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13.4.1.2 Hamilton's principle of least action

Now sufficient background has been given to return to the dynamics ofmechanical systems. The path through the C-space of a system ofbodies can be expressed as the solution to a calculus of variationsproblem that optimizes the difference between kinetic and potentialenergy. The calculus of variations principlesgeneralize to any coordinate neighborhood of . In this case, theEuler-Lagrange equation is

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

(13.125)

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The coming presentation will use (13.124) to obtaina phase transition equation. This will be derived by optimizing afunctional defined as the change in kinetic and potential energy.Kinetic energy for particles and rigid bodieswas defined in Section 13.3.1. In general, the kineticenergy function must be a quadratic function of . Itsdefinition can be interpreted as an inner product on , whichcauses to become a Riemannian manifold [156].This gives the manifold a notion of the ``angle' between velocityvectors and leads to well-defined notions of curvature and shortestpaths called geodesics. Let denote thekinetic energy, expressed using the manifold coordinates, whichalways takes the form

in which is an matrix called the mass matrixor inertia matrix.

The next step is to define potential energy. A system is called conservative if the forces acting on a pointdepend only on the point's location, and the work done by theforce along a path depends only on the endpoints of the path.The total energy is conserved under the motion of a conservativesystem. In this case, there exists a potential function such that , forany . Let denote the total potential energy ofa 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 withextremals of the functional,

(13.127)

using any coordinate neighborhood of . The principle can beseen for the case of by expressing Newton's secondlaw in a way that looks like(13.124) [39]:
in which the force is replaced by the derivative ofpotential energy. This suggests applying theEuler-Lagrange equation to the functional

(13.129)

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

The advantage of the Lagrangian formulation is that the C-space doesnot have to be , described in an inertialframe. The Euler-Lagrangeequation gives a necessary condition for the motions in any C-spaceof a mechanical system. The conditions can be expressed in terms ofany coordinate neighborhood, as opposed to orthogonal coordinatesystems, which are required by the Newton-Euler formulation. Inmechanics literature, the variables are often referred to asgeneralized coordinates. This simply means the coordinatesgiven 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 calculatedexpression for . The appearanceof seems appropriate because the resulting differentialequations are second-order, which is consistent with Newton-Eulermechanics.Example 13..11 (A Falling Particle) Suppose that a particle with mass isfalling in . Let denote the position of theparticle. Let denote the acceleration constant of gravity in the direction. The potential energy is .The kinetic energy is
The Lagrangian is

(13.132)

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

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

(13.134)

Pof Hamilton

Substitution into (13.130) yields three equations:
These indicate that acceleration only occurs in the direction,and this is due to gravity. The equations are consistent withNewton's laws. As usual, a six-dimensional phase space can be definedto obtain first-order differential constraints.

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

Steven M LaValle2012-04-20