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E-mail: berez.by@yandex.ru
Vasiliy Knyshev (Knysh)
On the symmetry of differentiable functions
Annotation: The introduction of notion of a non-degenerate function. Thanks to this received Newton's second law of the third order. As well as the equation of motion of a particle in a central field.
1. The function defined on an interval is degenerated if it is non-invertible on any subinterval of the interval definition. The example of a function degenerated is a constant function.
We assume that the function is non-degenerate. Then there exists at least one subinterval possibly larger where the function is invertible. After exclusion of all such inversion subintervals we have a set.
If it is discontinuous, then the function is called to as non-degenerated, for example sin x.
If it is not discontinuous then the set contains subintervals. On each of them the function is degenerated. We exclude all such subintervals of degeneration. The residue of the set is discontinuous. Then the function is called to as half-degenerated.
Since a half-degenerated (non-degenerated one is its special case) function is a negation of the degenerated one, any function is included in one of the above-mentioned classes.
The main theorem (measure). The interval is the sum of a finite or countablethe number of pairwise disjoint intervals.
A simple definition.
There exists a sum of intervals, on each of which the function is invertible (strictly monotonic). This is non-degenerate function.
For every sum of intervals there exists an interval, where the function is irreversible. This is a degenerate function.
Thus, a interval definition of non-degenerate function is a sum of subintervals (a theorem on the representation of interval sum subinterval). On each of which a function is strictly monotone. By Lebesgue's theorem on a function subinterval almost everywhere has a finite derivative. Thus we have the fundamental equation
(1) f(x)g(y)=1,
where f and g are adjoint derivatives of the inverse functions.
Any of these derivatives can not be zero almost everywhere, otherwise its adjoint derivative almost everywhere not finite, which contradicts the theorem of Lebesgue.
Let the function be non-degenerated, then on subinterval we have
(2) y(x)=g’(1/f(x)) and x(y)=f’(1/g(y)).
Taking into account a continuity of primitives we can extend the equations to include the discontinuous set. After differentiation on the subintervals we obtain the equations of invariability
(3) [f’(1/g(y))]/=g(y) and [g’(1/f(x))]/=f(x).
2.If the function is not strictly monotone on subinterval, then it can not be inverted, ie it is degenerate. For functions not strictly monotone on any subinterval, for the points where
exist at least one pair of points where
Pass to the limit for each type of points for included subinterval. Then, if the derivative exists and finite, then it must be zero.
3. Let the derivative be non-degenerated. Then on subinterval F(x)=g’(1/f(x)) and f have a strict sign and f is strictly monotonous as it is invertible together with g. Let g’(1/a), where the subinterval of f values of coincides with that of a, be represented by Taylor series in terms of powers of 1/a. We substitute a=f(x) and a0=f(x0) in the expansion
Let g’(1/a) be representable by Taylor series in a power of a, then
We find the coefficients. From (1) g’/(1/f)=g’/(g)=1/g/. The value g/ is found from g=1/f by differentiation with respect to y g/= - f // f 3<. From here
g’/(1/f)=-f 3/ f /.
For finding further coefficients (4) it is necessary to differentiate the last equation with respect to x and to divide by -f // f 2, and to execute the same operations with the result. We multiply the two sides of the last equation by -1/f 2, then
For finding further coefficients (5) it is necessary to differentiate the both sides of the last equation with respect to x and divide by f /, and execute the same operations with result.
Using the invariability equation it is possible to expand on a subinterval the increment of derivative f and 1/f in a power series of increment of the primitive.
4. Let us consider the non-degenerated derivative f(x). For finding on subinterval the adjoint g(y) we take into account the equation of invariability. We suppose a differentiation to be possible
We consider (6) as an equation operational with respect to g(y) with the iterations
,
where the value f /(x)/f(x) is invariable in the iterations and g0(y) is the initial point of the iterations. And x,y are arbitrarily admissible.
Let the conditions for which a iterations give a unique invariant point G(x,y) be fulfilled.
If x=x(y) in (6) is a primitive for desired g(y), then (6) is equivalent to (3). Then the unique g(y)=G(x(y),y).
We substitute G(x(y),y) in the fundamental equation
f(x)G(x,y)=1.
Then this equation defines implicitly the primitive on the subinterval. And it is possible to find a primitive without integration-summation.
If the derivative is degenerated on a subinterval then this method of finding a primitive may be not applicable. However its primitive together with the primitive of a non-degenerated derivative is an absolutely continuous function. Therefore the degenerated derivative can be Lebesgue- integrable .
5. We call fundamental equation (1) an equation of zero order. For determination the adjoint derivatives we suppose to be positive. Then
ln f(x) + ln g(y) = 0.
The equation fundamentals of further orders can be derived by differentiation of previous order fundamentals and by their reduction to the form with equiseparated f and g. We differentiate the last equation with respect to x ( the operations are analogous with respect to y)
f // f + g / / g 2 = 0.
We multiply it by - 1/ 2sqr(f) = - sqr(g) / 2. We receive the uniform representation of the fundamental equation of the first order
.
We differentiate it with respect to x or to y. After denoting operator 1/ sqr(...) = u(...) we receive fundamental equation of the second order
(8) uxxu+ uyy u = 0.
The fundamental equations of further orders can be derived by analogy. The notion of degeneration and non-degeneration is defined for a fundamental equation of any order. If it is non-degenerated then it is possible to define x or y and to pass to the corresponding equation of invariability of type (3). Let us understand the degeneration in a simplified way as an equality of a addend to a non-zero constant. The study of degenerated fundamental equations results in some functions. For example the study of a degenerated fundamental zero-order equation gives a linear function.
6.Let x (t) nondegenerate function with the corresponding derivatives. Then
Then we have the identity
If exist a corresponding function F, we have Newton's second law
In this
.
Likewise,
(11)
All the principles of mechanics are between Newton's second law of second and third order?
Galileo's law. If Fxx = Fyy = Fzz = 0, the uniformly accelerated motion. Wherein v ^ 2/2 = a + bx + cy + dz.
Let F = F (r), we differentiate both sides of (11) for x, y, z
7.General view of the gravitational force F(r)
The right side of this equation is a function of r. The solution of this linear inhomogeneous differential equation gives the general form of the gravitational force F '= const / r ^ 2 + something.
Addition. Likewise calculate the gravitational force not only the three-dimensional space: c / r, c / r ^ 2, c / r ^ 3, ... Is known that the c / r ^ 2 in three dimensions generates closed paths and others is not stable. Maybe that's why our physical world is three-dimensional?
8. F=F(r), we calculate the operator
The right side of this equation is a function of r. The solution of this linear inhomogeneous differential equation gives the general form of the gravitational force of the highest order F '= c1r + c2 / r ^ 6. Gravity center of the galaxy?
9. F'(r)=p/r^2, v^2/2=F+const
10. F'(r)=p/r^2, v^2/2=F+a+bx+cy+dz
The value (b, c, d) characterize an instantaneous change of power.
11. F'(r)=p/r^2, v^2/2=F+a+bx+cy+dz+exy+hxz+kyz+lxyz
This system of three equations for the three unknowns describes the particle motion in the general form.
12. New variables
13. Areas law with v^2/2 = F(r) + const.
14. Galilei's laws. The law on uniform rectilinear motion is the consequence of the second Law of Newton: Fx=Fy=Fz=0. The following law of Galilei is the consequence of the second Law of Newton of the third order: Fxx=Fyy=Fzz=0. Newton's laws of the following orders don't give new Laws of
Galilei.
Publication
1. http://journale.auris-verlag.de/index.php/EESJ/article/view/336/334
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