periodic Hip-Hop orbit in the 2N-body problem with equal masses, Discrete and Continuous Dynamical Systems -A, 33 (2013), 1137-1155. Google Scholar
[16]	J. E. Marsden, Lectures on Mechanics, LMS Lecture Note Series, 174, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624001. Google Scholar
[17]	M. Morse, The Calculus of Variations in the Large, American Mathematical Society Colloquium Publications, 18 American Mathematical Society, Providence, 1996. Google Scholar
[18]	D. Offin, A spectral theorem for reversible second order equations with periodic coefficients, Differential and Integral Equations, 5 (1992), 615-629. Google Scholar
[19]	D. Offin, Hyperbolic minimizing geodesics, Trans. AMS, 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1. Google Scholar
[20]	D. Offin and H. Cabral, Hyperbolic symmetric periodic orbits in the isosceles three-body problem, Disc. Cont. Dyn. Syst. Ser. S, 2 (2009), 379-392. doi: 10.3934/dcdss.2009.2.379. Google Scholar
[21]	G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284. Google Scholar

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