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
X } 
X } S ∞ ( X ) 
min{( x | z)x 0 , ( y | z ) x 0 }- δ 
, δ -hyperbolic space is abbreviatedly called hyperbolic. 
max{ d ( x, z ) + d ( y, w ), d ( x, w ) + d ( y, z ) } +2 δ 
[0, λ ] 
X is expressed by 
. 
xyz = 
is called geodesic triangle. 
X }.