We have performed a series of high-resolution ,N-body experiments on a connection machine CM-5 in order to study the stability of collisionless self-gravitating spherical systems. We interpret our results in the framework of symplectic mechanics, which provides the definition of a new class of particular perturbations: the preserving perturbations, which are a generalization of the radial ones. Using models defined by the Ossipkov-Merritt algorithm, we show that the stability of a spherical anisotropic system is directly related to the preserving or non-preserving nature of the perturbations acting on the system. We then generalize our results to all spherical systems. Since the 'isotropic component' of the linear variation of the distribution function cannot be used to predict the stability or instability of a spherical system, we propose a more useful stability parameter which is derived from the 'anisotropic' component of the linear variation.