Accessible Points of Mechanical Systems

Abstract

In this chapter we study the question of whether or not two points m₀ and m₁ in the configuration space of a mechanical system can be connected by a trajectory. It is known (see, e.g., [85]) that for a second-order differential equation on the Euclidean space such a trajectory exists provided that the right-hand side of the differential equation is bounded and continuous. More precisely, for any two points m_o and m₁ and any interval [a, b], there exists a solution m(t) such that m(a) = m_o and m(b) = m₁. When the right-hand side is linearly bounded, some similar results are known for small intervals of time. The situation becomes much more complex for a nonlinear configuration space. In Sect. 9, we illustrate this by three examples of mechanical systems on the two-dimensional sphere. In the first example, the force field is smooth and independent of time and velocity (and so it is bounded). However, none of the trajectories beginning at the North Pole reaches the South Pole. In the second example, the force field is still bounded, autonomous, and smooth but now depends on the velocity. In this case there is no trajectory connecting any two antipodal points on the sphere. In the third example, we consider a gyroscopic force on S². (Hence, the force field is linear in velocity.) The behavior of trajectories in this system turns out to be quite similar to the second example.