MATHEMATICAL CLASSIFICATION OF PHYSICAL QUANTITIES AND PHYSICS RELATIONS Thesis submitted to obtain the degree of Doctor of Philosophy Faculty of Sciences, Department of Mathematical physics and astronomy, University of Ghent, Belgium Philippe CHEVALIER Date: 13-11-2008 Version: draft 1 1 Table of contents 1 Table of contents ............................................................................................................................. 2 2 List of figures ................................................................................................................................... 6 3 Preface ............................................................................................................................................. 6 4 Introduction ..................................................................................................................................... 6 5 Definitions ....................................................................................................................................... 8 6 Base units in physics ........................................................................................................................ 8 6.1 Impact of the number of base units on physics .................................................................... 9 6.1.1 Description with one unit resulting in the graph 풗ퟎ풗ퟏ. ................................................. 9 6.1.2 Description with two units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ ........................................... 9 6.1.3 Description with three units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ . .................................. 9 6.1.4 Description with four units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ풗ퟒ . ............................... 9 6.1.5 Description with five units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ풗ퟒ풗ퟓ.............................. 9 6.1.6 Description with six units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ풗ퟒ풗ퟓ풗ퟔ ........................ 10 6.1.7 Description with seven units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ풗ퟒ풗ퟓ풗ퟔ풗ퟕ .............. 10 6.1.8 Description with n units resulting in the graph 풗ퟎ풗ퟏ풗ퟐ풗ퟑ풗ퟒ풗ퟓ풗ퟔ … 풗퐧 ................. 10 7 Properties of 풁ퟕ for mathematical modelling of physics equations ............................................ 11 7.1 General mathematical properties and definitions applicable for 풁ퟕ ................................... 11 7.2 풁ퟕ integer lattice as representation of physical quantities .................................................. 11 7.3 Vector operations in 풁ퟕ ........................................................................................................ 12 7.3.1 Addition in 풁ퟕ ............................................................................................................... 12 7.3.2 Dot product in 풁ퟕ .......................................................................................................... 13 7.4 Map SUM() ............................................................................................................................ 15 7.5 Subgroups of 풁ퟕ for classification of physical quantities ..................................................... 16 7.6 Equivalence relation R as a classifier ..................................................................................... 17 7.7 Classification using Principal Component Analysis (PCA) ...................................................... 20 7.8 Classification using RPM mapping ......................................................................................... 33 7.9 Classification using Sammon map ......................................................................................... 37 7.10 Classification using Curvilinear Component Analysis ............................................................ 40 7.11 Graph of dimensionless products in 풁ퟕ ................................................................................ 43 7.12 Graph spectrum and graph eigenvalues ............................................................................... 52 7.13 Minimal spanning tree of a graph representing the physical quantity “force” in 풁ퟕ .......... 53 7.14 Syntax and semantics of “energy equations” ....................................................................... 56 2 7.14.1 Energy syntax N°2 .......................................................................................................... 57 7.14.2 Energy syntax N°3 .......................................................................................................... 57 7.14.3 Energy syntax N°5 .......................................................................................................... 57 7.14.4 Energy syntax N°7 .......................................................................................................... 57 7.14.5 Energy syntax N°8 .......................................................................................................... 57 7.14.6 Energy syntax N°9 .......................................................................................................... 58 7.14.7 Energy syntax N°10 ........................................................................................................ 58 7.14.8 Energy syntax N°13 ........................................................................................................ 58 7.14.9 Energy syntax N°14 ........................................................................................................ 58 7.14.10 Energy syntax N°15 .................................................................................................... 58 7.14.11 Energy syntax N°16 .................................................................................................... 58 7.14.12 Energy syntax N°17 .................................................................................................... 58 7.14.13 Energy syntax N°18 .................................................................................................... 58 7.15 n-polytopes in 풁ퟕ .................................................................................................................. 58 7.16 Closest neighbour lattice points in 풁ퟕ for the quantity “energy “ ....................................... 61 7.17 Grassmann algebra of 풁ퟕ over 풁 for classification of physical quantities ............................ 62 8 Dimensional exploration of physics equations using dimensionless products ............................. 64 8.1 Dimensional analysis ............................................................................................................. 64 8.2 Buckingham 흅 theorem ........................................................................................................ 64 8.3 Dimensional matrix for SI units ............................................................................................. 64 8.4 Dimensional exploration of “a particle in a box “ ................................................................ 64 8.4.1 흅ퟏ .................................................................................................................................. 66 8.4.2 흅ퟐ .................................................................................................................................. 66 8.4.3 흅ퟑ .................................................................................................................................. 66 8.4.4 흅ퟒ .................................................................................................................................. 66 8.4.5 흅ퟓ .................................................................................................................................. 66 8.4.6 흅ퟔ .................................................................................................................................. 67 8.4.7 Approximation of 흎흅ퟏ, 흅ퟐ, 흅ퟑ, 흅ퟒ, 흅ퟓ, 흅ퟔ = ퟎ ......................................................... 67 8.4.8 Relations involving the dilution coefficient ................................................................... 70 8.4.9 Dimensional exploration of total angular momentum J ............................................... 71 8.4.10 Interaction of external fields on free particles .............................................................. 72 8.5 Dimensional exploration of “vacuum states”........................................................................ 73 8.6 Dimensional exploration of “a box containing charged particles” ....................................... 73 8.7 Dimensional exploration of “Casimir force” ......................................................................... 74 3 8.8 Dimensional exploration of “a rotating body” ..................................................................... 74 8.9 Parametric “dimensional exploration” of physics equations ................................................ 74 8.9.1 Solution for the quantity “Length” as function of the parametric set 풉, 풄ퟎ, 풎ퟎ, 푮 ..... 75 8.9.2 Solution for the quantity “Energy” as function of the parametric set 풉, 풄ퟎ, 풎ퟎ, 푮 ..... 76 8.9.3 Solution for the quantity “Force” as function of the parametric set 풉, 풄ퟎ, 풎ퟎ, 푮 ....... 77 8.9.4 Solution for the quantity “Length” as function of the parametric set 흁ퟎ, 휺ퟎ, 퐞, 퐆....... 79 8.9.5 Solution for the quantity “Frequency” as function of the parametric set 흁ퟎ, 휺ퟎ, 퐞, 퐆 . 79 8.9.6 Solution for the quantity “Energy” as function of the parametric set 흁ퟎ, 휺ퟎ, 퐞, 퐆....... 79 8.9.7 Solution for the quantity “Magnetic moment” as function of the parametric set 흁ퟎ, 휺ퟎ,