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[math]\displaystyle{ c }[/math]) nor are microscopically small ( [math]\displaystyle{ 10^{-9} m }[/math]).
The laws are formulated in terms of space, time, mass, and force:
Space and Time
Space
Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.
Cartesian
Vector Notation convention:
Position:
- [math]\displaystyle{ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i }[/math]
Velocity:
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + x\frac{d \hat{i}}{dt} + cdots }[/math]
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)
- [math]\displaystyle{ \frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt} }[/math]
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{i} + \frac{d z}{dt}\hat{i} }[/math]
Polar
Vector Notation convention:
Position:
- [math]\displaystyle{ \vec{r} = r \hat{r} }[/math]
- [math]\displaystyle{ \vec{r} \ne r \hat{r} +\phi \hat{\phi} }[/math]: \phi does not have the hits of length
Velocity:
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + x\frac{d \hat{i}}{dt} + cdots }[/math]
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)
- [math]\displaystyle{ \frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt} }[/math]
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{i} + \frac{d z}{dt}\hat{i} }[/math]
The unit vectors are changing in time. You could express the position vector in terms of cartesian unit vector in order to avoid this
- [math]\displaystyle{ \vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j} }[/math]
Spherical
Cylindrical
