The following terms are defined for M. Casco courses

A

Absolute Value

The absolute value of a number is -1 times the number if it is negative or +1 times the number if it is positive.

Acceleration

The rate of change of velocity with respect to time. Acceleration is a vector quantity. Acceleration is related to the force on an object and its mass by Newton's second law, stating that acceleration equals force divided by mass. A=F/m

Angular Frequency

The frequency of a periodic system, multiplied by 2p. The units are in radians per unit time but since radians are unitless, it comes out to be t^-1, the same as angular velocity of circular motion. Angular frequency is symbolized by the Greek letter omega (w).

Applied Force

We make a somewhat arbitrary distinction among the forces acting in a dynamical system. The categories are applied force, centering force and drag or friction force. The applied force is taken to be that force which is applied to the moving parts of a system by an outside agent as for example the force applied to a pendulum by someone pushing it or the force applied to a piece of metal by the magnetic field of an electromagnet. An applied force results in an energy transfer across the system boundary.

Attractor

An attractor is a particular state or set of states of a dynamical system that the system seeks as time passes. For instance a pendulum with friction will evetually come to rest hanging straight down with zero velocity. The point (0,0) then is an attractor for an undriven pendulum. Attractors may also be periodic, consisting of a set of states that the system visits, one after another, repeatedly. Periodic attractors may be finite or infinite sets. Another category of attractor is chaotic. Chaotic attractors consist of an infinite set of states that never repeat but are all contained in a finite volume of phase space.

B

Basin of Attraction

If a dynamical system eventually settles down to no motion, periodic motion or bounded chaotic motion, it is said to have settled on its attractor. It is possible that a system have multiple attractors and which long-term state of motion the systems seeks depends on its initial state. The set of all initial states that lead to a certain attractor is called that attractor's basin of attraction.

Bifurcation

In general a bifurcation is a splitting or branching of an object. With regard to an attractor, it refers to a doubling of the number of points in a finite periodic attractor as a control variable changes.

C

Centering Force

We make a somewhat arbitrary distinction among the forces acting in a dynamical system. The categories are applied force centering, sometimes called restoring, force and drag or friction force. The centering force is taken to be that force which tends to restore the system to some equilibrium state as for example the force applied to a pendulum by gravity. A centering force does not result in an energy transfer across the system boundary since the centering agent is customarily taken to be part of the system.

Chaotic

Chaotic is an adjective describing a dynamical system or mathematical function in which future states or values are not related in any simple way to the current state or value. It is not to imply that there is no connection between past and future states or values, only that the connection is so complex as to make prediction of the future from the past a practical impossibility. If the set of future states or values, though unpredictable, is limited to a finite range of values, the system or function is said to be bounded as well as chaotic.

Complex Number

Ordinary numbers that we use for accounting and simple calculations are called real numbers. There is no real number such that the square of that number is -1 since the product of any real number with itself is positive. To remedy this situation, the square root of -1 was defined and given the symbol i. Then a new class of numbers was invented, called imaginary numbers, made up of i times the set of real numbers. A complex number is a number comprised of a real part added to an imaginary part like a+b*i.

Complex Plane

The complex plane contains the set of complex numbers. It is a plane spanned by the set of real numbers, normally along the horizontal axis, and the set of imaginary numbers, normally along the vertical axis.

Conservative System

A dynamical system in which no energy is either lost or gained by the system. These are systems where friction is negligible.

Converge

If the dependent variable of a function under iteration gets closer and closer to a fixed value the function is said to converge under iteration.

Control Variable

In a dynamical system or mathematical function, the defining equation may contain parameters that are either constant, or subject to change. Those that are subject to change are called control variables. For example in the equation Y=A*exp(-(b-X)^2) if the parameter A can be adjusted between runs through the domain of X, A would be a control variable.

Cosine

In a right triangle, the ratio of the adjacent side to the hypotenuse. See the background page on trigonometric functions .

Cycle

The set of all the states or values visited by a periodic system or function during one period. In other words one cycle of anything that is repetitive is everything it does during one repetition.

D

Density

The mass per unit volume of an object. It would be measured in kilograms per cubic meter in the SI system of units.

Displacement

The difference between an initial position and a final one. Displacement is a vector quantity.

Diverge

If the dependent variable of a function under iteration increases without limit the function is said to diverge under iteration.

Domain

The domain is the set of values that the independent variable of a function may take on. A domain may be finite as in the set of numbers {1, 2, 3..n} or infinite as in all the mumbers between 0 and 1.

Dynamical System

A system that changes with the passage of time. Basically that is any system with moving parts.

