Note 1
Energy and Distance
1
Curve in 3-dimensional Euclidian space l : [0, 1] → R3
Longitude of l L ( l ) = 
dt
dt2
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
dt = dtgij is Riemann metric.
4
Longitude is defined by the next.
L ( x, xˑ ) = 
dt
dt5
Energy is defined by the next.
E ( x, xˑ ) =
∑I,j gi,j (x(t))xˑi(t)xˑj(t)dt
∑I,j gi,j (x(t))xˑi(t)xˑj(t)dt6
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]
Tokyo August 31, 2008
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