Generalized Coordinates

Definition: Configuration

Let $\mathcal{R}$ be a reference frame.

The configuration of a physical system with $n$ point particles $p_1,\dots ,p_n$ is its parametrization by the positions ${\mathbf{r}}_{p_1},\dots ,{\mathbf{r}}_{p_n}$ of $p_1,\dots ,p_n$:

$$({\mathbf{r}}_{p_1},\dots ,{\mathbf{r}}_{p_n})$$

Definition: Configuration Space

The phase space of this parametrization is known as configuration space.

In general, the configuration of a physical system with $n$ point particles has $3n$ degrees of freedom because each position has three components.

Definition: Generalized Coordinates

Suppose we have a physical system $S$ with $n$ point particles $p_1,\dots ,p_n$.

If $(q_1,\dots ,q_s)$ is a parametrization of $S$ with $s=3n$ degrees of freedom which is equivalent to the configuration parametrization of $S$, then we say that $q_1,\dots ,q_s$ are generalized coordinates for $S$.

Definition: Generalized Velocities

The derivatives ${\dot{q}}_1,\dots ,{\dot{q}}_s$ of the generalized coordinates $q_1,\dots ,q_s$ with respect to time are called generalized velocities.

Definition: Generalized Accelerations

The derivatives ${\ddot{q}}_1,\dots ,{\ddot{q}}_s$ of the generalized velocities ${\dot{q}}_1,\dots ,{\dot{q}}_s$ with respect to time are called generalized accelerations.

NOTATION

We often use the following shorthand notations:

• $q$ for $q_1,\dots ,q_s$
• $\dot{q}$ for ${\dot{q}}_1,\dots ,{\dot{q}}_s$
• $\ddot{q}$ for ${\ddot{q}}_1,\dots ,{\ddot{q}}_s$

Simply put, generalized coordinates are a set of values which allow us to uniquely determine the configuration of a physical system of point particles.

Definition: Path

Let $q_1,\dots ,q_s$ be generalized coordinates for a physical system $\mathcal{S}$ of point particles and let $t_1$ and $t_2>t_1$ be two moments in time.

А path in configuration space is a function $\mathbf{q}:[t_1;t_2]\to {\mathbb{R}}^s$ which is continuous on $[t_1;t_2]$ and continuously differentiable on $(t_1;t_2)$.

Each physical system follows only one path $\mathbf{q}$ between any two given moments $t_1$ and $t_2$. At any moment $t\in [t_1;t_2]$, the configuration of the physical system is completely described by $\mathbf{q}(t)={\left[\begin{array}{ccc}{q}_1& \cdots & q_s\end{array}\right]}^{\mathsf{T}}$.