**Energy Distance Theory**

**Note 1**

**Energy and Distance**

**TANAKA Akio**

1

Curve in 3-dimensional Euclidian space

**: [0, 1] → R***l*^{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***: [0, 1] → R***l*^{3}
Curve on

*U***(***x***)***t**Ω*(

*x*_{0},

*x*_{1}) = {

**: [0, 1] → R**

*l*^{3 }|

**(0 ) =**

*l*

*x*_{0},

**(1 ) =**

*l*

*x*_{1 }}

**(**

*x***)∈**

*t**Ω*(

*x*_{0},

*x*_{1})

**(**

*l***) =**

*t**φ*(

**(**

*x***) )**

*t***( 0 ) =**

*x*

*x*_{0}

**( 1 ) =**

*x*

*x*_{1}

**L**(

**) =**

*l**dt*=

*dt*

*g*is Riemann metric.

_{ij }
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**

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