basic_tools:configuration_space
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basic_tools:configuration_space [2018/04/13 11:22] bogumilvidovic [Concrete] |
basic_tools:configuration_space [2020/01/08 11:05] (current) jakobadmin [Intuitive] |
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| ====== Configuration Space ====== | ====== Configuration Space ====== | ||
| + | //see also [[basic_tools:phase_space|Phase space]] and [[basic_tools:hilbert_space]]// | ||
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | [{{ :basic_tools:configspace.png?nolink&400|adapted from page 174ff in “The Emperors new Mind” by R. Penrose}}] | ||
| The //configuration space// or //configuration manifold// is the collection of all the possible "snapshots" or descriptions that the system can take. | The //configuration space// or //configuration manifold// is the collection of all the possible "snapshots" or descriptions that the system can take. | ||
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| Formulated differently, the configuration space is the possible "positions" of a mechanical system. Take note that the states of motion, eg. velocities/momenta are not part of the configuration space. This is in contrast to the [[basic_tools:phase_space|phase space]], where we also take the states of motion into account. | Formulated differently, the configuration space is the possible "positions" of a mechanical system. Take note that the states of motion, eg. velocities/momenta are not part of the configuration space. This is in contrast to the [[basic_tools:phase_space|phase space]], where we also take the states of motion into account. | ||
| - | The state of a system is recorded in a configuration space point through all the locations and all the velocities that the objects in the system have at a given point in time. | + | The state of a system is recorded in a configuration space point through all the locations that the objects in the system have at a given point in time. |
| The time evolution of a system can then be represented as a path in configuration space. | The time evolution of a system can then be represented as a path in configuration space. | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
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| + | Each point of the configuration space represents one configuration our system can be in. For example, each specific way a double pendulum can be arranged is a point in our configuration space. The total configuration space consists of all possible configurations. | ||
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| + | The simplest example is an object moving in just one dimension. The configuration space is just a line, i.e. $\mathbb{R}$. Now, if we want to describe two objects which move in one dimension, we have two choices. Either we describe them using two location vectors: | ||
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| + | $$ \vec{r}_1 = (f(x)), \quad \vec{r}_2 = (g(x)) $$ | ||
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| + | {{ :basic_tools:configspace1.png?nolink&400 |}} | ||
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| + | or we glue the configuration spaces of the two objects together to one big configuration space and then only keep track of one location in this total configuration space: | ||
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| + | $$ \vec{r} = (f(x),g(x)). $$ | ||
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| + | When we glue two lines together we get a rectangle. At each possible location of the first object, we need to take into account the possibility that the second object could be anywhere. Hence we need a complete copy of $\mathbb{R}$ at each location of $\mathbb{R}$. Gluing a copy of $\mathbb{R}$ to each point of $\mathbb{R}$ yields a rectangle. Such a construction is known as a [[advanced_tools:product_spaces|product space]]. | ||
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| + | {{ :basic_tools:configspace2.png?nolink&600 |}} | ||
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| + | ---- | ||
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| [{{ :basic_tools:pendulumconfiguspace.png?nolink&300|Configuration space of a double pendulum. [[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Source]]}}] | [{{ :basic_tools:pendulumconfiguspace.png?nolink&300|Configuration space of a double pendulum. [[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Source]]}}] | ||
| - | Each point of the configuration space represents one configuration our system can be in. For example, each specific way a double pendulum can be arranged is a point in our configuration space. The total configuration space consists of all possible configurations. The total configuration space of a pendulum swinging just in two dimensions is a circle $S^1$, since a pendulum can swing once around its suspension. The configuration space of a pendulum swinging in 3-dimensions is the sphere $S^2$. | + | Another good example is the [[models:basic_models:pendulum|pendulum]]. The total configuration space of a pendulum swinging just in two dimensions is a circle $S^1$, since a pendulum can swing once around its suspension. The configuration space of a pendulum swinging in 3-dimensions is the sphere $S^2$. |
| This is useful, since now we can describe our system in terms of paths in the configuration space. The system starts in one specific configuration, which corresponds to one point in the configuration space. Then as it evolves this point starts moving around in configuration space. Thus, we only need to keep track of how this one point moves around. | This is useful, since now we can describe our system in terms of paths in the configuration space. The system starts in one specific configuration, which corresponds to one point in the configuration space. Then as it evolves this point starts moving around in configuration space. Thus, we only need to keep track of how this one point moves around. | ||
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| ---- | ---- | ||
| - | **Examples** | + | **Further Examples** |
| -->Free Particle# | -->Free Particle# | ||
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| **References** | **References** | ||
| - | 1. [Lanczos, C.] Principles of Classical Mechanics | + | * [Lanczos, C.] Principles of Classical Mechanics |
| - | + | * [Sussman G. J., Wisdom J. & Mayer M. E. ][[https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/toc.html#pref-1|Structure and Interpretation of Classical Mechanics]] | |
| - | 2. [Sussman G. J., Wisdom J. & Mayer M. E. ] [[https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/toc.html#pref-1|Structure and Interpretation of Classical Mechanics]] | + | |
| <tabbox Abstract> | <tabbox Abstract> | ||
| As the name indicates, the mathematical figure is a manifold, and we will denote it as $\mathcal Q$. We will use the letter $q$ to denote the points of the system, and $q^i$ will denote the coordinates. The link between both is called //configuration map// | As the name indicates, the mathematical figure is a manifold, and we will denote it as $\mathcal Q$. We will use the letter $q$ to denote the points of the system, and $q^i$ will denote the coordinates. The link between both is called //configuration map// | ||
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| [[basic_tools:phase_space|phase space]].<cite>[[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Lectures on Classical Mechanics]], by John Baez</cite></blockquote> | [[basic_tools:phase_space|phase space]].<cite>[[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Lectures on Classical Mechanics]], by John Baez</cite></blockquote> | ||
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| - | [{{ :basic_tools:configurationspacebaezclassicalmechanics.png?nolink&600 |[[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|Lectures on Classical Mechanics]], by John Baez}}] | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | In the [[formalisms:lagrangian_formalism|Lagrangian formalism]], we describe what happens in [[theories:newtonian_mechanics|classical mechanics]] by referring to points in the configuration space. In this sense, the configuration space is for the Lagrangian formalism, what the phase space is for the [[formalisms:hamiltonian_formalism|Hamiltonian formalism]]. | + | In the [[formalisms:lagrangian_formalism|Lagrangian formalism]], we describe what happens in [[theories:classical_mechanics:newtonian|classical mechanics]] by referring to points in the configuration space. In this sense, the configuration space is for the Lagrangian formalism, what the phase space is for the [[formalisms:hamiltonian_formalism|Hamiltonian formalism]]. |
| Configuration spaces are also the central objects in robotics, as the set of reachable positions by a robot. | Configuration spaces are also the central objects in robotics, as the set of reachable positions by a robot. | ||
basic_tools/configuration_space.1523611356.txt.gz · Last modified: 2018/04/13 09:22 (external edit)
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