In this paper, for any positive integer n, we study the Maslov-type index theory of iL0, iL1
Nearby in the stack
and
i−1L0
with
L0={0}×Rn⊂R2n
and
L1=Rn×{0}⊂R2n
. As 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
R2n
, 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
XT
with minimal period no less than
2n+2T
. Furthermore if
∫0TH"22(xT(t))dt
is positive definite, then the minimal period of
xT
belongs to
{T,2T}
. 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