([??], [H.sub.D]] = 0), implying that [??] is a constant of motion
At constant [theta] = 0 the geodesics may be obtained, and because [[partial derivative].sub.t] is a Killing vector, there will be a constant of motion
denoting now by q the constant of motion (the energy) associated with the timelike Killing vector and feeding it back into (14), we get
It is obvious that a constant of motion
is given by the total number of individuals; namely,
In an isolated system, total momentum is a constant of motion [3, 4]; starting from this invariance we will deduce the necessary condition for the solenoidality of [??].
The mechanical counterpart of momentum density, as obtained from Lorentz equation (1), together with the electromagnetic component of the total momentum, forms a constant of motion in two cases:
Is there any connection between the dynamical properties of Maxwell-Bloch equations and the geometry of the image of a vector valued constant of motion
(the energy-Casimir mapping in our case) and if yes, how can one detect as many as possible dynamical elements and dynamical behaviour (stability in our case)?
+ [m.sub.n]), for any N [less than or equal to] -2, the peakon ODEs (43) has a constant of motion
: [m.sub.1] + ...
where u = 1/r and h is a constant of motion
. The solution to equation (4.12) depicts the famous perihelion precession of planetary orbits
Let [??] = X(x) be a dynamical system on a differentiate manifold U, let [x.sub.0] be an equilibrium point, that is, X([x.sub.0]) = 0, and let C = ([C.sub.1], ..., [C.sub.j]): U [right arrow] [R.sup.j] be a vector valued constant of motion
for the above dynamical system with C([x.sub.0]) being a regular value for C.
where l is a constant of motion
. l physically corresponds to the angular momentum and hence equation (3.13) is the Law of Conservation of angular momentum in this gravitational field [7, 9].
implies that the phase flux is conserved along X: this means that a constant of motion
exists for L and a conservation law is associated to the vector X.