Phase space
The phase space describes the set of all possible states of a physical system [Note 1]. Each state of the system corresponds to a point in phase space. The phase space does not describe in contrast to the state-space of time.
For systems with up to three variables of the phase space can be represented graphically. This phase space portrait or phase portrait provides the opportunity to capture some characteristic structures such as zero klinai and fixed points as well as the vector field of the dynamics of the system without having to know the explicit solution of the equations of motion. Such a procedure is called phase space analysis.
A phase area can be divided into subspaces. For example forms in the phase space of a mass point of the space of possible position vectors of the real space and the lower space of possible momentum vectors of the momentum space.
Trajectories in phase space
The set of all points that determine a specific starting point of the temporal evolution of the system, ie trajectory. Trajectories in phase space crossing-free form curves so that you can of each of the points of a trajectory determine their further course unique. If there were two intersecting trajectories, the solution of the canonical equations would not be clear, and this would give no clear prediction. Closed curves, called orbits, however, are possible, they describe oscillatory systems.
A dynamic system whose trajectories fill the entire phase space, so come arbitrarily close to any point in phase space, it is called ergodic, see also ergodic hypothesis.
Despite their intersection freedom trajectories can differ are close in space. This is quantitatively described by the phase space density, which is also in the statistical mechanics of central importance. Important for the classification of a dynamic system, the development of the phase space of the phase space volume or density over time is:
- The distance between almost parallel trajectories into a bundle, as the phase volume is reduced; the system is called dissipative. Dissipative systems lose energy to their environment, so it is open systems.
- Systems with constant phase-space volume in fact called conservative. They have been completed, so the total energy obtained. The same is mathematically stated by the theorem of Liouville.
Phase space analysis
The phase space portrait is a way to analyze the temporal trends of dynamic systems graphically. To only the dynamic equations of the system are required, an explicit representation of the time-evolution, for example by dissolving an analytical differential equation, is not necessary.
As an example, some elements of the phase-space analysis follow a two-dimensional system by means of the differential equations
Is described:
- Drawing in the vector field of the dynamics: For a grid of dots, the direction of movement is indicated by arrows in the phase space. If you follow, starting from a given start point to the arrow, you come to a new point where you can repeat the procedure. So you can also draw on the basis of the vector field typical trajectories in the phase space portrait, help assess the qualitative behavior of the time evolution. When van der Pol oscillator, for example, all trajectories converge on a limit cycle, which can be illustrated within and outside of the cycle based on Beispieltrajektorien. For simple dynamical systems vector field and Beispieltrajektorien can often draw by hand, in more complex systems this can be done by computer programs.
- Drawing the zero klinai: A zero Kline, a curve in phase space along which does not change any of the dynamic variables. In the case of the above two-dimensional system, the x - Kline zero is defined by the condition, and the y - zero by the Kline. These equations can be often then solve for one of the variables when the overall dynamics can not be integrated analytically.
- Determining fixed points and their stability: states are referred to as fixed points, which do not change with time. Such fixed points correspond to the intersections of the zero klinai in phase space. In the above two-dimensional system explains the fact that at such an intersection point, the condition is met. By a linear stability analysis can also be determined, whether trajectories attracted or repelled in the vicinity of these points.
- Find by Separatrizen: As a separatrix (derived from the Latin separare " disconnect" ) is a curve, or ( hyper-) surface denotes the phase space regions separated from each other with different ( long-term) behavior. Example, are there two fixed points that attract trajectories, there may be a separatrix, which separates the two catchment areas from one another. With the villages and the stability of all fixed points, or with the vector field of the dynamics of the Separatrizen can be found without further calculations in appropriate cases.
Application
In the Hamiltonian mechanics of the phase space is the space of positions and momenta. In a number of free mass points of this space is thus -dimensional. The corresponding differential equation system is formed from the Hamilton 's equations of motion.
Since light of a specific wavelength is assigned to the pulse, it can be used in the geometrical optics, a "web" are assigned in the phase space.
Distinction to the configuration space
The configuration space is in contrast to the phase space only from the possible locations of the particles under consideration. For particles in three dimensions of the configuration space is thus -dimensional. However, the configuration space is not a phase space, since the indication of the place the system does not fully describe. In particular, trajectories may intersect in configuration space as often. The phase space in the sense of Hamiltonian mechanics is the cotangent bundle over the configuration space.