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Lie Group Lectures

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    a  r   X   i  v  :   1   1   0   4 .   1   1   0   6  v   2   [  m  a   t   h .   D   G   ]   7   A  p  r   2   0   1   1  Lecture Notes in Lie Groups Vladimir G. Ivancevic ∗ Tijana T. Ivancevic † Abstract These lecture notes in Lie Groups are designed for a 1–semester third yearor graduate course in mathematics, physics, engineering, chemistry or biology.This landmark theory of the 20th Century mathematics and physics gives arigorous foundation to modern dynamics, as well as field and gauge theoriesin physics, engineering and biomechanics. We give both physical and medicalexamples of Lie groups. The only necessary background for comprehensivereading of these notes are advanced calculus and linear algebra. Contents 1 Preliminaries: Sets, Maps and Diagrams 3 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Commutative Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Groups 63 Manifolds 8 3.1 Definition of a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Formal Definition of a Smooth Manifold . . . . . . . . . . . . . . . . 123.3 Smooth Maps Between Smooth Manifolds . . . . . . . . . . . . . . . 133.4 Tangent Bundle and Lagrangian Dynamics . . . . . . . . . . . . . . 143.5 Cotangent Bundle and Hamiltonian Dynamics . . . . . . . . . . . . . 17 ∗ Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Edin-burgh SA 5111, Australia (e-mail: Vladimir.Ivancevic@dsto.defence.gov.au) † Tesla Science Evolution Institute & QLIWW IP Pty Ltd., Adelaide, Australia (e-mail:tijana.ivancevic@alumni.adelaide.edu.au) 1  4 Lie Groups 18 4.1 Definition of a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 One-Parameter Subgroup . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 254.6 Actions of Lie Groups on Smooth Manifolds . . . . . . . . . . . . . . 264.7 Basic Tables of Lie Groups and Their Lie Algebras . . . . . . . . . . 284.8 Representations of Lie groups . . . . . . . . . . . . . . . . . . . . . . 314.9 Root Systems and Dynkin Diagrams . . . . . . . . . . . . . . . . . . 324.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.9.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.9.3 Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 344.9.4 Irreducible Root Systems . . . . . . . . . . . . . . . . . . . . 364.10 Simple and Semisimple Lie Groups and Algebras . . . . . . . . . . . 37 5 Some Classical Examples of Lie Groups 38 5.1 Galilei Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 General Linear Group . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Rotational Lie Groups in Human/Humanoid Biomechanics . . . . . 405.3.1 Uniaxial Group of Joint Rotations . . . . . . . . . . . . . . . 415.3.2 Three–Axial Group of Joint Rotations . . . . . . . . . . . . . 435.3.3 The Heavy Top . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Euclidean Groups of Rigid Body Motion . . . . . . . . . . . . . . . . 455.4.1 Special Euclidean Group  SE  (2) in the Plane . . . . . . . . . 465.4.2 Special Euclidean Group  SE  (3) in the 3D Space . . . . . . . 475.5 Basic Mechanical Examples . . . . . . . . . . . . . . . . . . . . . . . 505.5.1  SE  (2) − Hovercraft . . . . . . . . . . . . . . . . . . . . . . . . 505.5.2  SO (3) − Satellite . . . . . . . . . . . . . . . . . . . . . . . . . 515.5.3  SE  (3) − Submarine . . . . . . . . . . . . . . . . . . . . . . . . 525.6 Newton–Euler  SE  (3) − Dynamics . . . . . . . . . . . . . . . . . . . . 525.6.1  SO (3) : Euler Equations of Rigid Rotations . . . . . . . . . . 525.6.2  SE  (3) : Coupled Newton–Euler Equations . . . . . . . . . . . 535.7 Symplectic Group in Hamiltonian Mechanics . . . . . . . . . . . . . 56 6 Medical Applications: Prediction of Injuries 57 6.1 General Theory of Musculo–Skeletal Injury Mechanics . . . . . . . . 576.2 Analytical Mechanics of Traumatic Brain Injury (TBI) . . . . . . . . 636.2.1 The  SE  (3) −  jolt: the cause of TBI . . . . . . . . . . . . . . . 636.2.2  SE  (3) − group of brain’s micro–motions within the CSF . . . 642  6.2.3 Brain’s natural  SE  (3) − dynamics . . . . . . . . . . . . . . . . 