Forest UCM NLM

Newton's Laws of Motion

Limits of Classical Mechanic

Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).

It may be used to describe the motion of objects which are not moving at high speeds (0.1$c$) nor are microscopically small ( $10^{-9} m$).

The laws are formulated in terms of space, time, mass, and force:

Vectors

Vector Notation

A vector is a mathematical construct of ordered elements that represent magnitude and direction simultaneously.

$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i$

Vectors satisfy the commutative (order of addition doesn't matter) and associative ( doesn't matter which you add first) properties.

The multiplication of two vectors is not uniquely defined. At least three types of vector products may be defined.

Scalar ( Dot ) product

definition

$\vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3$

physical intepretation

$\frac{\vec{a} \cdot \vec{b}}{\left | \vec{b} \right |}$ is the length of $\vec{a}$ that is along the direction of $\vec{b}$ (a projection like the casting of a shadow)

Commutative property of scalar product

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

proof

$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$	definition of dot product
$a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3$	comutative property of multiplication
$b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a}$	definition of dot product

$\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}$

Distributive property of scalar product

$\vec{a} \cdot \left ( \vec{b} + \vec{c} \right ) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Vector ( Cross ) product

definition

$\vec{a} \times \vec{b} = \left( a_2b_3-a_3b_2\right) \hat{e}_1 +\left( a_3b_1-a_1b_3\right) \hat{e}_2 +\left( a_1b_2-a_2b_1\right) \hat{e}_3$

The vector product of $\vec{a}$ and $\vec{b}$ is a third vector $\vec{c}$ with the following properties.

$\left | \vec{c} \right | = \left | \vec{a} \right | \left | \vec{b} \right | \sin \theta$

$\vec{c}$ is $\perp$ to $\vec{a}$ and $\vec{b}$

the right hand rule convention is used to determine the direction of $\vec{c}$

physical interpretation

$A = \left | \vec{a} \times \vec{b} \right | =$ area of a parallelogram with vectors $\vec{a}$ and $\vec{b}$ forming adjacent edges

let $h$ represent the perpendicular distance from the teminus of $\vec{b}$ to the line of action of $\vec{a}$ ( a.k.a. the height)

then the area of the parallelogram is given by

$A=\left | \vec{a} \right | h$

the height $h$ is equivalent to $\left | \vec{b} \right | \sin \theta$ where $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$

thus

$A=\left | \vec{a} \right | h = \left | \vec{a} \right | \left ( \left | \vec{b} \right | \sin \theta \right ) = \left | \vec{a} \times \vec{b} \right |$

NON-Commutative property of vector product

$\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$

proof

$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$	definition of dot product
$a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3$	comutative property of multiplication
$b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a}$	definition of dot product

$\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}$

Distributive property of the vector product

$\vec{a} \times \left ( \vec{b} + \vec{c} \right ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

The scaler triple product

definition

scaler tiple product $\equiv \vec{a} \cdot \left (\vec{b} \times \vec{c} \right )$

A third vector product is the tensor direct product.

Space and Time

Space

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Forest_UCM_NLM_Ch1_CoordSys

Forest_Ugrad_ClassicalMechanics