Newton's laws of motion are often considered to be fundamental postulates for
describing the motion of particles in a gravitational field, at least from our daily
viewpoint. In a more general picture this is not so. Not only are they just a result
of the general theory of relativity, they can also be derived from a more general
principle, namely *Hamilton's principle*. Newton's laws of motion are just one
example of equations that can be deduced from Hamilton's principle, the equations of
motion for galaxies in an expanding Universe is another (see Section 3.3). The
following description of this principle is mainly taken from Gold.

Hamilton's principle is an ``integral principle'', which means that it considers the
entire motion of a system between time and . What is meant by this needs to
be specified somewhat. The instantaneous configuration of the system is described by the
values of *n* generalized coordinates , and corresponds to a particular
point in a Cartesian hyperspace where the *q*-s form the *n* coordinate axes. This
*n*-dimensional space is known as the configuration space. As the time evolves, the
system point moves in this configuration space, tracing out a curve. This curve
describes the path of motion of the system. The configuration space can be very
different from the physical three-dimensional space, where only three coordinates are
needed to describe a position at any give time. For example, a system that is being
described both by the spatial coordinates and the velocities would have a
six-dimensional configuration space at any given point in time.

Hamilton's principle is a version of the integral principle which considers the motion
of a mechanical system, described by a scalar potential that may be a function of the
coordinates, velocities and time. The integral, often also referred to as the
*action*, is, in an essential one-dimensional form from to , given by

where is the Lagrangian, given by , *T* and *V*
being the kinetic and potential energy, respectively. The dot indicate derivative with
respect to time. The dependence of *x* on *t* is not fixed; that is, *x*(*t*) is unknown.
This means that although the integral is from to , the exact path of
integration is not known. The correct path of motion of the system is such that the
action has a stationary value; i.e. the integral along the given path has the same value
to within first-order infinitesimals as that along all neighbouring paths. The
difference between two paths for a given *t* is called the *variation* of *x*,
, and is conventionally described by introducing a new function to
define the arbitrary deformation of the path and a scale factor to give the
magnitude of the variation. The function is arbitrary except for two
restrictions: firstly, it must satisfy the boundary values, ;
secondly, it must be twice differentiable. The paths can then be described as

We have a stationary value of the action when the derivative of *A* with respect to the
scale factor is zero:

The -dependence of the integral is contained in and , thus

By inserting equation (3.2) and integrating the second term by parts, we get

The integrated part vanishes due to the fixed end-points (boundary values). The condition for stationary values, equation (3.3), is therefore equal to the following relation:

In order to arrive at the equation of motion, the fundamental theorem of variational
calculus is needed. It state that if the integral in equation (3.6)
vanishes for every continuously differentiable in the interval ,
then the content of the brackets in the equation (3.6) must
identically vanish in the same interval; that is, for . It therefore
follows that *A* can have stationary values only if

which is the familiar Euler-Lagrange differential equation. By inserting the Lagrangian, one can then deduce the equation of motion for the mechanical system.

What is the point of going through this rather extensive deduction, just arrive at the well known Euler-Lagrange differential equation? In this case there are at least three motivations. Firstly, it shows that the Euler-Lagrange equations are in fact a result of the very compact Hamilton's principle. Secondly, it will be needed in the next section to justify the use of mixed boundary values. Thirdly, and most importantly, the principle is applied directly in the numerical implementation of the AVP.

The one-dimensional deduction presented in this section can easily be extended to any multidimensional case.

Mon Jul 5 02:59:28 MET DST 1999