Obi Theory Birth Again for PONTLYAGIN L. S.

 Obi Theory

 

 Birth

Again for PONTLYAGIN L. S.

 

TANAKA Akio

 

 

1

Obi’s birth of whirl starts at energy’s unbalance existence in universe.

2

 Now define vector field of smooth curved surface M.

Here touch plane of M is expressed by TxM.

Vector field is expressed by the following differential equation’s solution curve.

x(t) is contained in M

d/dt x(t)  is contained in Tx(t)M

3

Now define torus’ surface.

Here x is on surface M.

α limit set of x is expressed by α(x).

Now set a point (θ₁,θ₂) on M.

When differential equation system on M that becomes torus and the solution is expressed by double cycle, α(x) becomes all torus’ surface.

4

Now express obi.

Obi’s solution curb is expressed by polar coordinates (ρ, θ).

ρ = k exp(α/β  θ)    α is contained in e^αt, β is angular velocity, k is consonant

Obi is logarithmic helix.

5

Here suppose torus as obi’s differential-like figure.

For obi’s instantaneous figure, torus is examined by analytic geometry.

6

Obi’s whirl is also examined from torus’ movement  as obi’s instantaneous figure.

7

Obi’s birth is also examined from torus as obi figure’s origin.

 

Tokyo June 16, 2006
Sekinan Research Field of Language
www.sekinan.org