# Configuration Spaces

The set of all configurations of the system that can be assumed is called the configuration space of the system.

## Generalized Coordinates

1. In order to be able to talk about specific configurations we need to have a set of parameters that label the configurations. The parameters used to specify the configuration of the system are called the generalized coordinates.

2. The $\displaystyle{n}$-dimensional configuration space can be parameterized by choosing a coordinate function $\displaystyle{\chi}$ that maps elements of the configuration space to $n$-tuples of real numbers.

3. The motion of the system can be described by a configuration path $\displaystyle{\gamma}$ mapping time to configuration-space points.

4. Corresponding to the configuration path is a coordinate path $\displaystyle{q = \chi \circ \gamma}$ mapping time to tuples of generalized coordinates.

The function $\displaystyle{\Xi \chi}$ takes the coordinate-free local tuple $\displaystyle{( t, \gamma (t), \mathcal{D} \gamma (t), ... )}$ and gives a coordinate representation as a tuple of the time, the value of the coordinate path function at that time, and the values of as many derivatives of the coordinate path function as are needed.

\displaystyle{ \begin{aligned} \text{generalized coordinate representation} &= \Xi (\text{local tuple}) \\ (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \end{aligned} }

\displaystyle{ \begin{aligned} \text{generalized coordinates} &= q \\ &= \chi \circ \gamma \\ \\ q(t) &= \chi(\gamma(t)) \\ \end{aligned} }

\displaystyle{ \begin{aligned} t &\to \gamma: \text{configuration path} \to \chi: \text{generalized coordinates} = q \\ \end{aligned} }

\displaystyle{ \begin{aligned} (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \\ \Gamma[q](t) &= (t, q(t), Dq(t), ...) \\ \Gamma[q] &= \Xi_\chi \circ \mathcal{T}[\gamma] \\ \end{aligned} }

— 1.2 Configuration Spaces

— Structure and Interpretation of Classical Mechanics

— Me@2019-03-01 03:09:25 PM

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2019.03.01 Friday ACHK