phase portrait
A picture which plots the evolution of possible paths in PHASE SPACEthat a point in that space can have, starting from a certain set of initial conditions. In the case of SIMPLE HARMONIC MOTION the phase portrait consists of a set of ellipses which all have the same center. In the case of NON-LINEAR OSCILLATIONSmore compli- cated patterns appear in the phase portrait. The phase portrait is a convenient way of
showing the complicated behavior that can occur in dynamical systems, including the presence of ATTRACTORS, chaotic behavior, and LIMIT CYCLES.
phase space
A multi-dimensional space that can be used to define the state of a sys- tem. Phase space has coordinates (q1, q2,…p1, p2,…), where q1, q2, etc., are degrees of
freedom of the system and p1, p2, are the
momenta corresponding to these degrees of freedom. For example, a single particle has three degrees of freedom (correspond- ing to the three coordinates defining its po- sition). It also has three components of momentum corresponding to these degrees of freedom. This means that the state of the particle can be defined by six numbers (q1,
q2, q3, p1, p2, p3) and it is thus defined by a
point in six-dimensional phase space. If the system changes with time (i.e. the particle changes its position and momentum), then the point in phase space traces out a path (known as the trajectory). The system may consist of more than one particle. Thus, if there are N particles in the system then the state of the system is specified by a point in a phase space of 6N dimensions. The idea of phase space is useful in chaos theory. See also attractor.
phase speed
The speed with which the phase in a traveling wave is propagated. It is equal to λ/T, where T is the period. Com- pare group speed.phasor
/fay-zer/ See simple harmonic mo- tion.pi
/pÿ/ (π) The ratio of the circumference of any circle to its diameter. π is approxi- mately equal to 3.14159… and is a tran- scendental number (its exact value cannot be written down, but it can be stated to any degree of accuracy). There are several ex- pressions for π in terms of infinite series.pico-
/pÿ-koh/ Symbol: p A prefix denot- ing 10–12. For example, 1 picofarad (pF) =10–12farad (F).
pictogram
/pik-tŏ-gram/ (pictograph) A diagram that represents statistical data ina pictorial form. For example, the pro- portions of pink, red, yellow, and white flowers that grow from a packet of mixed seeds can be shown by rows of the appro- priate relative numbers of colored flower shapes.
piecewise
A function is piecewise contin- uous on S if it is defined on S and can be separated into a finite number of pieces such that the function is continuous on the interior of each piece. Terms such as piece- wise differentiable and piecewise linear are similarly defined.pie chart
A diagram in which proportions are illustrated as sectors of a circle, the rel- ative areas of the sectors representing the different proportions. For example, if, out of 100 workers in a factory, 25 travel to work by car, 50 by bus, 10 by train, and the rest walk, the bus passengers are repre- sented by half of the circle, the car passen- gers by a quarter, the train users by a 36° sector, and so on.pint
A unit of capacity. The US liquid pint is equal to one eighth of a US gallon and is equivalent to 4.731 8 × 10–4m3. In theUK it is equal to one eighth of a UK gallon and is equivalent to 5.682 6 × 10–4m3. The
US dry pint is equal to one sixty-fourth of a US bushel and is equivalent to 5.506 1 × 10–4m3.
pixel
/piks-ĕl/ See computer graphics.place value
the position of an integer in a number. For example, in the number 375 the 5 represents 5 units, the 7 represents 7 tens, and the 3 represents 3 hundreds.plan
An illustration that shows the ap- pearance of a solid object as viewed from above (vertically downward). See also ele- vation.planar
/play-ner/ Describing something that occupies a plane or is flat.plane
A flat surface, either real or imagi- nary, in which any two points are joined by a straight line lying entirely on the surface. Plane geometry involves the relationships between points, lines, and curves lying in the same plane. In Cartesian coordinates, any point in a plane can be defined by two coordinates, x and y. In three-dimensional coordinates, each value of z corresponds to a plane parallel to the plane in which the x and y axs lie. For any three points, there exists only one plane containing all three. A particular plane can also be specified by a straight line and a point.plane shape
A shape that exists in a plane, i.e. a two-dimensional shape that does not have a depth but has a width and height. Examples of plane shapes include triangles, squares, rectangles, kites, rhom- buses, and parallelograms. Plane shapes also include curves such as circles and el- lipses.Platonic solids
/plă-tonn-ik/ A name given to the set of five regular polyhedra. See polyhedron.plot
To draw on a graph. A series of indi- vidual points plotted on a graph may show a general relationship between the vari- ables represented by the horizontal and vertical axs. For example, in a scientific ex- periment one quantity can be represented by x and another by y. The values of y at different values of x are then plotted as a series of points on a graph. If these fall on a line or curve, then the line or curve drawnplot
50% bus 25% car 15% walk 10% trainthrough the points is said to be a plot of y against x.