In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in R n for n 2.

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least n 2 + 1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n satisfying the reversible condition N Σ = Σ, furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least n geometrically distinct brake orbits on it. As a consequence, we show that there exist at least n 2 + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n , furthermore, if all brake orbits in this domain are nondegenerate, then there are at least n geometrically distinct brake orbits in it. In the symmetric case,

In this paper, we prove that there exist at least n geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n satisfying the reversible condition N Σ = Σ with N = diag(−I n , I n ). As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer n.

In this paper, for any positive integer n, we study the Maslov-type index theory of i L0 , i L1As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in R 2n , which are semipositive, and superquadratic at zero and infinity we prove that for any T > 0, the considered Hamiltonian systems possesses a nonconstant T periodic brake orbit X T with minimal period no less than T 2n+2 . Furthermore if T 0 H ′′ 22 (x T (t))dt is positive definite, then the minimal period of x T belongs to {T, T 2 }. Moreover, if the Hamiltonian system is even, we prove that for any T > 0, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to {T, T 3 }.

In this paper, we prove that there exist at least n+1 2 + 1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n for n ≥ 2 satisfying the reversible condition N Σ = Σ with N = diag(−I n , I n ). As a consequence, we show that there exist at least n+1 2 + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n with n ≥ 2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n = 3. As an application, for n = 4 and 5, we prove that if there are exactly n geometrically distinct closed characteristics on Σ, then all of them are symmetric brake orbits after suitable time translation.