Dynamics

The study of motion and the forces which cause it.

E

Effective Mass

The mass in a dynamical system that must be included when we treat the moving parts of the system as though they were a particle , using the free body analysis in applying Newton's laws of motion .

Elastic Scattering

An interaction where two particles collide and the total kinetic energy of the two particles remains constant. The direction and speed of both particles will in general be different after the collision.

Energy

Energy is defined as the ability of an object to do work on its surroundings. It may be in the form of kinetic energy or of potential energy .

Equation

An equation is a mathematical expression with an equal sign (=) in it. It signifies that the numerical or vector value on one side of the = is the same as the numerical or vector value on the other side. An equation may include variables and parameters. If any of the variables are rates of change, the equation is called a differential equation.

Exponentiation

Raising a number to a power. The symbol ^ is used in this program to indicate this operation. The expression b^e means multiply b (the base) by itself e (the exponent) times.

F

Fluid

Fluid is material that takes the shape of its container. It may be a gas which expands to fill the entire volume of its container or liquid which settles into the bottom of its container. An alternative definition of fluid is that it is material that does not support shear forces.

Force

Quite simply a force is a push or a pull. Force is a vector quantity.

Free Body

An object that is unconstrained so that it may respond to forces in accordance with Newton's laws of motion .

Frequency

The number of cycles per unit time that a periodic system or function completes. The fequency (f) is related to the period (T) by f=1/T.

Friction

We consider two kinds of friction, sliding friction and turbulent friction. Sliding friction occurs when two solid objects maintain contact with one another while in relative motion. The small hills and valleys on the surfaces tend to get caught on one another as they pass, requiring force to be applied to keep the motion going. The reaction to this applied force is called the force of sliding friction. The work done by the applied force raises the temperature of the objects. The force perpendicular to the motion, holding the surfaces of the objects together is called the normal force. Turbulent friction occurs when an object moves through a fluid medium, stiring it up and losing some of its kinetic energy to the medium. This situation also requires the application of a force to keep the motion going and the reaction to that force is the force of turbulent friction. The work done by the applied force raises the temperature of the object and the surrounding fluid.

Function

A mathematical function is a rule relating two sets of objects. Here we will restrict ourselves to objects that are numbers or vectors. One of the sets is called the domain of the function, the other is called the range of the function. Functions are frequently expressed as equations as for example Y=X+2. This function is interpreted as follows. For every X in the domain, add 2 to it to get the corresponding Y in the range. Because we are free to choose any X we want, X is called the independent variable. Because once X is chosen Y is fixed, we call Y the dependent variable.

G

Gaussian

The bell shaped curve that is used to describe the distribution of quantities around some normal value, named in honor of Mr. Gauss we believe. This function is expressed as Y=A*exp(-(b-X)^2) , where "A" is the amplitude or height of the curve and "b" is the location of the peak of the curve on the X axis. The exp() symbol represents the number "e" (approximately equal to 2.7182818284), raised to the power of the stuff in its parentheses. For example exp(0)=1, exp(1)=e, exp(2)=e^2, exp(-1)=1/e, exp(-2)=1/(e^2), and so on. As you can see when X=b, Y=A in the Gaussian function. As X departs from b in either direction, the value of the exp() approaches zero, forcing Y to approach zero as well. See the Non-Linear Rate of Change display in the Rate of Change lesson for an illustration

H

I

Impulse Force

A force applied for a time which is short compared to the observation time, as for example the force between a bat and ball where the observation is over the entire flight of the ball from leaving the pitcher's hand to landing in the bleachers.

Iteration

Iteration is the process of taking the value of the dependent variable of a function and feeding it back into the function as the independent variable.

J

Julia Set

A Julia set, named for Gaston Julia, is the set set of all points in the complex plane such that the iterated function Z[n]=Z[n-1]^2+k, where k is a fixed complex number, does not diverge as n approaches infinity. It differs from the Mandelbrot set in that there is a different Julia set for each value of k. There is only one Mandelbrot set.

K

Kinetic Energy

The energy an object has as a result of its motion. Numerically the kinetic energy is equal to 1/2*m*v^2 where m is the mass of the object and v is the magnitude of its velocity .

Kinematics

The study of objects in motion without explicit consideration for the forces which produced the motion.

L

M

Magnitude

The size of a thing, without regard for its sign (+ or -) or direction. Similar to the absolute value of a number but applies to vectors as well.

Mandelbrot Set

The Mandelbrot set, named for Benoit Mandelbrot, is the set set of all points c in the complex plane such that the iterated function Z[n]=Z[n-1]^2+c does not diverge as n approaches infinity.

