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

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

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).

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. CappellR. 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. LewisD. OffinP.-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

show all references

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. CappellR. 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. LewisD. OffinP.-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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