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The Sixth Element

Having once opened Pandora’s Box, we might inquire if yet higher order derivatives might be applicable in Newton’s Second Law.  This makes particular sense when physics is faced with such anomalies as Sonoluminescence and other cases where there are radical changes in acceleration (and potentially radical changes in the rate of acceleration, etceteras).  The implication is if the addition of The Fifth Element (aka the third order differential term) can be so lucrative in terms of results, then what might happen if we added a fourth order differential term, or The Sixth Element?

In general, we may legitimately assume, as before, that Newton’s Second Law (or the electromagnetic circuit equivalent) can be written in the form:

F  =  m Sn=0  Dn Tn

                                                                                                                                                                (Equation 1)

where the summation is from n = 0 to some as yet undetermined integer value for n.  If we arbitrarily select a limit of n = 7, we obtain the Seventh Order equation of the form:  

F = D7 m T7 + D6 m T6 + D5 m T5 + D4 m T4 + D3 m T3 + m T2 + R T1  + k To

                                                                                                                                                                (Equation 2)

where each Dn is a constant, and Tn is the nth order derivative of the displacement, X.            

If we then let

To  º  X  =  +  a cos w t  +  b sin w t


T1   =  -  a w sin w t  +  b w cos  w t

T2   =  -  a w2 cos w t  -  b w2 sin w t

T3   =  +  a w3 sin w t  -  b w3 cos w t

T4   =  +  a w4 cos w t  +  b w4 sin w t  =  To w4 

T5   =  -  a w5 sin w t  +  b w5 cos w t  =  T1 w4

T6   =  -  a w6 cos w t  -  b w6 sin w t  =  T2 w4

T7   =  +  a w7 sin w t  -  b w7 cos w t  =  T3 w4

                                                                                                                                                                (Equations 3)

The most blatant observation is that:  

Tn+4   =  Tn w4   

and that we have only four terms (the zeroth through the third order differential) which are unique.  In effect, higher order differentials (aka Sixth+ Elements) are mathematically connected to the lower four differentials.  Note in particular, that Tn+8  = Tn w4, etceteras.

This would suggest that all the higher order differentials (above the third order) are no more than resonances of the first four differentials.  They will still have physical attributes of importance, but there seems little likelihood of anything profoundly unique.

At the same time, if  F = Fo cos w t, and we collect sine and cosine terms, we obtain:

D7 a w7 - D6 b w6 - D5 a w5 + D4 b w4 + D3 a w3 - b w2 - (R/m) a w + (k/m) b  =  0


 - D7 bw7 - D6 aw6 + D5  bw5 + D4 aw4 - D3 bw3 - aw2 + (R/m) bw + (k/m) a - Fo/m  = 0  

Collecting variables and solving simultaneously for a and b, we obtain:  

X  =  Fo cos (w t - f) / m Z


Z  =  [(D6 w6 - D4 w4 + w2 - k/m)2 + (- D7 w7 + D5 w5 - D3 w3 + w R/m)2] 1/2  


tan f  = (- D7 w7 + D5 w5 - D3 w3 + w R/m) / (-D6 w6 + D4 w4 - w2 + k/m)

                                                                                                                                                                (Equations 4)

Equations 4 are not easily deciphered, but we might note the grouping of terms as being possibly important.  For example, k/m and w2 derive from potential and kinetic energy terms, while R/m and D3 are “resistance” terms (viscosity and “delay time”).  This might influence, for example, our supposition for the higher order constants.  In fact, we might assume a situation where:  

tan f  =  “resistance” / “energy”

                                                                                                                                                                (Equation 5)

We can also neglect higher order terms to obtain slightly less complicated forms, i.e.:  Fourth Order:

x  =  Fo cos (w t - f) / m Z4


Z4  =  [(- D4 w4 + w2 - k/m)2 + (- D3 w3 + w R/m)2] 1/2


tan f4  = (- D3 w3 + w R/m) / (D4 w4 - w2 + k/m)

                                                                                                                                                                (Equations 6)

Fifth Order:

x  =  Fo cos (w t - f) / m Z5


Z5  =  [(- D4 w4 + w2 - k/m)2 + (D5 w5 - D3 w3 + w R/m)2 ] 1/2


tan f5  = (D5 w5 - D3 w3 + w R/m) / (D4 w4 - w2 + k/m)

                                                                                                                                                                (Equations 7)

Sixth Order:

x  =  Fo cos (w t - f) / m Z6


Z6  =  [(D6 w6 - D4 w4 + w2 - k/m)2 + (D5 w5 - D3 w3 + w R/m)2 ] 1/2


tan f6  = (D5 w5 - D3 w3 + w R/m) / (-D6 w6 + D4 w4 - w2 + k/m)

                                                                                                                                                                (Equations 8)

Fourth Order Differential -- The Sixth Element  

Equations 6 are essentially based on the equation:  

F  =  D4 m T4  +  D3 m T3  +  m T2  +  R T1  +  k To

                                                                                                                                                                (Equation 9)

If R and k can be neglected, we obtain:  

X4  =  Fo cos (wt - f4) / { m w2 [(1 - D4 w2)2   + D32 w2)] 1/2}  


tan f4  =  D3 w / (1  -  D4 w2)

                                                                                                                                                                (Equations 10)

We note immediately that the phase angle is modified from the third order derivative term, including the possibility that f4 may now be in excess of 90 degrees -- depending on the relative values of D4 w2 and 1 -- whereas previously, the f = p/2 point could not have been achieved.  This new factor makes for the possibility of experimentally verifying the need (or lack thereof) for the fourth order term.  At the same time, however, it may be that still higher order terms may intervene in some as yet unknown manner.

For the moment, however, we can stick with The Fifth Element, and temporarily, at least, neglect the higher order terms.  The idea is avoid forgetting about them altogether.  


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