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- Phase_space abstract "In mathematics and physics, a phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space). The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Willard Gibbs.A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition.In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system, or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase space trajectory for the system) through the high-dimensional space. The phase space trajectory represents the set of states compatible with starting from one particular initial condition, located in the full phase space that represents the set of states compatible with starting from any initial condition. As a whole, the phase diagram represents all that the system can be, and its shape can easily elucidate qualities of the system that might not be obvious otherwise. A phase space may contain a great many dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's x, y and z positions and momenta as well as any number of other properties.In classical mechanics, any choice of generalized coordinates q i for the position (i.e. coordinates on configuration space) defines conjugate generalized momenta pi which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the cotangent space of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.The motion of an ensemble of systems in this space is studied by classical statistical mechanics. The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. Within the context of a model system in classical mechanics, the phase space coordinates of the system at any given time are composed of all of the system's dynamic variables. Because of this, it is possible to calculate the state of the system at any given time in the future or the past, through integration of Hamilton's or Lagrange's equations of motion.".
- Phase_space thumbnail Focal_stability.png?width=300.
- Phase_space wikiPageID "191101".
- Phase_space wikiPageRevisionID "590702380".
- Phase_space hasPhotoCollection Phase_space.
- Phase_space id "p/p072590".
- Phase_space title "Phase space".
- Phase_space subject Category:Concepts_in_physics.
- Phase_space subject Category:Dimensional_analysis.
- Phase_space subject Category:Hamiltonian_mechanics.
- Phase_space subject Category:Dynamic_systems.
- Phase_space comment "In mathematics and physics, a phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space).".
- Phase_space label "Espace des phases".
- Phase_space label "Espacio fásico".
- Phase_space label "Espaço fásico".
- Phase_space label "Faseruimte".
- Phase_space label "Phase space".
- Phase_space label "Phasenraum".
- Phase_space label "Przestrzeń fazowa".
- Phase_space label "Spazio delle fasi".
- Phase_space label "Фазовое пространство".
- Phase_space label "فضاء الطور".
- Phase_space label "位相空間 (物理)".
- Phase_space label "相空間".
- Phase_space sameAs Fázový_prostor.
- Phase_space sameAs Phasenraum.
- Phase_space sameAs Espacio_fásico.
- Phase_space sameAs Espace_des_phases.
- Phase_space sameAs Spazio_delle_fasi.
- Phase_space sameAs 位相空間_(物理).
- Phase_space sameAs 위상_공간_(물리학).
- Phase_space sameAs Faseruimte.
- Phase_space sameAs Przestrzeń_fazowa.
- Phase_space sameAs Espaço_fásico.
- Phase_space sameAs m.01b3w6.
- Phase_space sameAs Q62542.
- Phase_space sameAs Q62542.
- Phase_space wasDerivedFrom Phase_space?oldid=590702380.
- Phase_space depiction Focal_stability.png.
- Phase_space isPrimaryTopicOf Phase_space.
