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An Introduction To Non-aristotelian Systems And General Semantics.

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THE NEWER 'MATTER*
703
For simplicity we will denote the 'time' derivatives by the chosen letter, but with a dot over it (newtonian method). Thus,
The coefficients in the expansion of c«, depend only on the values of the generalized co-ordinates and are independent of the value of the time-derivatives. The time-derivatives can be properly called generalized velocities, and we may denote them by g,-.2
In establishing formulae for the quantum theory we want to be as general as possible and not restrict ourselves to vibrational energy only. But we want to take into consideration any arbitrary point-mass, independently of whether we assume this point to be charged or not.
We define the momentum or impulse as the product of the mass and the velocity, or, p =mv. If, instead of denoting our co-ordinates by x, y, and z, we use the generalized co-ordinates g,-, we would have for the magnitude and direction of the velocities the time-derivatives of the co-ordinates; namely, , where, .
Ifrepresent the corresponding components of the momentum
or impulse, then we would have p, =mqi.                                                        (5)