Physicist's Introduction to Mathematica

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1 Physicist's Introduction to Mathematica Physics 200 Physics 50 Lab Revised for V6.0 Fall Semester 2000 Spring Semester 2006 Summer 2007 Nicholas Wheeler John Boccio John Boccio REED COLLEGE Swarthmore College Swarthmore College V4.0 Revised for V5.0 Laboratory 1 Part A Basic Graphics Pen-&-ink mathematicians have traditionally found graphics so labor intensive that they have tended to avoid figures whenever possible and, when a figure could not be avoided, to make do with a merely qualitative representation of the points at issue. Figures have tended to see service only as expository devices, used to express what was already known, and seldom to be used as aids to discovery. Mathematica makes the production of accurate figures~of many types~so quick and easy that it should/will become your habit to look to the graphical aspects of problems before you have fully understood them. The graphical aspects of your mathematical work will assume enlarged importance. Graphics~formerly an expository device~has become an exploratory tool. Resources Your principal resource is at your fingertips. Open Help > Find Selected Function > Visualization and Graphics, then look around to see what I mean. Basic graphics commands are simple and commonsensical, but you should be aware that if you go to File > Palettes > Plotting you will find buttons that produce things like Plot[,{,, }], and so remind you of the required syntax, and help you to get all the brackets and punctuation just right (Mathematica is unforgiving in those respects!). Your favorite Mathematica text is certain to contain a long/detailed graphics section. Keep a My Graphics notebook, recording your successful accomplishments; it is likely to become your primary reference, especially with regard to the correct management of the numerous "options" which are the main source of confusion in this area.

2 2 Laboratory 1A.nb Simple Plotting In[2]:=? Plot First Example: In[3]:= 8x, 0, 20<D Notice that Mathematica has no trouble at all with the infinities. Notice also that the output consists not of the figure, but of the not-too-informative expression -Graphics-. That output can be surpressed by appending a ; to our original command (try it). We will henceforth adopt that practice. We look now to a few basic elaborations of the preceding basic command: In[4]:= In[5]:= In[6]:= In[7]:= In[8]:= In[9]:= In[10]:= In[11]:= In[12]:= In[13]:= Plot@Tan@xD, 8x, 0, 20<, Frame Ø TrueD? Frame Plot@Tan@xD, 8x, 0, 20<, Frame Ø True, FrameTicks Ø 8Range@0, 6 p, pd, Automatic, None, None<D? FrameTicks? Range Plot@Tan@xD, 8x, 0, 13 p ê 2<, Frame Ø True, FrameTicks Ø 8Range@0, 6 p, pd, Automatic, None, None<, PlotStyle Ø 8AbsoluteThickness@2D<D Plot@Tan@xD, 8x, 0, 13 p ê 2<, Frame Ø True, FrameTicks Ø 8Range@0, 6 p, pd, Automatic, None, None<, PlotStyle Ø 8RGBColor@0.2, 0, 0.8D, Thickness@0.007D<D? RGBColor H*See the Color Charts, mentioned above*l? Thickness?? AbsoluteThickness REMARK: "Absolute thickness" is measured in printer's points (multiples of about 1/72nd of an inch). Second Example: In[14]:= PlotASin@tD -tê10, 8t, 0, 20 p<e Mathematica tries to show "the interesting part" of a figure, with the result that it has here clipped off some detail. I give two ways to override that defect: In[15]:= PlotASin@tD -tê10, 8t, 0, 20 p<, PlotRange Ø AllE

3 Laboratory 1A.nb 3 In[16]:= PlotASin@tD -tê10, 8t, 0, 20 p<, PlotRange Ø 8-1, 1<E Third Example: This example, borrowed provides a taste of "graphical discover/exploration." In[17]:= Plot@Sin@xD + ArcSin@xD, 8x, -0.85, 0.85<, Frame -> True, FrameTicks Ø , 0.8<, 8-1, 0, 1<, None, None<D The graph is remarkable for its unexpected linearity. To gain a sense of how linear it truly is, we might proceed as follows: In[18]:= In[19]:= Clear@aD a = Sin@.5D + ArcSin@.5D In[20]:= linear@x_d := ạ 5 x H*Equation of straight line passing through a couple of selected points: the origin and the point H0.5,aL*L In[21]:= Plot@8Sin@xD + ArcSin@xD, linear@xd<, 8x, -0.85, 0.85<, Frame -> True, FrameTicks Ø , 0.8<, 8-1, 0, 1<, None, None<D which reveals a slight departure from linearity at the ends. The mystery is explained when we enter In[22]:= Series@Sin@xD + ArcSin@xD, 8x, 0, 15<D and discover that the Maclaurin expansion has no terms of orders 2, 3, or 4. We made first use here of the following resource: In[23]:=? Series The dangling O@xD 16 is an uninformative annoyance, but is easily avoided: In[24]:= Series@Sin@xD + ArcSin@xD, 8x, 0, 7<D êê Normal FilledPlot Consider the following function In[25]:= sechsquared@x_d := 1 2 Sech@xD2 has all the properties required of a probability distribution In[26]:= - In[27]:= - In[28]:= - sechsquared@xd x x sechsquared@xd x x 2 sechsquared@xd x

