  A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path
which is everywhere tangent to the positive cone. These paths are generated by
negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean
space. A special case are autonomous positive paths, which are generated by
time-independent Hamiltonians, and which all lie in the set $\Uu$ of
diagonalizable matrices with eigenvalues on the unit circle. However, as was
shown by Krein, the eigenvalues of a general positive path can move off the
unit circle. In this paper, we extend Krein's theory: we investigate the
general behavior of positive paths which do not encounter the eigenvalue 1,
showing, for example, that any such path can be extended to have endpoint with
all eigenvalues on the circle. We also show that in the case $2n=4$ there is a
close relation between the index of a positive path and the regions of the
symplectic group that such a path can cross. Our motivation for studying these
paths came from a geometric squeezing problem in symplectic topology. However,
they are also of interest in relation to the stability of periodic Hamiltonian
systems and in the theory of geodesics in Riemannian geometry.
