The flow of

linear momentum ([T.sup.33]) is positive in the first region and then jumps up on the first boundary due to the jump of the current j.

The motion is assumed to be one-dimensional, due to the conservation of the

linear momentum [1, 7, 8, 14-17].

Let the entire space be divided into small regions of dimension [delta]x = [delta]y = [delta]z = [delta]l, where [delta]l is sufficiently small that any of the momentum densities, [[??].sub.pi] (x, y, z, t), can be considered constant within each region so that [[??].sub.pi](x, y, z, t) [delta][l.sup.3] is a

linear momentum propagating at c in a definite direction in space.

where [mathematical expression not reproducible] are, respectively, the components of the

linear momentum P in the spaces, [mathematical expression not reproducible] are all real.

Collecting all the results, Eqs (31)-(34), and substituting them into the balance of

linear momentum (29), we arrive at the dispersive wave equation in the form

As a result, the centroidal dynamics can be indirectly stabilized using a simple VRPbased control law to regulate the DCM position where the commanded VRP is mapped to a corresponding

linear momentum rate of change objective using Eq.

By referring to the resolved momentum control, the total momentum, including

linear momentum P and angular momentum L, can be defined as [13]

[K.sub.l] =

linear momentum resistance coefficient, 1/s

Linear momentum equation in the radial direction for the gaseous phase:

Analysis of reported experimental data [17] indicates that the forward angular momentum associated with a forward fall is neutralized primarily during the impact phase, whereas the forward and downward

linear momentum associated with a fall was reduced during the postimpact phase.

In this case, as the associated interaction term is negative, the particle phase delivers a

linear momentum to the gas in the axial direction, decreasing the velocity of the particles.

Because the yo-yo is moving in a downward line, it builds up

linear momentum (an object's mass times its velocity); but it also gains angular momentum, or momentum from rotating.