CANCELLED,   PROBLEM INDEFINITE
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INVENTED: 4/8/2010

Assume an isolated system of two gravitational masses, which are moving one against  the other, as shown below, at non relativistic
velocities.
 m₁                      m₂                                                                  

 ●                     ●     

    →                     ←

    U₀₁                    U₀₂

The potential energy of the first particle isV₁=-G m₁ /r, The corresponding energy is dE₁=-Gm₁. m₂ /r.ds₁, and the force on each particle is F₁ =Gm₁. m₂/r².
Similarly, for the second particle  is V₂= -Gm₂/r,  dE₂=-Gm₂. m₁ /r.ds₂,  F₂=Gm₂. m₁ /r²

Then the force on each particle  is
                                    F₁ = - F_2,  and dE₁  = dE₂     A
The work on each particle is dW₁ = (F₁ + F₁ + dF₁)/2.ds₁

= (F₁+1/2.dF₁).ds₁, dropping second order factors or cross products of differentials = F₁.ds₁. Thus
                                        dW₁= F₁.ds₁        B
The distance moved is ds₁= U₀₁.dt + (1/2)x(F₁/m₁).(dt)²

Similarly, dropping second order factors or cross products of differentials,
                                            dW₂ = F₂.ds_{2 ,} C
ds₂= U₀₂dt +(1/2)x(F₂/m₂).(dt)²

^dW₁+ dE₁ ^{+ dW}₂ + dE₂ = ^dW₁ ^{+ dW}₂ + 2dE=constant (?), i.e. the total work  plus the change of the potential energies which are common i.e. dE₁ = dE_2.=dE. See above equations  A.
or  ^dW₁ ^{+ dW}₂ + 2E= constant (?), with the factor 2 maintained. For, allegedly, the potential energy belongs to the system, e. to both of them and not to each particle. We may not accept this for as explained below in the concluding remarks the potential energy may conveniently chosen quantity to make the total energy conserved, not taking the reality that both particles are relative speaking equivalent. If not dropping the factor 2 makes the situation impossible for the conservation of energy.

  Then,  ds₁=U₀₁.dt + (1/2)x(F₁/m₁).(dt)² , 

ds₂ =U₀₂dt +  (1/2)x(F₂/m₂).(dt)^{2  , which are different FOR THEY ARE TWO POLYNOMIALS WITH RESPECT TO dt AND OF THE SECOND DEGREE, having arbitrary constants} U₀₁ ^and U₀₂,  ^{as well as arbitrary coefficients F/m}₁^,F/m₂^{. From that indeed we conclude different  moving distances for each particle. We may conclude that the energy (work) for the first particle plus the change of the corresponding common potential energy plus the work (different) for the second particle plus the corresponding change of the common potential energy, vary.  The energy is not conserved here, because the common potentials decrease (change) as
                        2/(r+ds}₁^+ds₂^{),
though the work or  the gained kinetic energy increases (counter changes) as 
            (ds}₁^+ds₂^{).
Making the two expressions which depend differently on 2/(r + ds}₁^+ds₂^{) and (ds}₁^+ds₂^{) and being unable to compensate one another in making their summation constant.  Therefore:
total energy}₁^{+ total energy}₂= not conserved. ^Q.E.D.

Note: the kinetic energies, which numerically are equal to the corresponding works, (for int(F.ds)= i nt(mgds)=int(mdu/dt.ds)= int(m.u.du)=1/2mu²), int meaning integral)  are included in the corresponding works and it should not be counted twice. Q.E.D.

with the power of logic and
mathematics.
P.T.P.
 

If the above is found in error, it is subject to 100.000 Euros reward.  This offer is not applied yet. It will be activated as soon as the sign under construction above is removed to allow detection of possible typo-mistakes

GENERAL DIDACTIC PARADIGMS- REMARKS AND CONCLUSION
We consider, in general, the potential energy which may or may not in the future,  produce a work equal to an assumed  potential energy, not being a real energy (substantial in the real sense), but  a man made intervention to make a changing energy, to be constant or conserved. In the above example of ours, the situation of man made saving the conservation,   potential energy, did not succeed for   two simultaneous energy changes, because, the corresponding algebra did not agree not to, with our choice of factor 2 for the potential energy!
A common mistake done by physicists and bibliography, is the arbitrary choice of the frame of references. For example, if once we chose the reference frame,  firstly attached to one particle, or, secondly. attached to the other particle, then we shall refer to the same, common relative velocity U and not to two different velocities-coefficients of our above analysis-U₀₁ and U_02. We could say having one and the same coefficient U and ignoring terms in the second order, then we would have the same variation for ds_1, ds₁, which we had in the above analysis and produced the non conservation of energy, making thus the energy to be conserved.
The mistake was, because we picked wishfully, purposely, the wrong reference frames (accelerating) for producing the conservation of energy. As observers, we should have picked our own (our office) and correct stationary reference frame, as we have implied so also in presenting the problem by the above picture. Notice it is not only Einstein's relativity which is relative. Also relative are classical mechanics with the velocity being relative - (in this case, the operation of addition of velocities is different than relativity's Theory).
The above result does not contradict celestial mechanics and, in particular, for a closed trajectory, the total energy is conserved. Our example above with the conventional choice of the factor 2, just shows correctly that the total energy temporally varies and temporally is not conserved.
 

 
Final report,
As it has been realized the validity of this presentation depends on the convention of a choice of a factor, for example, with the values 1 or 2 or something else, also on the prehistory of the particles, which depends on it's prehistory and so on¹. So, we decided not to activate the offer of this presentation for ever or any other offer related to it. For we may never have an unambiguous and objective thesis for every body, not stating the prehistory of the particles. The stated problem in a sense is indefinite, though we have chosen the factor 2. Conservation in time of Energy from time - infinity to time +infinity, remains, a problem.

¹ We may say for  an object that was raised to a height h, and then that we have spent a potential energy mgh plus the kinetic energy for the transportation.
It is different how ever to say the Earth moved downwards with the object stationary. Then we shall have spend mgh again plus the kinetic energy of the transportation of the Earth, which is relatively too much more.
This kinetic energy is unavoidable. It is not due to friction, though friction favorably might be used. This kinetic energy might be compared to the energy needed for  a spacecraft in breaking to land on the airless moon.
Then we have  to examine into the prehistory and ask: how the Earth was kept constant or the object was kept constant? The answer could be, because this and that. Then we may look into the prehistory of the prehistory. The answer could, again: This and that was done because.... and so on... Endlessly for time to -infinity.