The essence of this approach is to replace the standard Eulerian density-like variables for cloud and precipitation fields with a probabilistic

Lagrangian approach that applies point particles to represent the formation, growth, and fallout of cloud droplets, raindrops, and ice particles.

The stability of the

Lagrangian points make them prime candidates for building up future space based resources.

The

Lagrangian force density F comprises two parts of tension-compression [F.sub.S] and bending [F.sub.b] forces, i.e.

Eulerian materials and

Lagrangian elements can interact using the Eulerian-Lagrangian contact.

In this paper, we propose an alternative approach for the map from the canonical

Lagrangian variables to the Eulerian variables.

Then in this case we use the

Lagrangian derivative; for this we build the transported [bar.u]([theta]) on [[OMEGA].sub.0].

(ii) The subring of A* (X) containing divisors, Chern classes and

Lagrangian constant cycle subvarieties injects into cohomology.

ABSTRACT: We reformulated the

Lagrangian density for single fluid by using Caputo's fractional derivative, then from the fractional Euler-Lagrangian equation we obtained the equations that described the motion of single fluid in fractional form.

On the other hand, the choice N(t) = 1 for the gauge field on the

Lagrangian formulation represents no difficulty; however, if we want to implement the Hamiltonian formulation of the theory and thus the quantum formulation, we have to keep the gauge arbitrary.

Similar to GPS, a high-precision satellite navigation constellation which consists of libration point satellites in the Earth-Moon system is introduced to provide navigation information for deep space probes, which can be called, accordingly, the

Lagrangian point satellite navigation system.

The numerical computational model developed in this work for the transport of the particles is governed by the

Lagrangian approach, where the particles are located following a concentration exponential law or randomly located in the space.