The energy-momentum method (see, e.g., [R. Abraham and J. E. Marsden, Foundations of mechanics (1978; Zbl 0393.70001); J. E. Marsden and J. C. Simo, Act. Acad. Sci. Tau., 245–268 (1988); J. Marsden and A. Weinstein, Rep. Math. Phys. 5, 121–130 (1974; Zbl 0327.58005)]) is a fundamental tool in the analysis of autonomous symplectic nonlinear systems. This paper extends this method to the case of nonautonomous systems.
The paper is self-contained and as such will also be an ideal introduction in the literature on this subject. After an introduction that gives a well-defined context to the problem, the authors start with sections devoted to: Lyapunov stability of nonautonomous systems (Section 2), basics on symplectic geometry (Section 3), relative equilibrium points (Section 4). The fourth section contains also a theorem on time-dependent relative equilibria. The paper continues with foliated Lie systems and relative equilibrium submanifolds (Section 5) and the stability on the reduced space (Section 6). The main results obtained in Section 6 are then expanded and discussed in Section 7, which covers stability notions, reduced spaces, and relative equilibrium points. The paper is completed with a nice example which illustrates an application of the abstract theory, the “almost-rigid body”.
In the final Section 9 the authors focus on Poisson manifolds and study foliated Lie systems appearing in the study of relative equilibrium points of mechanical systems. They also consider several applications to interesting concrete systems, such as a ballet dancer turning around an axis, acrobatic diving into a swimming pool (the “falling cat problem”), or some celestial mechanics problems like stars passing through a nebula with a variable density. The unifying theme in these cases is the fact that the motion can be effectively described by a time-dependent inertia tensor.