Thus in a uniform and constant field the velocity is a linear function of time. The expression obtained for v shows that the particle moves in the plane defined by the force vector F and the initial velocity vector v0. Let us take this as the xy plane, and the y axis in the ...
To compute them explicitly, we first need the explicit expression of the relativistic scalar Green's function G(x). As we are deriving here the part of the gravitational interaction which is “conservative” (i.e., energy conserving), we must use the time-symmetric (half-advanced half-...
In the case of an arbitrary mass distribution, the gravitational force acting on a particle at a given point can be expressed as the product of the mass of the particle and the vector g, which is called the gravitational field strength at the given point. The larger the absolute value of...
While construct- ing the kinetic part of the vector matrix field Wˆ μ in (16) we kept in mind the analogy with the Ginzburg–Landau Lagrangian [60]. This analogy presupposed the replace- ment of the massive G-L scalar field by the massless vector field Wˆ μ and the transition ...
To be more specific, for a scalar field, Myers et al. [21] proposed that with a preferred frame defined by a a four-vector nα, the normal Klein–Gordon equation for a particle of mass m is replaced by, (◻+m2)Φ=icκ1Mp(n⋅∂)3Φ, (4) with Mp being the Planck mass...
From the observations, quantities that can be modeled have been removed, so that an element of the vector is given by: where: and yp is the mean value of xk over the area involved in the prediction, so that the residual observations involved in the prediction are centered. Δgkt(SH)...
(1), which is a second-order vector differential equation. The two-body equations of motion may be expressed in state-variable form by six scalar differential equations, and therefore, the complete solution requires six initial conditions: three initial position coordinates and three initial velocity...
Presumably, the criterion for choosing a partic- ular set of commuting vector fields could come from some more fundamental theory that has NC field theory as its low- energy limit. Gauge field theory actions in the NC setting have the same form as the corresponding commutative ones, and are ...
The field of interest in scientific work is the gradient of the dynamic gravitational force field. The gravitational radiation emitted by astronomical sources is of a force gradient or tensor type (rather than of a force or vector type as is electromagnetic radiation) and therefore requires an ...
(t) in a non-uniform magnetic vector field of flux density B(x, y, z) is subject to force F=I (t) {n×B(x, y, z)}, where n is the unit vector in the direction of current flow, in this case the z direction. The quantities Bx(0, t) and Bxz(0, t) therefore represent ...