Mass

The property of an object which determines its resistance to changes in velocity . In the presence of a gravitational field, as near the surface of a planet, the mass of an object is proportional to its weight, the force exerted on the object by the planet.

Mechanics

The study of objects in motion. Mechanics is normally limited to a small number of large slow objects, as opposed to statistical mechanics which deals with large numbers of objects, relativistic mechanics which deals with objects moving near the speed of light and quantum mechanics which deals with objects more or less the size of atoms. Mechanics encompasses the topics of kinematics and dynamics .

N

Normal

Another word for perpendicular. Normal in this sense is usually used in refering to a vector's orientation relative to some surface. For example a vertical vector is normal to a horizontal surface.

O

Object

A thing. The term "object" is the most general form of thingness. There are physical objects like baseballs and uranium atoms, and mathematical objects like numbers and vectors . It will be clear from the context what sort of object we are talking about.

Origin

The point in a reference frame from which measurements are made. It is the location of the zero value for each axis in the frame.

P

Parameter

In an equation those elements that are not variables are parameters. If a parameter is multiplied times a variable it may be called a coefficient. Parameters may be fixed or adjustable. Fixed parameters are called constants. Adjustable parameters are called control parameters. In the equation Y=A*exp(-(b-c*X)^2) A, b and c are parameters. If A takes on different values during different runs through the the values of X, it is considered a control parameter. If b is fixed during all runs through the values of X, it is a constant. The parameter c is a coefficient of X.

Particle

An object whose size is negligible in the context of our observation of it. For example the Earth might be considered a particle if we were studying its orbit around the Sun, but not if we want to know anything about its rotation about its axis. The nucleus of an atom might be a particle in an experiment on elastic scattering , but not in considering nuclear fission.

Period

The interval of time between the occurrence of identical states in a periodic system.

Periodic

A dynamical system or function which at some point returns to the same state or value. If a system or function ever revisits the identical state or value it will continue to come back to it again and again in equal intervals of time. That is why we call such a system periodic.

Phase Angle

The offset from the origin of a periodic function like the sine or cosine. For example in the function x=A*sin(w*t + f), f is the phase angle. The units on f are radians.

Phase-Control Space

If in a phase space representation of a dynamical system we exchange the roles of a control parameter and the independent variable, allowing the former to vary while holding the latter fixed, the resulting graph is plotted in a mixed space called phase-control space. See the online course for more information.

Phase Space

Sometimes as an aid in understanding what is happening in a dynamical system it is useful to mentally create a new kind of space. This space has dimensions of the state variables for our dynamical system. In the case of the pendulum, for example, it would be one dimension measured in position and one dimension measured in velocity. Time might be considered a third dimension of this "phase space", just as it might be considered a fourth dimension of ordinary space.

pi

The ratio of the circumference of a circle to its diameter is named by the Greek letter pi (p). The numerical value of pi is approximately 3.1415926.

Position

The location of an object relative to some point we have chosen to be the reference point. Position is a vector quantity.

Potential Energy

The potential energy of an object is the energy that object has as a result of its position relative to other objects. The numerical value of potential energy depends on the nature of the interaction of the object with its surroundings and the choice of a position to be the zero energy point.

Property

A characteristic that is inherently associated with the object which is said to have that property. For example the mass of an object is one of its properties. So also might be color, density and many other characteristics. Properties are classified as extensive or intensive. Extensive properties increase in proportion to the size of the object, as mass does for example. Intensive properties are independent of the size of the object. The density for examples remains the same if I cut an object in half and throw half of it away. Things like an object's position or velocity are not considered to be properties of the object. They are not a characteristic of the object only but are also dependent on the reference frame in which the object is located.

Q

Quadratic

A function involving the second and lower power, and none higher, of the independent variable. A quadratic function may contain x^2 explicitly or it may contain terms like x*(1-x), where the second power of x is implied. In general a quadratic may be written as y=a*x^2+b*x+c . For an illustration of a quadratic function, see the Quadratic Derivative display in the Rate of Change lesson.

Quantity

A numerical value either scalar or vector , which describes some attribute of an object like its position or its velocity . We sometimes speak of physical quantities to signify that we are talking about an object's properties or attributes as opposed to a purely mathematical quantity.

R

Radian

An angular unit of measure. A radian is an angle subtended by an arc whose length equals one radius. Since the circumference of a circle is 2* PI *radius and a radian spans an arc of one radius, there are 2*PI radians in a complete circle. So 1 radian equals 360/(2*PI) degrees. This is illustrated below.

