Generalized Coordinates

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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