4 4 Laboratory 1A.nb and when plotted looks very Gaussian (or "normal"): In[29]:= 8x, -4, 4<D Direct visual comparison with the Gaussian with the same variance is enhanced if one makes use of a resource provided as a Standard Add-On: In[30]:= gaussian@x_, s_d := s 1 2 p ExpB- 1 2 x s 2 F In[32]:= PlotB:sechsquared@xD, gaussianbx, p2 F>, 8x, -4, 4<, Filling Ø 81 Ø 82<<F 12 Polar Plot Not infrequently the result of physical calculation lends itself better to polar than to Cartesian graphical representation. For example, the theory of planetary motion (Kepler problem) leads to an equation of the form constant r = 1 + eccentricity µ cos HqL The relevant software is provided as an add-on to Mathematica: In[33]:= In[34]:= In[35]:= Needs@"BarCharts`"D; Needs@"Histograms`"D; Needs@"PieCharts`"D? PolarPlot 1 PolarPlotB, 8q, 0, 2 p<f Cos@qD Slight adjustment causes the orbit to precess: In[36]:= 1 PolarPlotB, 8q, 0, 20 p<f Cos@1.01 qd Try these further examples of polar plots: In[37]:= In[38]:= PolarPlotA9t, t 1.1, t 1.2 =, 8t, 0, 2 p<e PolarPlot@Sin@5 qd, 8q, 0, 2 p<d List Plot In[39]:=? ListPlot The familiar Fibonacci numbers are available as a built-in resource: In[40]:= In[41]:=? Fibonacci Fibonacci@50D Make a table of the first 10 Fibonacci numbers

5 Laboratory 1A.nb 5 In[42]:= fib = Table@Fibonacci@nD, 8n, 10<D and use ListPlot to display that data graphically: In[43]:= ListPlot@fib, GridLines Ø Automatic, Prolog Ø AbsolutePointSize@5DD Note the techniques used to turn on the grid lines, and to adjust the point size. Parametric Plot In[44]:=? ParametricPlot First Example: Simple Cycloid A circle of unit radius rolls along the x-axis. We are interest in the curve traced by a point P marked on the circumference of the circle. Working from a sketch, we are led to define In[45]:= In[47]:= x@q_d := q - Sin@qD y@q_d := 1 - Cos@qD ParametricPlot@8x@qD, y@qd<, 8q, 0, 6 p<d The figure is misleading because different scales are used on the two axes; we now correct that defect, place the frame ticks at more natural places, and highlight the curve itself: In[48]:= ParametricPlot@8x@qD, y@qd<, 8q, 0, 4 p<, AspectRatio Ø Automatic, Frame Ø True, FrameTicks Ø 880, 2 p, 4 p<, 80, 1, 2<, None, None<, PlotStyle Ø 8RGBColor@0, 0, 1D, Thickness@0.007D< D NOTE: Sometimes Mathematica produces a figure that is unaccountably tiny. In such cases, click on the figure and drag the handles (now visible) to rescale the figure. "Hypocycloids" and "hypercycloids" result when the point P is placed interior/exterior to the unit circle. Here we make the appropriate modifications: In[49]:= In[50]:= In[52]:= Clear@rD x@q_, r_d := q - r Sin@qD y@q_, r_d := 1 - r Cos@qD ParametricPlot@88x@q, 0.5D, y@q, 0.5D<, 8x@q, 1.0D, y@q, 1.0D<, 8x@q, 1.5D, y@q, 1.5D<<, 8q, 0, 4 p<, AspectRatio Ø Automatic, Frame Ø True, FrameTicks Ø 880, 2 p, 4 p<, 80, 1, 2<, None, None<, PlotStyle Ø 8RGBColor@1, 0, 0D, 8<, RGBColor@0, 0, 1D<D Second Example: Epicycles A wheel of given radius turns with given angular velocity. Pinned at its circumference is a second wheel of given radius that turns with given angular velocity. And pinned to its circumference is a third wheel. Etc. We are interested in the motion of a point marked on the circircircumference of the last wheel.

6 6 Laboratory 1A.nb In[53]:= ParametricPlotBEvaluateB Cos@7 qd, Sin@7 qd< :CosB-17 q + p 2 F, SinB-17 q + p 2 F>F, 8q, 0, 2 p<, AspectRatio Ø Automatic, Frame Ø TrueF Adjustment of the numerics would, of course, result in different curves. It is not immediately obvious why the present numerics (1, 7 & 17) generate 6-fold symmetry. Note the use here of In[54]:=? Evaluate without which the argument of ParametricPlot[]would not be a list {,}, as it is required to be. Third Example: Lissajous Figures The following construction is familiar to anyone who has ever played with an oscilloscope: In[55]:= In[56]:= In[58]:= In[59]:= Clear@a, b, x, yd x@t_, a_, a_d := a Cos@a td y@t_, b_, b_, d_d := b Cos@b t + dd ParametricPlot@8x@t, 2, 2D, y@t, 1, 3.00, p ê 2D<, 8t, 0, 16 p<, AspectRatio Ø AutomaticD ParametricPlot@8x@t, 2, 2D, y@t, 1, 3.01, p ê 2D<, 8t, 0, 16 p<, AspectRatio Ø AutomaticD Fourth Example: Cornu Spiral The Cornu integrals arise in physical optics (diffraction theory). The following pretty figure involves some pretty heavy calculation; we'll ask Mathematica how long it takes. t t In[60]:= cornu@t_d := : 0 SinAu 2 E u, 0 CosAu 2 E u> In[61]:= ParametricPlot@Evaluate@cornu@tDD, 8t, -10, 10<, AspectRatio Ø AutomaticD êê Timing Now Save, Quit, and in a freshly opened Mathematica proceed to Part B.

Physicist's Introduction to Mathematica

Physicist's Introduction to Mathematica Laboratory 6 Part B Fitting Curves to Data Preliminary Remarks It is increasingly rare for your activity in the laboratory to be describable as "hands-on" production