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Some Introduction to Applied Mathematics (Applied Mathematics Books)
The Laws of Motion
The three laws that govern motion are usually attributed to Isaac Newton.
1. Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.
2. Rate of change motion is proportional to the impressed force, and is in the direction in which the force acts.
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3. To every action there is always an equal an opposite reaction. The bodies referred to in the First Law should be taken to be particles. Particles are idealizations of objects that have no size. The Laws of Motion can, in most cases, be generalized to systems of interacting particles, although care must be taken in some specific instances. The tendency of a particle to continue in a state of rest or uniform motion is called inertia. Mass is the quantitative measure of inertia of a body and it is given numerical value by comparison to a standard. An inertial reference frame is one in which Newton’s First Law holds strictly. Such a frame is also an idealization rather than a physically realizable frame. Changes in position are measured in terms of distances and times. Distances and times are given quantitative meaning by comparison to standards.
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The concept of motion in the Second Law can be given quantitative measure as the momentum of a particle. If m is the mass and v is the rate of change of position, then the momentum p is given by p = mv. Newton’s Second Law then can be expressed as a vector equation,
F = dp
dt = m
Conceptually, forces are pushes or pulls. The fundamental forces of classical mechanics are either the gravitational force or the electromagnetic force or some manifestations (friction, viscosity, contact, etc.) of these which may obscure their true origins. Forces are given quantitative meaning through Newton’s Second Law which connects force to quantitative measures of mass, length and time. Newton’s three Laws of Motion are based on experiment and cannot b proved or derived.
There are two aspects to motion. The first is called kinematics and the second is called dynamics. Kinematics is purely the description of motion independent of the laws of physics which govern it. The latter study i dynamics. We are particularly interested in the dynamics of particles and systems, but before we turn to dynamics, we have to develop a mathematical language for describing motion We will be concerned with two kinds of quantitites. The first, called scalar quantities, are characterized by magnitude only and are represented by ordinary real numbers. Time and temperature are scalar quantities. Other quantitites have the combined characteristics of magnitude and direction.
These are represented by mathematical objects called vectors. Relative position, velocity, force and acceleration are vector quantities. In three-dimensional space, you can think of a vector as a directed line segment having length (magnitude) and direction. You may also think of it in terms of an ordered set of three scalar components. In texts, vectors are usually represented in bold face.
a = (a1, a2, a3)
The essential characteristics of vectors are the following:
1.2.1 Equality of Vectors
Two vectors a and b are equal if and only if their respective components are equal, i.e. a = b means that a1 = b1, a2 = b2 and a3 = b3. 1.2.2 Vector Addition
a + b = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3)
1.2.3 Vector Subtraction
a − b = a + (−1)b = (a1 − b1, a2 − b2, a3 − b3)
1.2.4 Null Vector
There exists a null vector 0 such that a + 0 = a.
1.2.5 Commutative Law of Addition
a + b = b + a
1.2.6 Associative Law of Addition
a + (b + c) = (a + b) + c
In component form this can be written:
(a1 + (b1 + c1), a2 + (b2 + c2), a3 + (b3 + c3)) = ((a1 + b1) + c1,(a2 + b2) + c2,(a3 + b3) + c3)
1.2.7 Multiplication of a Vector by a Scalar
There are several kinds of multiplication defined for vectors. Let p be a scalar
and let a be a vector. Then,
pa = p(a1, a2, a3) = (pa1, pa2, pa3)
1.2.8 Scalar Multiplication of Vectors; Dot Product
There are two kinds of multiplication defined for vectors with vectors. The
first is called the dot product.
a · b = |a||b| cos θ = !aibi = aibi.
Here, |a| is the magnitude of a, i.e.,
|a| = a2
1 + a2
2 + a2
3 = √aiai,
and θ is the angle between the directions of a and b. Very often, for brevity, an index such as i in the expression aiai that is repeated exactly twice will imply a summation on that index. In such cases the summation sign # is not written and the index i is said to be a dummy index because when you actually write out the expression, it is replaced by 1s, 2s and 3s. Such is the case for
aiai ≡ !aiai ≡ a2
1 + a2
2 + a2
3 ≡ ajaj .
The dot product of two vectors is itself a scalar; hence this form of multipli-
cation is described as “scalar multiplication” of vectors.
The following are theorems for the dot product:
a · b = b · a
a · (b + c) = a · b + a · c
p(a · b) = (pa) · b = a · (pb) = (a · b)p
1.2.9 Vector Multiplication of Vectors; Cross Product
The second kind of multiplication produces a vector. Again θ is the angle between the directions of a and b. We then have, c = a × b = |a||b|sin θnˆ, where n is a vector of unit length and direction perpendicular to the plane which contains both a and b. The direction is further defined by a right hand rule, i.e., the direction of nˆ is the direction the thumb of our right hand would be pointing if you pointed the fingers of your right hand along the direction of a and curled them toward the direction of b. In truth, the
cross product is not a true vector, but rather is a pseudovector. It does have magnitude and components like a vector, but its direction is ambiguous and has to be defined artificially by a right-hand-rule. Physical quantities, such as the magnetic field, that are represented by pseudo vectors do not have a true direction, but are given one artificially by a right-hand-rule.
We may also write,
ci = !
δijkaj bk ≡ δijkaj bk
where, again, the repeated indices j and k indicate a sum and are dummies. The symbol δijk is a 3 × 3 × 3 matrix called the Levi-Civita tensor. If i, j and k have the values 1, 2 and 3 respectively or any even permutation of (123), then δijk = +1. If the values of i, j, and k respectively are an odd permutation of (123) then δijk = −1. Otherwise (such as in the case of two indices being the same), δijk = 0. (Think of the indices as labeled beads on a loop of string. An even permutation is one you can get by simply moving the beads counterclockwise or clockwise along the loop to change their order as they sit in your hand. Thus (312) and (231) are even permutations of (123). An odd permutation, on the other hand, is one for which you would have to exchange the positions of two adjacent indices. Thus (213) and (132) are odd permutations of (123).) While this form takes a little getting used to,it can be a powerful way of writing and manipulating vectors. Please note that it is the ith component and not the full pseudovector that is given by the subscript form.
We have the following theorems:
a × b = −b × a
a × (b + c) = a × b + a × c
p(a × b) = (pa) × b = a × (pb) = (a × b)p