The thorny way of truth - Free Energy Community

68What results from equation (2) is that no term can occur in(dS/dt)-H(dD/dt) that docs not contain one of the velocity components asa factor. Derivative dS/dt satisfies this condition. In order that the same isthe case with dD/dt, no term can be present in D in which the velocitieswould only occur in the first power. For then, the second component ofdD/dt would be loaded with terms which would be free of x ' ,y' , z ' . So,one sees that in D the magnitudes x ' ,y ' , z ' must be at least contained inthe second power.As the simplest example, we will take a homogeneous function of thesecond degree of x',y',z' for D:Coefficients Ai,j. . .F,, are functions of the coordinates of all the points.Derivative dD/dt will then consist of a homogeneous function of the thirddegree of x ' ,y', z ' and a homogeneous function of the first degree of thesame variable, and the coefficients that occur are functions of coordinatesx,y,z. However, the homogeneous linear function of x',y',z' thatoccurs in dD/dt^ just like function dS/dt, already has form (1) by itself,and cannot be put into this form in any other way. But, on the other hand,the function of the third degree occurring in dD/dt can be put into form(1) through a manifold of ways. So, the forces in motion are not totallydetermined by the expression for work.The theory of the conservation of kinetic energy is expressed in theformula T-S-D= const. We will now inquire how the motion mustproceed so that this theory is valid.We have t clue in section 43 as to how to answer this question. There,it is proven that:When P is only dependent on the coordinates qi, qi, ... and thisfunction's expression explicitly does not contain time t and, furthermore,when T is a homogeneous function of the second degree of qi', q2', . ..,thenIff{T+P)dt=./'