Generalized Coordinates#
Definition: Configuration
Let \(\mathcal{R}\) be a reference frame.
The configuration of a physical system with \(n\) point particles \(p_1, \dotsc, p_n\) is its parametrization by the positions \(\boldsymbol{r}_{p_1}, \dotsc, \boldsymbol{r}_{p_n}\) of \(p_1, \dotsc, p_n\):
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 \(p_1, \dotsc, p_n\).
If \((q_1, \dotsc, q_s)\) is a parametrization of \(S\) with \(s = 3n\) degrees of freedom which is equivalent to the configuration parametrization of \(S\), then we say that \(q_1, \dotsc, q_s\) are generalized coordinates for \(S\).
Definition: Generalized Velocities
The derivatives \(\dot{q}_1, \dotsc, \dot{q}_s\) of the generalized coordinates \(q_1, \dotsc, q_s\) with respect to time are called generalized velocities.
Definition: Generalized Accelerations
The derivatives \(\ddot{q}_1, \dotsc, \ddot{q}_s\) of the generalized velocities \(\dot{q}_1, \dotsc, \dot{q}_s\) with respect to time are called generalized accelerations.
Notation
We often use the following shorthand notations:
- \(q\) for \(q_1, \dotsc, q_s\)
- \(\dot{q}\) for \(\dot{q}_1, \dotsc, \dot{q}_s\)
- \(\ddot{q}\) for \(\ddot{q}_1, \dotsc, \ddot{q}_s\)
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 \(q_1, \dotsc, q_s\) be generalized coordinates for a physical system \(\mathcal{S}\) of point particles and let \(t_1\) and \(t_2 \gt t_1\) be two moments in time.
А path in configuration space is a function \(\boldsymbol{q}: [t_1; t_2] \to \mathbb{R}^s\) which is continuous on \([t_1; t_2]\) and continuously differentiable on \((t_1; t_2)\).
Each physical system follows only one path \(\boldsymbol{q}\) 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 \(\boldsymbol{q}(t) = \begin{bmatrix}q_1 & \cdots & q_s\end{bmatrix}^{\mathsf{T}}\).