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In general, it is accepted

where
dP
is the differential of the linear momentum,
v and dv
are the speed and its differential,
M and dM
are the mass and its differential.

Travelling together with an object X of the mass m moving in frame M at a speed vx/m , we cannot detect speed vx/m. Our speed with respect to spacetime structure is now vm/x. Thus the linear momentum of the object X of the rest mass m, motionless with respect to frame X, but moving at a speed vm/x with respect to the spacetime structure of the frame M may be defined as pm/x. It is the same linear momentum, as if the object X of a non-relativistic mass m was moving at a speed vx/m with respect to frame X :
(47)

Looking from frame M, we can see our own frame M moving at a contemporary speed

It means, vx/m represents galileian projection in M-frame, corresponding to the speed vm/x in X-frame. Further we obtain (see also equation (17)):
(47a)

Consequently,

(47b)

Equations (47) and (47b) define the linear momentum of the object in its own reference frame. In spite of the fact, that object is without motion with reference to its own frame, it moves with respect to spacetime structure of any other reference frame, and all kinds of interactions influencing motion of the object, are realized by affecting of this relative motion.

With reference to frame M :

where
dpx/m
stands for the linear momentum differential of the object X in M-frame, and,
Mmx and dMmx
stand for galileian projection of the mass and its differential from X to M-frame.

Substituting for Mmx from equation (46a), we can derive
(48)

In case when dvx/m is in line with vx/m, we obtain
(49)

The galileian projection of the linear momentum from M- into N-frame:

Substitution for Mn(mx) from equation (42), and for

yields :
(50)

Applying equation (12), we can express dm/n / (m/n )2 in equation (50) :
(50b)

It is dvm/n = 0 at vm/n = const., if M-frame moves without acceleration, and without rotation with reference to N-frame, and, d = 0 if the acceleration of the object X in M-frame is in line with the speed vx/m . Under these circumstances, we receive:
(50c)

Substitution for

in equation (50c), yields:
(50d)

Applying equation (49), we have in common case:
(51)

(52)

In common motion configuration X/M/N, we may write
(53)

Under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame, and that acceleration of the object X in M-frame is in line with the speed vx/m , we receive
(54)

The force is defined as
(55)

Dividing equation (47) by time differential dtx, we receive the force responsible for acceleration of the object in its own frame.
(56)

Dividing equation (47b) by time differential dtx yields

Since

we have
(56a)

Comparing equations (56) and (56a), we receive:
(56b)

The force acting on the object moving in M-frame, derived from equation (48):
(57)

If acceleration is acting in direction of the speed, equation (57) gives :
(57a)

If acceleration is acting in opposite to direction of the speed, equation (57) gives :
(57b)

The galileian projection of the force from M- to N-frame, derived from equation (50) :

(58)

Applying equation (50b), we obtain :
(58a)

where (in equations 58 and 58a)
Fn(x/m)
stands for the galileian projection of the force from M- to N-frame. Direction of the force is defined by the angle nm according to equation (10), giving galileian projection of the angle from M- to N-frame resulting from vectors' superposition in equation (58),
ax/m
stands for acceleration of the object X in M-frame,
am/n
stands for acceleration of the frame M with reference to N-frame,
vx/m
stands for speed of the object X in M-frame at the angle a from vm/n,
vm/n
stands for speed of the frame M with reference to frame N,
m/n
stands for spacetime density at angle a in M- with reference to N-frame, and,
stands for angular speed of the speedvx/m vector in M-frame
Under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame, and that acceleration of the object X in M-frame is in line with the speed vx/m, we receive(see equation 50d) :
(58b)

Applying equations (51) and (52) :
(59)

(60)

In comon motion configuration X/M/N :
(62)

And finally, under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame, and that acceleration of the object X in M-frame is in line with the speed vx/m, we receive:
(62)

Remark: Direction of the vector ax/m is defined by the angle nm according to equation (10).
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