  In this work, we propose a geometric framework for analyzing mechanical
manipulation, for instance, by a robotic agent. Under the assumption of
conservative forces and quasi-static manipulation, we use energy methods to
derive a metric. In the first part of the paper, we review how quasi-static
mechanical manipulation tasks can be naturally described via the so-called
force-space, i.e. the cotangent bundle of the configuration space, and its
Lagrangian submanifolds. Then, via a second order analysis, we derive the
control Hessian of total energy. As this is not necessarily positive-definite,
from an optimal control perspective, we propose the use of the squared-Hessian,
also motivated by insights derived from both mechanics (Gauss' Principle) and
biology (Separation Principle). In the second part of the paper, we apply such
methods to the problem of an elastically-driven, inverted pendulum. Despite its
apparent simplicity, this example is representative of an important class of
robotic manipulation problems for which we show how a smooth elastic potential
can be derived by regularizing mechanical contact. We then show how graph
theory can be used to connect each numerical solution to `nearby' ones, with
weights derived from the very metric introduced in the first part of the paper.
