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PHY3603: Classical MechanicsChapter 1: Matrices and Vectors

Contents1.1: Concept of Scalar1.2: Coordinate Transformations1.3: Properties of Rotation Matrices1.4: Matrix Operations1.5: Geometrical Significance of Transformation Matrices1.6: Scalars and Vectors Operations1.7: Scalar product of Two Vectors1.8: Unit Vectors1.9: Vector Product of Two VectorsSummary

Learning Outcomes

To describe matrices and matrix notation

To apply matrices in coordinate transformation and vectors

To differentiate scalars and vectors operations

1.1: Concepts of Scalar

Consider the Figure 1. We can label the coordinate as M(x,y) where M referring to mass. Example: M(x=2, y=3)=4.

What happen to the coordinates in (b) if M=4?

In general we can write: M(x,y)=M(x’,y’) - (1.1)

Quantities that are invariant under coordinate transformation (CT) - those that obey Eq. 1.1, they are called scalars. Scalar- phy. quantity that has magnitude. Example: mass, temperature, density, volume, energy and etc.

How about those that cannot be describe in this way? For example direction of motion and forces. These require vectors.

Vector has magnitude and direction. It requires a set of no. for its complete specification. Example: velocity , acceleration and displacement.

Vector usually is written as A or just bold face A. Italics A is for scalars.

1.2: Coordinate Transformations

Consider a point Px1, x2, x3 for Cartesian (rectangular) coordinate system. If we rotate P anticlockwise through angle q, then the coordinates become Px1 ', x2 ', x3 ' .

x1 ' is the sum of line Oa + (ab +bc); that is,

x1 ' = x1 cos q + x2 sin q

= x1 cos q + x2 cosp 2 - q (1.2a)

x2 ' is the sum of line Od - de ; that is,

x2 ' = -x1 sin q + x2 cos q

= x1 cos p 2 + q + x2 cos q(1.2b)

Introduce notation: Angle btwn x1 ' axis and x1 axis given as (x1 ' , x1). Writing in a general form:

(xi ' , xj). We then defined a set of numbers lij by lij ª cos xi ' , xj -(1.3).

Simplify Fig. 1.2, we have

2 PHY3603 Lecture1.nb

1.4

Eq. of transformation (Eq. 1.2) become,

1.5 a, b

Thus in general for 3D:

1.6

In summation notation:

1.7

The inverse transformation is

1.8

The quantity lij is called the direction cosine of xi ' axis relative to the xj

axis and is represented in a square matrix given as follows :

PHY3603 Lecture1.nb 3

1.9

Once lij is found, Eq. 1.7 + 1.8, will give the general rule for specifiying the coor. of a point in the

system. It is then called a transformation matrix or a rotation matrix.

1.3: Properties of Rotation Matrices


From the Fig., we choose an origin. The quantities of interest are the cosines of the angles, cos a,cos b and cos g - direction cosines of the lines.

Therefore, the 1st identity we get

1.10

2nd identity: If we have two lines like Fig. b), the cosine of angle q is given by

1.11

Suppose now we perform a rotation about the origin. The new position is given as x1’ , x2’ and x3’ - can be specified in the form lij. Since angle btw x1 ' and x2’ -axis is q=p/2, Eq. 1.11 becomes

1.12 a

This eq. gives 3 out of 6 relation among the lij. When i=1, k=2; i=2,k=3; i=3,k=1.

From Eq 1.10, we can write:

1.12 b

These are the 3 remaining relation among lij.

Combining Eq. 1.12a+1.12b, we have

1.13

1.14

where dik is the Kronecker delta symbol. Eq. 1.13 is true only if the coordinate axes are mutually perpendicular to each other. Such sys. are said to be orthogonal and Eq. 1.13 is the orthogonality condition.


If we consider the other way round; xi - axes as lines in xi ’ coordinate sys.. The the relation would be

1.15

In this case, we consider the point P to be fixed and allowed the coordinate axes to be rotated. This interpretation is not unique. We can maintain the axes and allowed the part to rotate. In either event, the transformation matrix is the same.

1.4: Matrix Operations

What is a square matrix, column matrix and row matrix?

For multiplying two matrices, we have the following equivalent expressions (refer Eq. 1.7 + 1.8)

1.18

Multiplication of matrix A and B is defined only if the no. of column of A is equal to the row of B (**Bold character denotes the totality of the matrix**)


1.19

Matrix multiplication is not commutative ie. AB∫BA.

