Energy Distance Theory Note 1 Energy and Distance

Energy Distance Theory

Note 1

Energy and Distance

TANAKA Akio

1

Curve in 3-dimensional Euclidian space     l : [0, 1] → R³

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     x₁, x₂

φ = (φ₁, φ₂, φ₃ )

A =φ ( x₀ )

B =φ ( x₁ )

3

Curve in S     l : [0, 1] → R³

Curve on U    x ( t )

Ω(x₀, x₁) = { l : [0, 1] → R³ | l (0 ) = x₀, l (1 ) = x₁ }

x(t)∈Ω(x₀, x₁)

l ( t ) =φ ( x ( t ) )

x ( 0 ) = x₀

x ( 1 ) = x₁

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))xˑⁱ(t)xˑ^j(t)dt

6

2 E ( x, xˑ ) ≥ (L ( x, xˑ ) )²

7

Theorem

For x∈Ω(x₀, x₁), 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