Now sufficient background has been given to return to the dynamics of mechanical systems. The path through the C-space of a system of bodies can be expressed as the solution to a calculus of variations problem that optimizes the difference between kinetic and potential energy. The calculus of variations principles generalize to any coordinate neighborhood of . In this case, the Euler-Lagrange equation is

in which is a vector of coordinates. It is actually scalar equations of the form

The coming presentation will use (13.124) to obtain
a phase transition equation. This will be derived by optimizing a
functional defined as the change in kinetic and potential energy.
Kinetic energy for particles and rigid bodies
was defined in Section 13.3.1. In general, the kinetic
energy function must be a quadratic function of . Its
definition can be interpreted as an inner product on , which
causes to become a *Riemannian manifold* [156].
This gives the manifold a notion of the ``angle'' between velocity
vectors and leads to well-defined notions of curvature and shortest
paths called *geodesics*. Let
denote the
kinetic energy, expressed using the manifold coordinates, which
always takes the form

in which is an matrix called the

The next step is to define potential energy. A system is called *conservative* if the forces acting on a point
depend only on the point's location, and the work done by the
force along a path depends only on the endpoints of the path.
The total energy is conserved under the motion of a conservative
system. In this case, there exists a *potential function*
such that
, for
any . Let denote the total *potential energy* of
a collection of bodies, placed at configuration .

It will be assumed that the dynamics are time-invariant.
*Hamilton's principle of least action* states that the trajectory,
, of a mechanical system coincides with
extremals of the functional,

using

in which the force is replaced by the derivative of potential energy. This suggests applying the Euler-Lagrange equation to the functional

in which it has been assumed that the dynamics are time-invariant; hence, . Applying the Euler-Lagrange equation to (13.127) yields the extremals.

The advantage of the Lagrangian formulation is that the C-space does
not have to be
, described in an inertial
frame. The Euler-Lagrange
equation gives a necessary condition for the motions in any C-space
of a mechanical system. The conditions can be expressed in terms of
any coordinate neighborhood, as opposed to orthogonal coordinate
systems, which are required by the Newton-Euler formulation. In
mechanics literature, the variables are often referred to as
*generalized coordinates*. This simply means the coordinates
given by any coordinate neighborhood of a smooth manifold.

Thus, the special form of (13.124) that uses (13.129) yields the appropriate constraints on the motion:

Recall that this represents equations, one for each coordinate . Since does not depend on time, the operator simply replaces by in the calculated expression for . The appearance of seems appropriate because the resulting differential equations are second-order, which is consistent with Newton-Euler mechanics.

The Lagrangian is

(13.132) |

To obtain the differential constraints on the motion of the particle, use (13.130). For each from to ,

(13.133) |

Since does not depend on , the derivative for each . The derivatives with respect to potential energy are

(13.134) |

Substitution into (13.130) yields three equations:

(13.135) |

These indicate that acceleration only occurs in the direction, and this is due to gravity. The equations are consistent with Newton's laws. As usual, a six-dimensional phase space can be defined to obtain first-order differential constraints.

The ``least'' part of Hamilton's principle is actually a misnomer. It is technically only a principle of ``extremal'' action because (13.130) can also yield motions that maximize the functional.

Steven M LaValle 2012-04-20