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

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

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

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.

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