Jan 28, 1999 - understanding of the important subject of linear algebra. This ... GA is an index free language. Denote ... Associative so no need for brackets.
PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA LECTURE 5
SUMMARY Today will see how GA simplifies and improves our understanding of the important subject of linear algebra. This has many applications, and is crucial for the gauge theory of gravity. As a major application we will look in detail at Hamiltonian mechanics and will uncover a geometric framework which forms the natural setting for Hamilton’s equations. Linear functions of vectors and multivectors. The determinant and its geometric meaning. Non-orthonormal frames and a selection of useful algebraic identities. The adjoints and the inverse. Hamilton’s equations in a geometric setting. Conservation equations. Canonical Transformations. 1
L INEAR F UNCTIONS GA is an index free language. Denote linear functions . Defining property
mapping vectors to vectors as
Combine two linear functions
and , get a third (cf matrix
multiplication). Write
Associative so no need for brackets. Extend action of
to entire GA by
Right-hand side also a blade with same grade as the original argument. Extended linear functions preserve grade
They are also multilinear
for any multivectors and . This is the way to understand linear algebra! 2
have seen
Example — Rotations. With
extended action to multivectors has same law, so
Key result. Take a product function
. See that
Extension of product = product of extended functions. Still write
and right-hand side is unambiguous.
T HE D ETERMINANT
¿ ¿
¾
¾ ½
½
3
Unit cube transformed to parallelepiped, sides and
¿ . Volume is
½ , ¾
½ ¾ ¿ Define determinant as the volume scale factor. Linear functions grade preserving for all multivectors. But highest grade element is unique up to scale. Define
Now prove a key result for determinants. Take
.
Get
Just used multilinearity and extension results. Have proved
The simplest proof anywhere!
N ON -O RTHONORMAL F RAMES Very useful. Unavoidable for special relativity. Take set of linearly independent vectors . Not necessarily orthogonal. Any vector
decomposes uniquely
How do we find the components? Need a second set 4
related to first by
Æ
The reciprocal frame. With these get
Æ
Note position of indices. To construct reciprocal frame, see ½ orthogonal to ¾ . ½ perpendicular to hyperplane ¾ ¿ . Find by dualisation — multiplication by . Have
½
¾ ¿
found by dotting with ½ ½ ½
½ ¾
Define
½ ¾
so
½ ½ . Arrive at useful formula
·½ ½ ½
term missing from product. Purely geometric reasoning led quickly to an algebraic formula. Can be directly applied. Will use arbitrary frames and reciprocals where frames needed. 5
S OME U SEFUL R ESULTS Basic identity
Build up useful results. First
Extends inductively to
for grade- multivector. Next use
symmetric on only get scalar contribution
where is dimension of vector space. Follows that
Combining above gives
6
Recovering a Rotor Two arbitrary non-orthonormal frames and related by a rotation,
How do we find ? Work in 3-d, so
¾
Find that
Now form
is scalar multiple of , so
where . Recovers the directly from frame vectors.
7
T HE A DJOINT The reverse map
Decomposing
in a frame
Same as transpose of a matrix/tensor. Construct extension
½ ¾
Extension of adjoint adjoint of extension. Write
Extend to mixed grades, e.g.
Similar argument, get remarkably useful formulae
8
T HE I NVERSE Preceding formulae quickly yield the inverse function! Set
in second formula,
Write as
½
½
The green terms undo effect . Must represent the inverse function. Therefore have
½ ½
½ ½ ½
½
No simpler proof anywhere else! And very useful, can be coded in symbolic algebra packages (Maple).
Example - Rotations
Rotation
. Adjoint
found from
Extends to
. Inverse given by
½
as
½ ½
. Inverse = adjoint — an orthonormal
transformation. 9
H AMILTONIAN M ECHANICS Possess necessary ideas to geometrise Hamiltonian dynamics. Start with Lagrangian , are arbitrary coordinates. Lagrange’s equations
Equivalent to Hamilton’s equations:
Hamiltonian
given by
with expressed in terms of the
2nd order equations first order equations. Natural setting is -d ‘doubled’ space . Define point in phase space by the vector
10
Hamiltonian is function of this vector,
, so that
where is gradient operator
Hamilton’s equations specify a phase space trajectory
Recover the bivector ! Hamilton’s equations now become
Number of advantages 1. Easy to prove e.g. conservation theorems and Liouville’s theorem. 2. Canonical transformations understood geometrically. 3. Poisson bracket naturally incorporated (later). 4. Extends to more complicated systems. Phase space manifold.
symplectic bivector. Equation structure
unchanged. 5. Natural setting for instability and chaos. 11
D ERIVATIVES AND F LOWS Introduce -d fixed frame . Write
Have
Follows from chain rule that for functions of only
Take scalar function
on phase space. (Independent of .) Evolution along phase space trajectory determined by
. If invariant along constant Immediately get
direction
in phase space,
where
, get
¼
¼ . From above see that
so
¼
is conserved quantity. 12
C ANONICAL T RANSFORMATIONS Equation
is geometric. Can decompose in any
coordinate frame. Gives passive transformations. Useful, but not the whole story. Suppose have different set of coordinates ! . Form a different vector
and canonical momenta
¼
!
