The second variation of functional length is zero on geodesic path

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The second variation of functional length is zero on geodesic path From: STALKER2002 Subject: General Date/Time 2008-07-10 01:56:38 Remote IP: 79.139.140.38 Message This is the definition of the geodesic path. Let's consider a simple circle on the plane XY.Points A,B,C on it. Arcs AB, BC and AB@BC=AC going through B from A to C are geodesic in the sense that the second variation of functional of the length of the arc is equal to zero on geodesic path. Even if the way AC through B is not the shortest way in comparison with the second arc AC, going not through B! Geometrical Interpretation of Quaternions&MULTIPLICATION: (366) STALKER2002 (346) - - 2008-07-01 4:04 pm Re: Geometrical Interpretation of Complex Numbers&MULTIPLICATION: (278) STALKER2002 (346) - - 2008-07-07 00:06 am Multidimensional Generalization of Complex Numbers&Quaternions: (259) STALKER2002 (346) - - 2008-07-07 01:08 am Does anybody understand that this is a new DISCOVERY?: (314) STALKER2002 (346) - - 2008-07-07 2:59 pm NEW?: (274) Steven (694) - (Training Log) - 2008-07-08 4:20 pm Re: NEW? after quaternions and octanions there was NO other generalization of associative algebra: (262) STALKER2002 (346) - - 2008-07-08 4:39 pm Frobenius theorem (real division algebras): (281) STALKER2002 (346) - - 2008-07-08 5:13 pm Frobenius theorem (real division algebras): (256) Steven (694) - (Training Log) - 2008-07-08 5:18 pm Commutativity is not needed for quaternions and their generalization: (262) STALKER2002 (346) - - 2008-07-08 5:33 pm I never said that: (256) Steven (694) - (Training Log) - 2008-07-08 9:17 pm Not true: (265) Steven (694) - (Training Log) - 2008-07-08 5:14 pm The path AC is not the same as the pair of two arcs AB and BC: (243) STALKER2002 (346) - - 2008-07-08 5:22 pm Then how is your multiplication defined?: (250) Steven (694) - (Training Log) - 2008-07-08 9:31 pm Re: Then how is your multiplication defined? Look how!: (256) STALKER2002 (346) - - 2008-07-09 01:08 am Re: Then how is your multiplication defined? Look how!: (251) STALKER2002 (346) - - 2008-07-09 01:09 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (256) STALKER2002 (346) - - 2008-07-09 01:28 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (261) Steven (694) - (Training Log) - 2008-07-09 3:23 pm Not at All! The group of arcs deffers slightly from SO(4): (255) STALKER2002 (346) - - 2008-07-09 4:01 pm Not at All! The group of arcs differs slightly from SO(4): (266) STALKER2002 (346) - - 2008-07-09 4:05 pm However, some questions are still to be solved..: (263) STALKER2002 (346) - - 2008-07-09 5:00 pm Even in the AB@BC case, you have a well-defined problem (nt): (270) Steven (694) - (Training Log) - 2008-07-09 5:18 pm I know that; I wasn't talking about that (nt): (265) Steven (694) - (Training Log) - 2008-07-09 5:17 pm This is a problem: (271) Steven (694) - (Training Log) - 2008-07-09 3:13 pm The shortest way to Tokyo from Moscow is through China: (271) STALKER2002 (346) - - 2008-07-09 3:49 pm No, you can't force the continuity: (271) Steven (694) - (Training Log) - 2008-07-09 5:32 pm The second variation of functional length is zero on geodesic path: (276) STALKER2002 (346) - - 2008-07-10 01:56 am The FIRST, not the second variation of functional length is zero on geodesic path: (263) STALKER2002 (346) - - 2008-07-10 06:36 am That's my point!: (278) Steven (694) - (Training Log) - 2008-07-10 2:52 pm Of course Imust be more accurate and I care about geodesic preservation: (252) STALKER2002 (346) - - 2008-07-10 3:45 pm For any 3 points A, O, C there exists one and only one plane AOC,containing them,: (271) STALKER2002 (346) - - 2008-07-10 4:12 pm Yes: (256) Steven (694) - (Training Log) - 2008-07-10 11:15 pm Correct Idea! Because in 3D case I have defined multiplication of arcs via parallel translation of o: (267) STALKER2002 (346) - - 2008-07-11 00:17 am Antisymmetric 4-DYADS ARC[i,j]=N1(i)N2(j)-N2(i)N1(j). A New idea: (250) STALKER2002 (346) - - 2008-07-15 4:00 pm For simple 4-DYADS ARC1[i,j]=N1(i)N2(j) and ARC2[j,k]=N2(j)N3(k) ARC1@ARC2=N1(i)N3(k) : (235) STALKER2002 (346) - - 2008-07-15 4:51 pm This is why vector and tensor math is superior to and replaced quaternion interpretations (nt): (252) Steven (694) - (Training Log) - 2008-07-16 01:14 am By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (266) STALKER2002 (346) - - 2008-07-10 5:07 pm By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (263) Steven (694) - (Training Log) - 2008-07-10 10:49 pm [Top of List] [Previous Thread] [Next Thread] This Forum is for posting on topics of interest to the Taoist (Daoist) community. Tao is exceptionally broad. 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