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We investigate the detailed dynamics of multidimensional Hamiltonian systemsby studying the evolution of volume elements formed by unit deviation vectorsabout their orbits. The behavior of these volumes is strongly influenced by theregular or chaotic nature of the motion, the number of deviation vectors, theirlinear (in)dependence and the spectrum of Lyapunov exponents. The differenttime evolution of these volumes can be used to identify rapidly and efficientlythe nature of the dynamics, leading to the introduction of quantities thatclearly distinguish between chaotic behavior and quasiperiodic motion on$N$-dimensional tori. More specifically we introduce the Generalized AlignmentIndex of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped,whose edges are $k$ initially linearly independent unit deviation vectors fromthe 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 moredetailed information on the local dynamics, allows for a faster and clearerdistinction between order and chaos than SALI and works even in cases where theSALI method is inconclusive.

Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method