![]() ![]() ![]() Main articles: phase line (mathematics) and phase planeįor simple systems, there may be as few as one or two degrees of freedom. 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. ![]() 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. The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. The motion of an ensemble of systems in this space is studied by classical statistical mechanics. 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. coordinates on configuration space) defines conjugate generalized momenta p i which together define co-ordinates on phase space. In classical mechanics, any choice of generalized coordinates q i for the position (i.e. The initially compact ensemble becomes swirled up over time. The systems are a massive particle in a one-dimensional potential well (red curve, lower figure). in robotics, like analyzing the range of motion of a robotic arm or determining the optimal path to achieve a particular position/momentum result.Įvolution of an ensemble of classical systems in phase space (top). Phase spaces are easier to use when analyzing behavior of mechanical systems restricted to motion around and along various axes of rotation or translation - e.g. 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. A phase space may contain a great number of dimensions. 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. ![]() 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. The system's evolving state over time traces a path (a phase space trajectory for the system) through the high-dimensional space. 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. 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.
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