basic_tools:configuration_space
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basic_tools:configuration_space [2018/05/05 11:51] jakobadmin ↷ Links adapted because of a move operation |
basic_tools:configuration_space [2020/01/08 11:05] jakobadmin [Intuitive] |
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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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| [{{ :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]]}}] | ||
| - | Another good example is the [[models:basis_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 $S^2$. | + | 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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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | In the [[formalisms:lagrangian_formalism|Lagrangian formalism]], we describe what happens in [[theories:classical_mechanics: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.txt · Last modified: 2020/01/08 11:05 by jakobadmin
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