Energy Distance Theory Note 1 Energy and Distance

Energy Distance Theory

Note 1

Energy and Distance

TANAKA Akio

1

Curve in 3-dimensional Euclidean space   l : [0, 1] → R ³

Longitude of l   L ( l ) = dt

2

Surface   S

Curve combines A and B in S   l

Coordinate of S     φ : U → S

Coordinate of U   x ₁ , x ₂

φ = ( φ ₁ , φ ₂ , φ ₃ )

A = φ ( x ₀ )

B = φ ( x ₁ )

3

Curve in S   l : [0, 1] → R ³

Curve on U   x ( t )

Ω ( x ₀ , x ₁ )= { l : [0,1] → R ³ | l (0) = x ₀ , l (1 ) = x ₁ }

x ( t ) ∈ Ω ( x ₀ , x ₁ )

l ( t )= φ ( x ( t ) )

x ( 0 )= x ₀

x ( 1 ) = x ₁

L ( l ) = dt   =   dt

g _ij is Riemann metric.

4

Longitude is defined by the next.

L ( x, xˑ )   =   dt

5

Energy is defined by the next.

E ( x, xˑ )   =   ∑ _I,j g _i,j ( x ( t )) xˑ ⁱ ( t ) xˑ ^j ( t ) dt

6

2 E ( x, xˑ ) ≥ ( L ( x, xˑ ) ) ²

7

Theorem

For x ∈ Ω ( x ₀ , x ₁ ), the next two are equivalent.

(i) E takes minimum value at x .

(ii) L takes minimum value at x .

8

What longitude is the minimum in curve is equivalent what energy is the minimum in curve.

9

Longitude L is corresponded with distance in Distance Theory.

[References]

Distance Theory / Tokyo May 4, 2004

Property of Quantum / Tokyo May 21, 2004  

Mirror Theory / Tokyo June 5, 2004

Mirror Language / Tokyo June 10, 2004

Guarantee of Language / Tokyo June 12, 2004

Reversion Theory / Tokyo September 27, 2004

Tokyo August 31, 2008

Sekinan Research Field of Language