Editing Phase space (section)

== Principles ==
In a phase space, every [[degrees of freedom (physics and chemistry)|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 (mathematics)|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 number of dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's ''x'', ''y'' and ''z'' positions and momenta (6 dimensions for an idealized monatomic gas), and for more complex molecular systems additional dimensions are required to describe vibrational modes of the molecular bonds, as well as spin around 3 axes. Phase spaces are easier to use when analyzing the behavior of mechanical systems restricted to motion around and along various axes of rotation or translation{{snd}} e.g. in robotics, like analyzing the range of motion of a [[robotic arm]] or determining the optimal path to achieve a particular position/momentum result.

[[File:Hamiltonian flow classical.gif|frame|left|Evolution of an [[statistical ensemble (mathematical physics)|ensemble]] of classical systems in phase space (top). The systems are a massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

=== Conjugate momenta ===

In classical mechanics, any choice of [[generalized coordinates]] ''q''<sub>''i''</sub> for the position (i.e. coordinates on [[Configuration space (physics)|configuration space]]) defines [[conjugate momentum|conjugate generalized momenta]] ''p''<sub>''i''</sub>, which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the [[cotangent bundle]] 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.

=== Statistical ensembles in phase space ===
The motion of an [[Statistical ensemble (mathematical physics)|ensemble]] of systems in this space is studied by classical [[statistical mechanics]]. The local density of points in such systems obeys [[Liouville's theorem (Hamiltonian)|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.