656.2.4 Brain’s traumatic dynamics: the  SE  (3) −  jolt . . . . . . . . . 686.2.5 Brain’s dislocations and disclinations caused by the  SE  (3) −  jolt 69 1 Preliminaries: Sets, Maps and Diagrams 1.1 Sets Given a map (or, a function)  f   :  A  →  B , the set  A  is called the  domain   of   f  , anddenoted Dom f  . The set  B  is called the  codomain   of   f  , and denoted Cod f.  Thecodomain is not to be confused with the  range   of   f  ( A ), which is in general only asubset of   B  (see [8, 9]). A map  f   :  X   → Y   is called  injective  , or 1–1, or an  injection  , iff for every  y  in thecodomain  Y   there is  at most   one  x  in the domain  X   with  f  ( x ) =  y . Put another way,given  x  and  x ′ in  X  , if   f  ( x ) =  f  ( x ′ ), then it follows that  x  =  x ′ . A map  f   :  X   → Y  is called  surjective  , or  onto , or a  surjection  , iff for every  y  in the codomain Cod f  there is  at least   one  x  in the  domain   X   with  f  ( x ) =  y . Put another way, the  range  f  ( X  ) is equal to the codomain  Y  . A map is  bijective   iff it is both injective andsurjective. Injective functions are called  monomorphisms  , and surjective functionsare called  epimorphisms   in the  category of sets   (see below). Bijective functions arecalled  isomorphisms  .A  relation   is any subset of a  Cartesian product   (see below). By definition, an equivalence relation   α  on a set  X   is a relation which is  reflexive, symmetrical   and transitive  , i.e., relation that satisfies the following three conditions:1.  Reflexivity  : each element  x ∈ X   is equivalent to itself, i.e. , xαx ;2.  Symmetry  : for any two elements  a,b  ∈ X  ,  aαb  implies  bαa ; and3.  Transitivity  :  aαb  and  bαc  implies  aαc .Similarly, a relation  ≤  defines a  partial order   on a set  S   if it has the followingproperties:1.  Reflexivity  :  a ≤ a  for all  a ∈ S  ;2.  Antisymmetry  :  a  ≤ b  and  b ≤ a  implies  a  =  b ; and3.  Transitivity  :  a ≤ b  and  b ≤ c  implies  a ≤ c .A  partially ordered set   (or  poset  ) is a set taken together with a partial order onit. Formally, a partially ordered set is defined as an ordered pair  P   = ( X, ≤ ), where X   is called the  ground set   of   P   and  ≤  is the partial order of   P  .3  1.2 Maps Let  f   and  g  be maps with domains  A  and  B . Then the maps  f   + g ,  f   − g ,  fg , and f/g  are defined as follows (see [8, 9]):( f   + g )( x ) =  f  ( x ) + g ( x ) domain =  A ∩ B, ( f   − g )( x ) =  f  ( x ) − g ( x ) domain =  A ∩ B, ( fg )( x ) =  f  ( x ) g ( x ) domain =  A ∩ B,  f g  ( x ) =  f  ( x ) g ( x ) domain = { x ∈ A ∩ B  :  g ( x )  = 0 } . Given two maps  f   and  g , the composite map  f   ◦ g , called the  composition   of   f  and  g , is defined by( f   ◦ g )( x ) =  f  ( g ( x )) . The ( f  ◦ g ) − machine is composed of the  g − machine (first) and then the  f  − machine, x →  [[ g ]] →  g ( x ) → [[ f  ]] → f  ( g ( x )) . For example, suppose that  y  =  f  ( u ) =  √  u  and  u  =  g ( x ) =  x 2 + 1. Since  y  is afunction of   u  and  u  is a function of   x , it follows that  y  is ultimately a function of   x .We calculate this by substitution y  =  f  ( u ) =  f   ◦ g  =  f  ( g ( x )) =  f  ( x 2 + 1) =   x 2 + 1 . If   f   and  g  are both differentiable (or smooth, i.e.,  C  ∞ ) maps and  h  =  f  ◦ g  is thecomposite map defined by  h ( x ) =  f  ( g ( x )), then  h  is differentiable and  h ′ is given bythe product: h ′ ( x ) =  f  ′ ( g ( x )) g ′ ( x ) . In Leibniz notation, if   y  =  f  ( u ) and  u  =  g ( x ) are both differentiable maps, then dydx  =  dydududx. The reason for the name  chain rule   becomes clear if we add another link to the chain.Suppose that we have one more differentiable map  x  =  h ( t ). Then, to calculate thederivative of   y  with respect to  t , we use the chain rule twice, dydt  =  dydududxdxdt. Given a 1–1 continuous (i.e.,  C  0 ) map  F   with a nonzero  Jacobian   ∂  ( x,... ) ∂  ( u,... )   thatmaps a region  S   onto a region  R , we have the following substitution formulas:4
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