Fun with Formulas !

1. Fun with Formulas ! Werner Joho Paul Scherrer Institute (PSI) CH5232 Villigen, Switzerland 4.July 2003 (updated 1.10.2010) W.Joho

2. Introduction Formulas can be fun. They often can be made to look simple, transparent and thus beautiful (in the spirit of Einstein and Chandrasekhar). This can be achieved with some simple rules and a few tricks of the trade.This paper is a sample of some simple formulas, collected during my career as a physicist. The following material was presented (but not published) at a seminar talk given at the CERN Accelerator School on Synchrotron Radiation and Free Electron Lasers, Brunnen, Switzerland, 2-9 July 2003 (this file is available on the WEB with google: „JOHO PSI“) W.Joho

3. Content • philosophy for formulas • capital growth • new interpretation of Ohm‘s law • logarithmic derivatives • the relativistic equations of Einstein • the magic triangle formed by the logarithmic derivatives of the relativistic parameters • Alternative Gradient Focusing, constructed by hand • binomial curves everywhere, approximation of a variety of functions, like beam profiles, the fringe field of magnets, the flux and brightness of synchrotron radiation etc. • simple representation of phase space ellipses • how to win money with statistics ! • design of beautiful tables with a Hamiltonian W.Joho

4. philosophy for formulas simplify formulas, they should look „beautiful“ formula should indicate the proper dimensions use units of 1'000 (cm should not exist in formulas!) choose right scales for plots (e.g. logarithmic) example: c = 3·108 m/s ?? better is: c = 0.3·109 m/s or 300 m/ms or 0.3 mm/ps !! for comparison of electric forces: q·Є (kV/mm) with magnetic forces: q·v·B = q·β·c·B c = 300 (kV/mm)/T !! m0 = 4p 10-7 Vs/Am ?? better is: m0 = 0.4p mH/m = 0.4p T/(kA/mm) W.Joho

5. how to avoid akward numbers in electrodynamics W.Joho

6. use logarithmic derivatives ! W.Joho

7. Einstein triangle W.Joho

8. „Magic Triangles” (W.Joho)with logarithmic derivatives of relativistic parameter multiplication factors form inverse triangle W.Joho

9. trigonometric functions for relativistic formula ! W.Joho

10. highly relativistic case W.Joho

11. IR-FEL l* = 2‘500 mm K-edges l* = 1.5 mm SLS ESRF Undulator Radiation produced by an electron beam of energy E = gmc2 TESLA W.Joho

12. AG-focusing • simple example of alternative gradient focusing: • FODO-lattice with thin lenses (focal length f) • if L = 2f => construction is possible by hand ! • it takes 6 periods to get a 3600-oscillation • i.e. the phase advance/period is  = 600 exact solution with transfer matrices gives for L = 2f =>  = 600 (graphic example) for L = 4f =>  = 1800 (instability !) W.Joho

13. magnetic fringe field with binomial W.Joho

14. Flux Spectrum of synchrotron radiation spectral flux F of electrons with energy E and current I from a bending magnet with magnetic field B. F = 2.46 ·1013 E[GeV] I[A] G1(x) (photons/(s ·mrad ·0.1% bandwidth) x=ε/εc , ε = photon energy , εc = critical photon energy G1 x W.Joho

15. Flux-Spectrum of Synchrotron Radiation from Bending Magnet with Field B Fit with G1(x) = A x1/3 g(x) , fit of binomial g(x) with 8 data points to ±1.5%: A = 2.11 , N = 0.848 , xL = 28.17 , S = 0.0513 W.Joho

16. general binomial curves W.Joho

17. Classification of binomial curves any 3 reference points will give a fit for N, S and xL. But the chosen points top, mid and bottom allow a convenient classification in the (A,B)-Diagram W.Joho

18. properties of binomials W.Joho

19. 1 20 17 18 13 21 2 3 14 5 6 7 8 15 4 16 19 typical profiles y(u) in (A,B)-plot =square 12 short range region Gaussian 11 exponentials B=A4 10 long range region unaccessible region 9 W.Joho

20. representations of beam profiles with binomials Profiles Tails of Profiles (full width at 10% level : ≈ 4.4 σ for large range of m) W.Joho

21. clipped binomial phase space densities big trick: plot fractionwhich is outside of ellipse! Sacherer, Lapostolle εp/ε for m1.5 the curves have a crossing point at εp ≈ ε and p ≈ 13%; i.e. ca. 87% of all particles are inside an ellipse with emittance ε=(2σ)·(2σ‘) , independent of m. For a Gaussian distribution we have p=exp(-2εp/ε), which gives a straight line in this diagram (m=). W.Joho, 1980 PSI report TM 11-4 W.Joho

22. correlations x y • example: • income and research for 50 US companies in 1976 • (from journal „Physics Today“, march and september 1978 ) • x = income / sales • y = research budget / sales • There are 3 possibilities to show a correlation: • linear fit of y(x) : income stimulates research ! • linear fit of x(y) : research stimulates income ! • correlation ellipse from <x y> : high income  strong research Y= fit: y(x) fit: x(y) X= W.Joho

23. Representation of rms beam ellipse in phase space (x, x‘) W.Joho

24. The parametric representation of the rms beam ellipse in phase space (x, x‘) W.Joho

25. Dictionary for Beam Parameters W.Joho

26. Convolution of two ellipses W.Joho

27. in the the same spirit one can write: This formula from Euler combines beautifully 3 fundamental numbers in mathematics another „gem“ from Euler is: or from Ramanujan comes: (I figured this out myself, but I am sure it exists somewhere in the literature, but I could not find yet the proper reference) W.Joho

28. treacherous predictions ! 1) If you see a series of numbers: 2, 4, 6, 8, 10, 12, … created by a formula F(n), for n=1, 2, 3, …6 you probably guess, that the next term is 14 !? Now give me the number Y, your year of birth. I give you below a formula F(n), where the next term in the series (for n=7) is not 14, but exactly Y ! F(n)=2n+ (Y-14)(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/6! ------------------------ The above example was easy to construct. But what about the next treacherous example? 2) The following formula was constructed by the Swiss Physicist Leonard Euler: P(n)=n(n-1)+41 Believe it or not, but for n=1, 2, 3….up to 40 this formula gives a prime number ! It fails the first time at n=41, where P(41)=41*41=1‘681(It then fails further at n=42, 45, 50, 57, 66 etc.) W.Joho

29. winning money with statistics ! • throw simultaneously 6 dices: • If all 6 dices show different numbers (1, 2, 3, 4, 5, 6) • I give you 20 times your betting sum …..but chance is only 1.5% !! to be fair, I should offer you 65 times your betting sum! W.Joho

30. winning money with statistics ! what is the chance, that in a group of n persons, 2 people have the same birthday (disregarding the year of birth and the 29th february)? with 23 people the chance is already 50%, with 40 people it is 87% and with 80 people a double coincidence is a „sure bet“ and a triple coincidence has a 42% chance! W.Joho

31. exponential growth with compound interest With an interest rate of 2% it takes 35 years to double the income (50 years without compound interest) How can we get this result very quickly? For a quick estimate of exponential growth one can use: e7 ≈ 210 ≈ 103 example: With an interest rate of p(%) it takes T2 years to double an initial capital investment C0. (70≈100 ln2) T2 = 70 years/p(%) To have an increase by a factor of 1‘000 (≈210) it takes T1000 years: T1000 = 10 T2 = 700 years/p(%) W.Joho

32. Growth of Capital • William Tell deposited 1 Fr.in a bank account, 700 years ago! • => assume he gets 3% interest before taxes, and 2% netto after taxes • The difference goes to the government (which does not pay taxes)!! • after 700 years the government has 109 Fr. the descendents of William Tell „only“ 106 Fr. !? W.Joho

33. „Hamiltonian Table“ You want to construct a nice table for your living room with this shape? W.Joho

34. „Hamiltonian“ Plots Try to plot a curve F(x,y)=const.=c , where F(x,y) can be quite complex like W.Joho

35. Example of „Hamiltonian“ Plotswith F(x,y)=const. You want to construct a nice table for your living room with one of these shapes? I give you the corresponding parameters for a modest royalty! W.Joho

36. „real fun“ with formulas ! two very famous equations are: a2 + b2 = c2Pythagoras (500 BC) E = mc2 Einstein (1905) with some easy Algebra we get: a2 + b2 = E/m Pythagoras–Einstein-Joho (2008) The Swiss physicist Paul Scherrer gave a beautiful analogy for the famous Einstein equation: „This energy E (=mc2) is deposited on a blocked bank account“! (this analogy is mentioned by the author Max Frisch in his book „Stiller“) W.Joho

37. Varia W.Joho

38. photon energy   wavelength  use of magic numbers to memorize the relation  = 1240 eV nm (=hc)trick=>takesquare root ! 35 eV  35 nm VUV-region soft X-rays 1.1 keV  1.1 nm „old fashioned“ 3.5 keV  3.5 Å „infrared-people“ use wavenumber k in [cm-1] 100 cm-1 100 m (correlations for arbitrary numbers are then quickly estimated by multiplication resp. division) W.Joho

39. Graphical solution of the lens equation the lens equation (Newton) solved for a thin lens with focal length f same graph for resistances in parallel, capacitances in series etc.! W.Joho

40. Brightness of Synchrotron Radiation from Bending Magnet with Field B Fit with H2(x) = A x2/3 h(x) , fit of binomial h(x) with 8 data points to ±2%: A = 2.95 , N = 1.11 , xL = 1.336 W.Joho

41. binomials W.Joho

42. Heart  Motor Who is more reliable, your heart or the motor of your car ? • Assumptions: • a car makes about 200’000 km with an average speed of 40km/h => runs for about 5’000 h or 300’000 min. • the motor runs at an average of 2’000 cycles/min • => the motor makes about 0.6 ·109cycles, If you live 80 years, your heart has made about 2.5 ·109heart beats=> your heart will make about a factor 4 more cycles than the motor of a car!! by the way: during your life you experience some special dates: after ≈ 11 years and 41 weeks you lived 100’000 h after ≈ 27 years and 20 weeks you lived10’000 days after ≈ 31 years and 36 weeks you lived 109 s W.Joho