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Using Math Symmetry Operations to Solve a Problem in Elementary

PhysicsSubmitted to The Physics Teacher as:

“Applying Symmetry and Invariance to a Problem with Two Parallel Current

Carrying Wires”Sandy Rosas and Marc Frodyma

San Jose City College

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Symmetry Operations: A Quantity of Interest is Left Invariant by the

Operation

Operations leaving the roots of equations invariant:

1: Linear Equations:

Ax + B = 0

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Apply a Translation and Magnification:

Make the Substitution for x:

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Substitute x’ = 0 Into the Transformation Equation

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2: Quadratic Equations:

or

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Translation Eliminates Linear Term:

With This Substitution, Equation Becomes:

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Solution to Transformed Equation:

Undo the Translation:

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Scipione del Ferro (1465 – 1526)

Niccolo Tartaglia (1499 - 1557)

Girolamo Cardano (1501 - 1576) .

A good secondary source:Stillwell, J. (1994). Mathematics and Its History, New York, NY: Springer-Verlag,p54.

3. Cubic Equations:

Who Gets the Credit?

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Cubic Equation:

Creative Step: Do a Translation to Eliminate Quadratic Term:

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New Equation:

Coefficients p, q Combinations Of A, B, and C

Good Algebra Exercise for Students!

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Results for p and q:

and

Check My Algebra!

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Change of Variables on Left Side!

So we have:

Algebra!

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Recall x’ = u + v

We Claim:

and

Why! Hint: x’ is Arbitrary!

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Eliminate v Between Equations for p and qGet a Quadratic in u Cubed

Solution:

More Good Algebra for Students!

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But We Had:

and

Top Equation Symmetric With Respect to Interchanging u and v

So u and v Have Same Solutions!

with

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Solution for v:

Use “+” for u and “-” for v To Satisfy

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But Recall:

So We Have:

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Finally, Undo the Translation:

We Could Also Examine The Quartic But Alas, Not the Quintic or Higher

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http://www.storyofmathematics.com/images2/diophantus.jpg

Diophantus, 2nd Century AD

Hellenistic Mathematician

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The Most Famous Diophantine Equation!

The Pythagorean Theorem

Find Integer Solutions to:

How Do We Find Them?

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Example Problem Requiring Pythagorean

Triples for its Solution

A right triangle with integer sides has its perimeter numerically equal to its area. What is the largest possible value of its perimeter?

Problem 11 Math Contest 2015 Round One www.AMATYC.org

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Is There A Formula That Generates

All of the Pythagorean Triples?

YES!

And It Is Very Clever!

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Here is the Formula:

With m, n Arbitrary IntegersGregory Melblom, Private Communication

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Previous Formula Generates All Primitive Triples (no common factors other than one)

But It Misses Some Non-Primitive Triples,For Example: (9, 12, 15)

So We Add A Common Factor!

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Add a Common Factor of k:

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A Right Triangle With Perimeter = Area

Set A = P and Cancel Common Factors

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With Common Factors Canceled In the Equation A = P We have:

k = 1, n = 1, and m = 3 gives x = 8, y = 6, and z = 10

k = 1, n = 2, and m = 3 gives x = 5, y = 12, and z = 13

k = 2, n = 1, and m = 2 gives x = 6, y = 8, and z = 10

The Only Possibilities!

Largest Perimeter is 5 + 12 + 13 = 30

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Consider the Following Equations

What is the Ratio n/m?

Problem 20 Math Contest 2015 Round One www.AMATYC.org

Another Great Problem!

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Use the Definition of Logs:

Original Equation Becomes:

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Divide Both Sides By Smallest Term:

Rewrite As:

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Make a Substitution:

And We Have:

Equation for the Golden Section!

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And the Answer to the Problem is:

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An Example of Symmetry Transformations In Physics

Reduction to the Equivalent One-Body Problem

Consider a Binary Star System

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"Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif

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Two Stars Orbit in Ellipses About Common Center of MassQuantity of Interest: Separation R(t) Two-Body Problem

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Apply Transformation:“Mass” m Orbits Fixed Force Center. Separation R(t) and Force Left Invariant

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Magnetic Field, North and South Poles

https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A

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Magnetic Field of a WireNo North and South Poles!

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Magnitude of the B Field

Field Strength Proportional to Ratio of Current to Distance

To Keep Field Invariant, ChangeCurrent and Distance by Same Factor

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Original Version of Problem

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Symmetry Transformation: Move Wire 2 Into Symmetric Position

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Questions

Why is I2’ Up? (I2 was Down)

Claim: I2’ = I1 = 10A. Why?

Distance of I2’ Reduced By Factor of Three. What was Original I2?

Original Problem Solved!

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More General Problem, Point POut of the Plane of the Wires

Find the Unknown Current I2

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Find the Base Angles With Laws of Cosines and Sines

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Move I1 Towards Point P To Make BNet HorizontalBut No Change in Magnitude

Base Angles Change by 5 Degrees Why?

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New Base Angles

The New Distance a of I1

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Original Distance of I1 was 4

New Distance of I1’ is 3.05

So To Keep BNet Invariant:

Recall I1 was 10A

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By Symmetry:

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Apply Correction Factor 1.25 To I1’ to Make B1 = B2

Then Apply Rotation With Point P as AxisP is Back in the Wire Plane

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Net Field At P is Now Zero

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Complete Symmetry Restored!

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References with Slide Numbers1. Find information about The Physics Teacher at: https://www.aapt.org/Publications/2. Photo Courtesy of Jamie Alonzo, Dean of Math/Science, SJCC.6. Stillwell, J. (1989). Mathematics and its History, New York, NY: Springer-Verlag, New York, p51.9. Stillwell, p54.20. Image Retrieved from: http://www.storyofmathematics.com/images2/diophantus.jpg22. A limited number of past exam questions are available to the public at:http://www.amatyc.org/?page=SMLPastQuestions23. Melblom, G., EVC, Private Communication. Material on Pythagorean formula used with permission.29. See Reference for #22.33. See a definition of the Golden Ratio in: Huntley, H.E (1970). The Divine Proportion,New York, NY: Dover, pp24-27.36. "Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons – https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif37. Image Retrieved from: https://www.google.com/search?q=binary+orbit&ie=utf-8&oe=utf-8 38. Image Retrieved from: https://www.google.com/search?q=elliptical+orbit&tbm=isch&tbo=u&source=univ&sa=X&ved=0CEkQsARqFQoTCLn6n43YtccCFcqViAodHpkE6Q39. Image Retrived from: https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A40. Image Retrieved from: https://www.google.com/search?q=magnetic+field+of+a+wire&tbm=isch&tbo=u&source=univ&sa=X&ved=0CC0QsARqFQoTCKXtlYjatccCFUk5iAodtBUMYw

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References, Cont.42. See for example: Giambattista, A., Richardson, B. & Richardson, R. (2007).College Physics, 2nd ed. Boston, MA: McGraw-Hill, p728; Walker, J. Physics, 3rd ed. (2007). New Jersey, NJ: Pearson/Prentice Hall, p766; Serway,R. & Vuille, C. College Physics, 10th ed. (2015). Boston, MA: Cengage Learning, p695; Young, H. D., Freedman, R. A. & Ford, A. L. University Physics, 13th ed. (2014).Boston, MA: Pearson, p951.

44. Image Retrieved from: http://www.amazon.com/Brilliant-Blunders-Einstein-Scientists-Understanding/dp/1439192375/ref=sr_1_1?s=books&ie=UTF8&qid=1449596241&sr=1-1&keywords=Brilliant+Blunders