# Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

Chapter

## Abstract

In this chapter we study the question of whether or not two points *m* _{0} and *m* _{1} in the configuration space of a mechanical system can be connected by a trajectory. It is known (see, e.g., Hartman (1964)) that for a second order differential equation (i.e., in particular, for Newton’s law) on 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* _{0} and *m* _{1} and any interval [*a*,*b*], there exists a solution *m*(*t*) such that *m*(*a*)=*m* _{0} and *m*(*b*)=*m* _{1}. When the right-hand side is linearly bounded, some similar results are known for small time intervals.

## Keywords

Mechanical System Accessible Point Lower Semicontinuous Conjugate Point Complete Riemannian Manifold
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## References

- 144.Hartman, P.: Ordinary Differential Equations. John Wiley and Sons, N.Y.-L.-Sydney (1964) Google Scholar

## Copyright information

© Springer-Verlag London Limited 2011