In this thesis we investigate the existence and stability of periodic solutions of Hamiltonian systems with a discrete symmetry. The global existence of periodic motions can be proven using the classical techniques of the calculus of variations; our particular interest is in how the stability type of the solutions thus obtained can be determined analytically using solely the variational problem and the symmetries of the system -- we make no use of numerical or perturbation techniques. Instead, we use a method introduced in [41] in the context of a special case of the three-body problem. Using techniques from symplectic geometry, and specifically the Maslov index for curves of Lagrangian subspaces along the minimizing trajectories, we verify conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle. We study the applicability of this method in two specific cases. Firstly, we consider another special case from celestial mechanics: the hip-hop solutions of the 2N-body problem. This is a family of Z_2-symmetric, periodic orbits which arise as collision-free minimizers of the Lagrangian action on a space of symmetric loops [14, 53]. Following a symplectic reduction, it is shown that the hip-hop solutions are brake orbits which are generically hyperbolic on the reduced energy-momentum surface. Secondly we consider a class of natural Hamiltonian systems of two degrees of freedom with a homogeneous potential function. The associated action functional is unbounded above and below on the function space of symmetric curves, but saddle points can be located by minimization subject to a certain natural constraint of a type first considered by Nehari [37, 38]. Using the direct method of the calculus of variations, we prove the existence of symmetric solutions of both prescribed period and prescribed energy. In the latter case, we employ a variational principle of van Groesen [55] based upon a modification of the Jacobi functional, which has not been widely used in the literature. We then demonstrate that the (constrained) minimizers are again hyperbolic brake orbits; this is the first time the method has been applied to solutions which are not globally minimizing. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-25 10:47:53.257

Neste trabalho, apresentamos um método algébrico para obter formas normais de campos vetoriais reversíveis equivariantes. Adaptamos o método clássico de Belitskii-Elphick, usando ferramentas da teoria invariante para estabelecer fórmulas que consideram as simetrias e antissimetrias como ponto de partida. Mostramos que este método, mesmo sem simetrias, possui uma estreita relação com o método da transversal completa da teoria de singularidades. Com as ferramentas desenvolvidas nesta tese, a forma normal obtida e uma série formal que não depende do cálculo do kernel do chamado operador homológico. Formas normais para duas classes de campos, ressonantes e não ressonantes, são apresentadas, para diferentes representações do grupo \'Z IND. 2\' x \'Z IND. 2\' cuja linearização tem uma parte nilpotente de dimensão 2 e uma parte semi-simples com autovalores puramente imaginários / We give an algebraic method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method by Belitskii-Elphick using tools from invariant theory to establish formulae that take symmetries into account as a starting point. We show that this method, even without symmetries, has a close relation to complete transversal of singularities theory. Applying the method developed in this thesis, the resulting normal form is a formal series which does not depend of the computation of the kernel of the so called homologic operator. Normal forms of two classes of non-resonant and resonant cases are presented, for dierent representations of the group \'Z INT. 2\' x \'Z INT. 2\' - with linearization having a 2 - dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues

Neste trabalho, apresentamos um método algébrico para obter formas normais de campos vetoriais reversíveis equivariantes. Adaptamos o método clássico de Belitskii-Elphick, usando ferramentas da teoria invariante para estabelecer fórmulas que consideram as simetrias e antissimetrias como ponto de partida. Mostramos que este método, mesmo sem simetrias, possui uma estreita relação com o método da transversal completa da teoria de singularidades. Com as ferramentas desenvolvidas nesta tese, a forma normal obtida e uma série formal que não depende do cálculo do kernel do chamado operador homológico. Formas normais para duas classes de campos, ressonantes e não ressonantes, são apresentadas, para diferentes representações do grupo \'Z IND. 2\' x \'Z IND. 2\' cuja linearização tem uma parte nilpotente de dimensão 2 e uma parte semi-simples com autovalores puramente imaginários / We give an algebraic method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method by Belitskii-Elphick using tools from invariant theory to establish formulae that take symmetries into account as a starting point. We show that this method, even without symmetries, has a close relation to complete transversal of singularities theory. Applying the method developed in this thesis, the resulting normal form is a formal series which does not depend of the computation of the kernel of the so called homologic operator. Normal forms of two classes of non-resonant and resonant cases are presented, for dierent representations of the group \'Z INT. 2\' x \'Z INT. 2\' - with linearization having a 2 - dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues