This condition is briefly investigated to illustrate the effects of accelerative forces on the direction of motion of a point mass in D₀ . In doing so however, it also enables the reason for the original designations of "longitudinal" and "transverse" mass to be simply demonstrated (see Sect. 3.1 ) . The spatial situation can most easily be described by Fig. 4.1. where, for clarity, spatial cartesian co-ordinates are now represented by X and Y.

The momentum equations are:

(4.1 )

where m is the energy mass of the point mass and M_x and M_y are the respective spatial, and M_t the temporal, existence momentums. The initial velocity at t = 0 is v₀, the spatial acceleration is a, and the other terms are as shown in Fig. 4.1. Differentiating with respect to t gives the force equations thus:

(4.2)

If in (4.2), F_t is zero, m can be determined by simple integration to give, as in Sect. 3:

(4.3)

so that also

(4.4)

but in this case with

(4.5)

then

(4.6)

and this in (4.4) gives:

(4.7)

Substitution from (4.14) for dv_x /dt and dv_y /dt then gives:

(4.8)

Finally substitution for v_x and v_y from relationships implicit in Fig. 4.1, yields:

(4.9)

which clearly shows that a variation in mass only results from that element of applied accelerative force acting along the velocity vector.

Determination of dv_x /dt and dv_y /dt and Associated Inertial Masses.

Substitution of (4.3) and (4.7) into the spatial part of (4.2) yields after reduction:

(4.10)

so that from the F_y half of this equation:

(4.11)

With substitution of this into the F_x half of (4.10), there is after reduction:

(4.12)

but from Fig. 4.1, F_y = F_x tan.x which when substituted into (4.12) yields:

(4.13)

Finally substitution for v_x and v_y from relationships implicit in Fig. 4.1, gives:

(4.14)

Interpretation of these equations is quite simple when (4.3) and (4.9) are introduced. In both cases (4.14) reduces to the spatial parts of (4.2), showing that the first term inside the respective brackets is the normal acceleration resulting from the ratio of force to energy mass, while the second term is the retardation due to the reaction between mass rate and spatial velocity. Consequently from (4.14), m_ax and m_ay, the inertial masses in the respective directions, may be expressed thus:

(4.15)

The difference between these two terms is solely due to the different mass rate reaction forces generated in the respective directions. Equality occurs when the applied force and velocity vectors are coincident i.e. when x and .h are equal (but not zero ) . Then:

(4.16)

and is of course equal to the inertial mass of rectilinear motion.

Angular Relationship Between Applied Force and Acceleration.

This can most easily be inspected by comparing the respective angular relationships between both the a and F directions and the X axis. Thus from (4.14) directly:

(4.17)

which shows that the resultant spatial acceleration does not lie in the same direction as the applied force. The reason is again the difference in the mass rate reaction terms, in the X and Y directions.

Determination of dv/dt and dv_n /dt

These terms are defined as the accelerations produced both along and normal to the velocity vector. Substitution of (4.14) into (4.6) yields:

(4.18)

Note that inertial mass along the velocity vector, is, as would be expected by virtue of (4.9), equal to that in rectilinear acceleration.

To determine dv_n /dt, note that from Fig. 4.1

(4.19)

after expansion, substitution from (4.17) produces, after some reduction:

(4.20)

so that substitution for dv/dt from (4.18 ) then yields:

(4.21)

The point about this result is of course the appearance of the energy mass, there being no mass rate reaction involved because the direction concerned is normal to the velocity vector.

Other terms, such as acceleration along and normal to the direction of applied force, angular rate and angular acceleration of the velocity vector and, associated energies, exhibit relationships of a similar nature to the above.

The two boundary conditions of x are, x = 0 and x = p /2, at which the following apply:

(i ) x = 0 e.g. F lies parallel to the Velocity Vector, along the X axis

Motion in the Y direction is non-existent while that in the X direction is of course that of rectilinear motion of the main text, i.e. h is also zero. (Note that this condition is a special case of x = h used to obtain (4.16) above ) . Of particular interest however is from (4.15)

(4.22)

m_ax is now the inertial mass of rectilinear motion and applies only in the X direction, i.e. in the direction of the now coincident force, acceleration and velocity vectors. It was for this reason that m_ax, in the above form was in the literature originally termed "longitudinal" mass. Under this condition m_ay is the mass applicable normal to the accelerated motion and was consequently originally termed "transverse" mass. It is equal to the value of energy mass because of course no mass rate reaction term exists in the transverse direction.

The most interesting result from this situation emerges from (4.14) and (4.15) i.e.

(4.23)

This simply states that with all the accelerative force applied normal to the direction of initial velocity, a very small deceleration in that direction occurs due to the reaction of mass rate with v_x. Inertial mass in the X direction consequently takes the `hypothetical' value of zero as a deceleration occurs without the application of an external force in that direction.

Summary

It is clear that the primary difference between the motion described herein and that of Classical Mechanics, apart from the main relativistic effects, is due to the mass rate reaction terms. Most particular in this respect is the non-coincidence of the force and acceleration vectors. In the next Section it is shown that a primary result of this effect upon a trajectory is to cause it to rotate.

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