Transcript Coordinates

Coordinates
Basis

A basis is a set of elements
that generate a group or
field.
• Groups have a minimum set
that generates the group.
• Cyclic groups have a single
element basis.

Vector spaces use the
scalars and basis vectors to
generate the space.
Example
 A basis Bi for M2(R) is
1 0 0 1 0 0 0 0
0 0 , 0 0 , 1 0 , 0 1

 
 
 


The vector equation
1B1   2 B2   3 B3   4 B4  0
has only one solution
1   2   3   4  0
so they are linearly
independent.
Cartesian Coordinates

Three coordinates
• x1, x2, x3
• Replace x, y, z
• Usual right-handed system

x3

e1
A vector can be expressed in
coordinates, or from a basis.
• Unit vectors form a basis

r  ( x1 , x2 , x3 )





r  x1e1  x2e2  x3e3  xi ei
Summation convention used
x1
x2
Cartesian Algebra

Vector algebra requires
vector multiplication.
• Wedge product
• Usual 3D cross product
 

a  b  e ijk ai b j ek

The dot product is also
defined for Cartesian
vectors.
 
a  b  ai bi
Kronecker delta:
•
•
dij = 1, i = j
dij = 0, i ≠ j
Permutation epsilon:
•
•
eijk = 0, any i, j, k the same
eijk = 1, if i, j, k an even
permutation of 1, 2, 3
• eijk = -1, if i, j, k an odd
permutation of 1, 2, 3
e ijke klm  d ild jm  d imd jl
Coordinate Transformation

x3
x3
x2
x2
x1
x1
A vector can be described by
many Cartesian coordinate
systems.
• Transform from one system
to another
• Transformation matrix L
x j l ij xi
xi  lij x j
A physical property that transforms
like this is a Cartesian vector.
General Transformation

Transformation and inverse

For a small displacement
x
dxi  i dqm
qm

If

Then the inverse exists
• qm = qm(x1, x2, x3, t)
• xi = xi(q1, q2, q3, t)

Generalized coordinates
need not be distances.

For a small displacement a
non-zero determinant of the
transformation matrix
guarantees an inverse
transformation.
xi
0
qm
dqm 
qm
dxi
xi
Other Coordinates

Polar-cylindrical coordinates
•
•
•


r: q1 = (x1 + x2)1/2
q: q2 = tan-1(x2/x1)
z: q3 = x3
Translation with constant
velocity
• q1 = x1 – vt
• q2 = x2
• q3 = x3
Spherical coordinates
•
•
•
r: q1 = (x1 + x2 + x3)1/2
q: q2 = cot-1(x3/ (x1 + x2)1/2)
f: q3 = tan-1(x2/x1)

Translation with constant
acceleration
• q1 = x1 – gt2
• q2 = x2
• q3 = x3
Constraints

Coordinates may be constrained to a manifold
• Surface of a sphere
• Spiral wire

A function of the coordinates and time: holonomic
• f(x1, x2, x3, t) = 0
• If time appears the constraint is moving.
• If time does not appear the constraint is fixed.

Non-holonomic constraints include terms like velocity
or acceleration.
System of Points

Coordinates with two indices: xri
•
•

r represents the point
i represents the coordinate index (use i through n)
A rigid body has k holonomic constraints.
• fj(xri, t) = 0
• System has f = 3N – k degrees of freedom
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