- 1.1 Basic notions of mechanics
1.1 Basic notions of mechanics
Definition 1.1 System
A mechanical system is comprised of
particles labeled by an index $i\!\in\!\mathbb{I}$,
that have masses $m_i$,
reside at the positions $\mathbf x_i$,
and move with velocities $\mathbf v_i$.
Remark. We say that the system has $N$ particles when $\mathbb I = \{1, \dots, N\}$.
Remark. Bold-face symbols indicate here that $\mathbf x_i$ describes a position in space. For a $D$-dimensional space one needs $D$ numbers1) to specify the position, and $\mathbf x_i$ may be thought of as a vector in $\mathbb{R}^D$. We say that $\mathbf x_i$ is a $D$-vector. In Chapter 2 we will take a closer look at vectors and their properties.
Remark. In order to emphasize the close connection between positions and velocities, the latter will also be denoted as $\dot{\mathbf x}$.
Remark. In hand writing vectors are commonly denoted by an arrow, i.e., $\vec x$ rather than $\mathbf x$.
Example 1.1 A piece of chalk
We wish to follow the trajectory of a piece of chalk through the lecture hall (i.e., when it is thrown, not when somebody is using it to write on a blackboard).
In order to follow its position and orientation in space, we decide to model it
as a set of two masses that are localized at the tip and at the tail of the chalk.
The positions of these two masses $\mathbf x_1$ and $\mathbf x_2$
will both be vectors in $\mathbb{R}^3$.
For instance we can indicate the shortest distance to three walls
that meet in one corner of the lecture hall.
In this model we have $N=2$ and $D=3$.