  We investigate the detailed dynamics of multidimensional Hamiltonian systems
by studying the evolution of volume elements formed by unit deviation vectors
about their orbits. The behavior of these volumes is strongly influenced by the
regular or chaotic nature of the motion, the number of deviation vectors, their
linear (in)dependence and the spectrum of Lyapunov exponents. The different
time evolution of these volumes can be used to identify rapidly and efficiently
the nature of the dynamics, leading to the introduction of quantities that
clearly distinguish between chaotic behavior and quasiperiodic motion on
$N$-dimensional tori. More specifically we introduce the Generalized Alignment
Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped,
whose edges are $k$ initially linearly independent unit deviation vectors from
the studied orbit whose magnitude is normalized to unity at every time step.
The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI)
(GALI$_2$ $\propto$ SALI). However, GALI$_k$ provides significantly more
detailed information on the local dynamics, allows for a faster and clearer
distinction between order and chaos than SALI and works even in cases where the
SALI method is inconclusive.