Radius or Curvature

The radius of the largest circle containing the point at which the radius of curvature is to be determined and fitting within the curve. See the illustration below.

Range

The range is the set of values that the dependent variable of a function may take on. A range may be finite as in the set of numbers {1, 2, 3..n} or infinite as in all the mumbers between 0 and 1.

Reference Frame

A mathematical object which is used to allow comparison of the positions in space of physical objects like particles, or the comparison of one particle's positions at different times. For examples see the Measurement in Mechanics lesson. The reference frame may be made up of any set of coordinates which uniquely specify a point in space.

S

Scalar

A scalar quantity is one having only magnitude , not direction information. This is as opposed to a vector quantity which has both magnitude and direction.

Set

In mathematics a set is a collection of related objects. The mathematical usage is similar to the ordinary English meaning of the word. The objects that make up a set are called the elements of the set. If a set contains an unlimited number of elements it is an infinite set. Otherwise it is a finite set.

Shear Force

Imagine a solid rectangular block of material with the bottom face held fixed on the surface on which the block was resting and a force applied along the top face... Sort of the same force you might apply to an Oreo cookie to slide the top cookie off the bottom one to get at the cream filling. The applied force in this situation is called a shear force. A solid block subject to this shear force would be deformed so that its front face that was a rectangle becomes a parallelogram.

Significant Figures

The number of digits in a numerical value that are reliably known. If the numbers being used in a calculation are measured values, there will always be a limit on the accuracy of the measurement. The results of any calculations based on those numbers should not be reported with more significant figures than the least acurate of the measured values. For example if the length of a rectangle is measured to within 0.1 cm to be 25.3 cm and its width to within 0.1 cm to be 6.6 cm, multiplying shows the area to be 166.98 cm^2. The result however should be reported only to 2 significant figures since the width is only known to that accuracy, giving an area of 170 cm^2.

Sine

In a right triangle, the ratio of the opposite side to the hypotenuse. See the background page on trigonometric functions .

State

Dynamical systems evolve over the course of time. The state of the system at any instant may be identified by the values of certain variables at that instant. For example specifying the angle from the vertical and the velocity of a frictionless pendulum allows us to predict its position and velocity at any future time. Therefore the state of the pendulum at any instant is its position and velocity. In this example the position and velocity are known as state variables.

State Variable

An observable quantity which must be specified in order to determine how a Dynamical systems changes over the course of time. In conservative systems if all the state variables are known at any instant, the state of the system is determined for all future time.

Systéme International (International System)

The most commonly accepted system of units in scientific work. The fundamental units in this system are the meter, kilogram and second.

Spacetime

An extension of the concept of space to include an additional dimension perpendicular to the normal axes spanning our familiar three-dimensional space. This additional dimension measures time so that a point in spacetime locates an event. Since full four-dimensional spacetime is difficult to picture, we frequently work in spacetime consisting of one or two spatial dimensions and the time dimension.

T

Tangent

A straight line which touches a curve in one and only one point. The slope of a tangent is the slope of the curve at that point. Slope is the change in the vertical coordinate divided by the corresponding change in the horizontal coordinate. See the Rate of Change lesson for more on slopes.

Also, in a right triangle, the ratio of the opposite side to the adjacent. See the background page on trigonometric functions .

Trajectory

The path an object takes through space. Frequently associated with a projectile like a bullet or a missle.

U

Universality

Universality refers to the fact that a certain number, approxinately 4.669, discovered by Mitchell Feigenbaum and called Fiegenbaum's number, keeps popping up in what appear to be totally unrelated circumstances. See the online course for more information.

V

Vector

A quantity having both magnitude and direction. The direction may be expressed as an angle from a single axis in two dimensions. In three dimensions, the direction must be a pair of angles measured from different axes.

Velocity

The speed of an object in a given direction. Velocity is a vector quantity.

W

Wavelength

The distance covered by a travelling wave in one period . It is the distance between points of the same phase angle in a travelling wave. For example the distance between peaks, or the distance between valleys of a wave train. The wavelength is frequently symbolized by the Greek letter lambda l .

Work

Work is defined as the application of force over some displacement . Numerically the work done is the product of the force and the distance moved in the direction of that force. This may be calculated as force times displacement times the cosine of the angle between force and displacement. The angle gets involved because things do not always move in the direction in which you push them.

Worldline

A worldline is a series of events in spacetime that comprise the history of an object or a system. For example the series of events that define the world line of a tossed ball might be the ball was at (x,y,z,t) for all x, y z and t between the ball tossed event and the ball caught event.

X

Y

Z