The ! are functions of original . View new vector ¼ as function of the old, . Write
¼
This is an active transformation — a displacement. Points actually moved around in phase space. No restriction on form of
other than invertible.
Assume
independent of . Form ¼
Define differential of
A linear function of . Also position dependent. Sometimes write
but suppress the 13
where possible.
Now have
¼
Next relate gradients with respect to and ¼ . Have
¼
¼
¼
¼ ¼
¼
¼
Find that
So
¼ ¼
¼
¼
¼
¼
¼ . Very neat again! Now get
¼
½
But transformed Hamiltonian is ¼
½
¼ , so
¼ ¼ ¼
Equations of motion after transformation are now
¼
¼ ¼
Will still be Hamiltonian in form if
Defines a canonical transformation.
is a symplectic
transformation. Examples include unitary transformations. 14
Mar 5, 1999 - To find the field equations we must establish covariant forms of the field strength ... The second field equation. .... had quadratic д µ ½. ¾.
Feb 16, 1999 - Same transformation law as for vectors. Very efficient. EXAMPLES. 1. Stationary charges in ¼ frame set up field. Ь ½ · Р¾. Second observer ...
Jan 21, 1999 - In the final section we will start to look at the GA ... (ع¼ is the velocity of the centre of mass.) We need the ... Angular momentum density is ´ ¡ µ.
Feb 9, 1999 - Projective splits for observers. ¯ Handling Lorentz transformations with rotors. ¯ Photons and redshifts. ¯ The structure of the Lorentz group. 1 ...
Feb 4, 1999 - Dot Цwith , get directional derivative in direction. ÐЦ ´Ьµ РС. ¯ ¼. ´Ь · ¯µ ´Ьµ. ¯. ´Ьµ a multivector-valued function of position. Scalar field ´Ьµ ...
Jan 26, 1999 - Complex structures and doubling bivectors. 6. Unitary groups .... Familiar? it is the polar decomposition of a complex number back again.
levels. Expanding to next order get. ÐÐ. С. ´ «µ¾. ¾Т¾. С. ´ «µ. ¾Т. Т. Р· ½. ¿. Binding energy increased slightly, and get dependence on Ð . Lifts degeneracy in ...
Jan 19, 1999 - Now add a third vector ¿. , orthogonal to ½ and ¾ . Generate. 3 independent bivectors. ½ ¾. ¾ ¿. ¿ ½. The expected number of independent ...
formulated as a gauge theory, but thought to be difficult and unnatural. Geometric algebra .... Lamb shift. Dirac equation accounts for fine structure. The hyperfine.
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Particle in a magnetic field. ... Decompose into Pauli basis, then and. Ð. Get multivector equivalent. ½. ¾. ÒР· ¿µ. 2 .... We first write the equation in the form. Ш. ½.
Hamilton introduces his quaternions, which generalize complex numbers. But confusion persists over the status of vectors in his algebra â do ´. µ constitute the.
This property is exploited in the theory of rotations. Having established a complete geometric interpretation for the product of two vectors, we turn to consider the ...
Apr 29, 2009 - Complex numbers. Let i be the pseudoscalar of a plane in Rn. A complex number in the plane is a scalar + bivector: a+ib. Since i2 = â1, GA ...
Page 1. Geometric Algebra Primer. Jaap Suter. March 12, 2003. Page 2 .... practice. . . ...After perusing some of these, the computer scientist may well ... can express any geometric relation or concept. ...... A = (4, 8, 5, 6, 2, 4, 9, 3) â Cl3 we
Mar 12, 2003 - Adopted with great enthusiasm in physics, geometric algebra slowly emerges in computational .... A computer scientist first pointed to geometric algebra as a ...... In Cl4 a bivector takes six scalars, and a trivector takes four scalar
Maxplus scalars and matrices in Scilab. 3. Input-Output Max-Plus Linear Systems. 3.1. Transfer Functions. 3.2. Rational Series. 4. Dynamical Maxplus Linear ...
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ations. We present results on i) linear algebra, ii) system theory, iii) duality between .... may be called dynamic programming with independent instantaneous costs (c de- ..... Feller, W.: An Introduction to Probability Theory and its Applications.
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