![]() ![]() 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 the 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. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. It is the direct product of direct space and reciprocal space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state corresponding to one unique point in the phase space. ![]()
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