What is a transposed matrix?

For Eq. 1.8, it may be written as


What is identity matrix?

Example: Consider a 2D orthogonal rotation matrix l

Using the orthogonality condition (Eq. 1.13) we get:

1.26

Inverse matrix - defined as matrix when multiplied by the original matrix produces the identity matrix - denoted as l-1.

1.27

Comparing Eq. 1.26 + 1.27, we get:

1.28

Summary of rules of matrix algebra:

matrix multiplication is not commutative in general. Special case like identity matrix and inverse - always commute.

Matrix multiplication is associative

Matrix addition is performed by adding corresponding elements of the two matrices.


1.5: Geometrical Significance of Transformation Matrices

From Fig. 1.6, after a rotation we have x1’ = x2, x2’ = -x1 and x3’ = x3. The only non vanishing cosines are:

From Fig. 1.7: x1’ = x1, x2’ = x3 and x3’ = -x2. The transformation matrix is:


Transformation matrix for the combined transformation for rotation about x3-axis followed by rotation about the new x1’ - axis:

(1.32)

(1.33)

(1.34)

So what is the final orientation?

The order of multiplication is important because they are not commutative. Please show.


Consider the coordinate rotation in Fig. 1.10. the elements of transformation matrix in 2 D are given by the following cosines:

If rotation is 3D with x3’ = x3:


A reflection through the origin of all axes -- inversion. x1’ =-x1, x2’ = -x2 and x3’ = -x3.

Now, consider back l3. We know it is formed from two orthogonal transformation. We want to proof that succesive application of orthogonal transformation -> orthogonal transformation.

From this eq. we get a transformation matrix ml. With this xiØ xi”

This combined transformation will be shown orthogonal if mlt = ml-1. The transpose of a matrix is the product of the transposed matrices taken in reverse order. ABt = Bt At.

1.38


t 1 1.39 Finding determinants of all the rotation matrices:

We find that all determinants from rotations starting from original set of axes have +1. Determinant from inversion is -1 - imply that inversion can not be generated by any series of rotation.

In orthogonal trans.: determinant = +1 -- called proper rotations while determinant = -1 --- called improper rotations.

All orthogonal matrices must have determinant : ±1

1.6: Scalars and Vectors Operations

Consider:


If under such transformation, a quantity is unaffected then that quantity is called a scalar.

If a set of quantities A1, A2, A3) is transformed by l with the result

then quantities Ai transforms as the coordinates of a point and A=A1, A2, A3) is termed a vector.

Consider A and B are vectors; f , y and x are scalars; for addition operation:

1.45 1.46

1.47 1.48

Multiply by scalar x

1.49 1.50

Can you proof them?

1.7: Scalar product of Two Vectors

Multiplication of two vectors A and B forms the scalar product and is defined as

1.52

Magnitude of a vector, A is given as

1.53

From Eq. 1.52, divide both side with AB, we have:


1.54

1.55

1.56

Show that A · B is indeed a scalar.

The distance from origin to a point x1, x2, x3) is defined by vector A called the position vector.

Distance between A and B

Since magnitude is the square root of a scalar product, it is invariant to a coordinate transformation. This show that orthogonal transformations are distance preserving transformation. Angle is also preserve btwn 2 vectors.


1.8: Unit Vectors

Unit vectors are vectors having a length equal to the unit of length used along the particular

coordinates axes. E.g. Unit vector along the radial direction described by the vector R is eR=R / R What are the commom sets of unit vector? How do we express a vector?

We use ( e1, e2 , e3) because easy for summation. Components of the vector A by projection onto the axes

1.62

1.63

1.64

1.9: Vector product of Two Vectors

Vector product of A and B is denoted as

1.65

Components of C are defined by

1.66

where eijk is the permutation sysmbol or Levi-Civita density and has the following properties:


1.67

Example:

Now we evaluate components of C:

1.68

The magnitude of C=A×B is

1.71

Can you proof it?

Properties of the vector product:


The orthogonality of the unit vectors ei requires the vector product to be

1.79

For vector product C, we have

1.8

Finally, some identities of vector:

Summary


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Documents

x1 cos q x2 sin q

x1 sin q x2 cos q

x1 cos q x2 cosp

x1 cos p

x2 axis

lij cos xi

cosine of angle q

angle btwn x1 axis