ToeQuest

AntonioLao · Quote #1 (**permalink**) 05-05-2006, 02:20 PM

The principle of least action in analytical mechanics (in complex domain of quantum mechanics this is equivalent to Lagrangian formalism) raises the fact that nature is frugal not lazy. Nature does not waste any unnecessary expenditure of its reserved energy sources. Moreover, in order to conserve and optimize these energy expenses, nature is wise; it uses minimum principles describable by mathematics. One of these found in the real domain is given by the following: There exist two real numbers, a and b such that their products are equal to their differences, ab = a - b if and only if a = b / (1 - b) and b = a / (1 + a).

If a and b are dimensions of force then equality of forces implies that a = b = 0. If a = ∞ then b = 1 and if a = 1 then b = ∞.

If a and b are dimensions of time then the interval Dt = t(a) – t(b) is meaningful only if t(a) ≠ t(b) otherwise Dt = t(a) Ä t(b) = 0. This suggest that the physical definition of acceleration as distance per unit time per unit time implies t(a) Ä t(b) = t(a) – t(b). This can be used to remove time parameterization in quantum mechanics and quantum field theories. Hence it proved the independence of the phase factor exp(iq) in all quantum mechanical wavefunctions.

If a and b are dimensions of length then if a metric (distance) is defined as m = a - bthen m = aÄbcan be used to prove the differential form of Stoke’s theorem in vector analysis.

If a and b are dimensions of energy then if a= 1is the kinetic energy and b= ½is the potential energy then ab = a - bsatisfies the virial theorem. Furthermore, square of energy as ab implies that ab = (a - b)(a + b) where the factor (a + b) is normalized into a unit circle a + b = 1 implying that a + b = a + b = 1 which is true if and only if a= 1and b= 1/∞ or if a= 1/∞and b= 1.

If a and b are dimensions of speed then the g-factor of special relativity becomes the real part of an imaginary number.

Guille · Quote #2 (**permalink**) 05-05-2006, 02:35 PM

Quote:

Originally Posted by AntonioLao

ab
If a and b are dimensions of energy then if a= 1is the kinetic energy and b= ½is the potential energy then ab = a - bsatisfies the virial theorem. Furthermore, square of energy as ab implies that ab = (a - b)(a + b) where the factor (a + b) is normalized into a unit circle a + b = 1 implying that a + b = a + b = 1 which is true if and only if a= 1and b= 1/∞ or if a= 1/∞and b= 1.

But this means that a+b is equal to zero (for the multiplication of ab is the same as their difference and at the same time the same as the multiplication of their difference and their sum. But this also doesn't make sense because it would impply that ab is something times nothing. Of course, that is the meaning of the paragraph, 1 and 1/∞, but it doesn't make sense) and thus also a + b equals zero, and thus also 1=0. Any explanation???

AntonioLao · Quote #3 (**permalink**) 05-05-2006, 02:42 PM

Quote:

Originally Posted by GUILLE

Any explanation???

If a = b then it implies both are zero and they repels. If a ≠b then they attrack each other. This implies that 0 and 1/infinity is different mathematically. We cannot substitute one for the other.

AntonioLao · Quote #4 (**permalink**) 05-07-2006, 03:38 PM

Fora + b = 1 to hold, let a= sinqand b = cosqsuch that the trigonometric identity sinq + cosq = 1 is valid. Furthermore, extension to nth powers $\alpha^n \beta^n = \alpha^n - \beta^n$ with polar form $r^n = \sec^n \theta - \csc^n \theta$ and normalized duals $\alpha^n + \beta^n = 1$ with polar form $\sec^n \theta - \csc^n \theta = 1$ .

Guille · Quote #5 (**permalink**) 05-07-2006, 06:34 PM

Quote:

Originally Posted by AntonioLao

If a ≠b then they attrack each other.

What exactly means to 'attrack'? I had never read it before.

Quote:

Originally Posted by AntonioLao

Fora + b = 1 to hold, let a= sinqand b = cosqsuch that the trigonometric identity sinq + cosq = 1 is valid. Furthermore, extension to nth powers

with polar form

and normalized duals

with polar form

.

Could this also derive a sort of the nth power version of Euler's identity:

In yours, you take away, but what happens if you take the inverse of yours, then they would sum, the cosine and the sine? If so, would it also give e^ix?