1. Using Math Symmetry Operations to Solve a Problem in Elementary Physics Submitted 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 1

2. 2

3. 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 3

4. Apply a Translation and Magnification: Make the Substitution for x: 4

5. Substitute x’ = 0 Into the Transformation Equation 5

6. 2: Quadratic Equations: or 6

7. Translation Eliminates Linear Term: With This Substitution, Equation Becomes: 7

8. Solution to Transformed Equation: Undo the Translation: 8

9. 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? 9

10. Cubic Equation: Creative Step: Do a Translation to Eliminate Quadratic Term: 10

11. New Equation: Coefficients p, q Combinations Of A, B, and C Good Algebra Exercise for Students! 11

12. Results for p and q: and Check My Algebra! 12

13. Change of Variables on Left Side! So we have: Algebra! 13

14. Recall x’ = u + v We Claim: and Why! Hint: x’ is Arbitrary! 14

15. Eliminate v Between Equations for p and q Get a Quadratic in u Cubed Solution: 15 More Good Algebra for Students!

16. But We Had: and Top Equation Symmetric With Respect to Interchanging u and v So u and v Have Same Solutions! with 16

17. Solution for v: Use “+” for u and “-” for v To Satisfy 17

18. But Recall: So We Have: 18

19. Finally, Undo the Translation: We Could Also Examine The Quartic But Alas, Not the Quintic or Higher 19

20. 20 http://www.storyofmathematics.com/images2/diophantus.jpg Diophantus, 2 nd Century AD Hellenistic Mathematician

21. 21 The Most Famous Diophantine Equation! The Pythagorean Theorem Find Integer Solutions to: How Do We Find Them?

22. 22 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

23. 23 Is There A Formula That Generates All of the Pythagorean Triples? YES! And It Is Very Clever!

24. 24 Here is the Formula: With m, n Arbitrary Integers Gregory Melblom, Private Communication

25. 25 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!

26. 26 Add a Common Factor of k:

27. 27 A Right Triangle With Perimeter = Area Set A = P and Cancel Common Factors

28. 28 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

29. 29 Consider the Following Equations What is the Ratio n/m? Problem 20 Math Contest 2015 Round One www.AMATYC.org Another Great Problem!

30. 30 Use the Definition of Logs: Original Equation Becomes:

31. 31 Divide Both Sides By Smallest Term: Rewrite As:

32. 32 Make a Substitution: And We Have: Equation for the Golden Section!

33. 33

34. 34 And the Answer to the Problem is:

35. An Example of Symmetry Transformations In Physics Reduction to the Equivalent One-Body Problem Consider a Binary Star System 35

36. "Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif 36

37. Two Stars Orbit in Ellipses About Common Center of Mass Quantity of Interest: Separation R(t) Two-Body Problem 37

38. Apply Transformation:“Mass”  Orbits Fixed Force Center. Separation R(t) and Force Left Invariant 38

39. Magnetic Field, North and South Poles https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A 39

40. Magnetic Field of a Wire No North and South Poles! 40

41. Magnitude of the B Field Field Strength Proportional to Ratio of Current to Distance To Keep Field Invariant, Change Current and Distance by Same Factor 41

42. Original Version of Problem 42

43. Symmetry Transformation: Move Wire 2 Into Symmetric Position 43

44. 44

45. Questions Why is I 2 ’ Up? (I 2 was Down) Claim: I 2 ’ = I 1 = 10A. Why? Distance of I 2 ’ Reduced By Factor of Three. What was Original I 2 ? Original Problem Solved! 45

46. More General Problem, Point P Out of the Plane of the Wires 46 Find the Unknown Current I 2

47. Find the Base Angles With Laws of Cosines and Sines 47

48. Move I 1 Towards Point P To Make B Net Horizontal But No Change in Magnitude Base Angles Change by 5 Degrees Why? 48

49. New Base Angles The New Distance a of I 1 49

50. Original Distance of I 1 was 4 New Distance of I 1 ’ is 3.05 So To Keep B Net Invariant: Recall I 1 was 10A 50

51. By Symmetry: 51

52. Apply Correction Factor 1.25 To I 1 ’ to Make B 1 = B 2 Then Apply Rotation With Point P as Axis P is Back in the Wire Plane 52

53. 53

54. Net Field At P is Now Zero 54

55. Complete Symmetry Restored! 55

56. 56 References with Slide Numbers 1.Find information about The Physics Teacher at: https://www.aapt.org/Publications/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.jpghttp://www.storyofmathematics.com/images2/diophantus.jpg 22. A limited number of past exam questions are available to the public at: http://www.amatyc.org/?page=SMLPastQuestions 23. 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.gif 37. 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=https://www.google.com/search?q=elliptical+orbit&tbm isch&tbo=u&source=univ&sa=X&ved=0CEkQsARqFQoTCLn6n43YtccCFcqViAodHpkE6Q 39. Image Retrived from: https://www.google.com/search?q=magnetic+dipole&tbm=https://www.google.com/search?q=magnetic+dipole&tbm isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A 40. Image Retrieved from: https://www.google.com/search?q=magnetic+field+of+a+wire&tbm=https://www.google.com/search?q=magnetic+field+of+a+wire&tbm isch&tbo=u&source=univ&sa=X&ved=0CC0QsARqFQoTCKXtlYjatccCFUk5iAodtBUMYw

57. 57 References, Cont. 42. See for example: Giambattista, A., Richardson, B. & Richardson, R. (2007). College Physics, 2 nd ed. Boston, MA: McGraw-Hill, p728; Walker, J. Physics, 3 rd ed. (2007). New Jersey, NJ: Pearson/Prentice Hall, p766; Serway,R. & Vuille, C. College Physics, 10 th ed. (2015). Boston, MA: Cengage Learning, p695; Young, H. D., Freedman, R. A. & Ford, A. L. University Physics, 13 th ed. (2014). Boston, MA: Pearson, p951. 44. Image Retrieved from: http://www.amazon.com/Brilliant-Blunders-Einstein-Scientists-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