MATHPHYSICSThe Combination of Physics and MathMotion of double pendulum1/16 College of the Redwoodshttp://online.redwoods.cc.ca.usThe Double PendulumLiHong Huang Hermanemail: [email protected]/16 MATHAssuming the there exists and admissible function y(x)that minimizes the integralI =Zx2x1f(x, y, y0)dx. (1)Let η(x) be any function with the properties that η00(x) iscontinuous andη(x1) = η(x2) = 0. (2)If α is a small parameter, then¯y(x) = y(x) + αη(x). (3)3/16 And if the well-defined real number I is in terms of α,thenI(α) =Zx2x1f(x,¯y,¯y0)dx=Zx2x1f[x, y(x) + αη(x), y0(x) + αη0(x)]dx.If we differentiate function I with respect to α1, we haveI0(α) =Zx2x1∂∂αf(x, ¯y, ¯y0)dx. (4)1We are trying to find the minimum point.4/16 After series of calculation, we obtainddx(∂f∂y0) −∂f∂y= 0, (5)which is Euler Lagrangian Equation.5/16 PHYSICSAfter some sophisticated argument in physics, we ob-tain the Lagrange’s equations,ddt(∂L∂˙qk) −∂L∂qk= 0 (6)where L is the Lagrange’s equations describing the motionof a particle in a conservative force field. AndL = T − V. (7)T is the kinetic energy, V is the potential energy, qkis thedisplacement in any direction.6/16 The Combination of Physics and MathThis is a figure of double pendulum.7/16 xy+θ1θ2M1L1M2L2Figure 1:8/16 If we model the double pendulum, we obtainx1= l1sin θ1(8)x2= l1sin θ1+ l2sin θ2(9)y1= l1cos θ1(10)y2= l1cos θ1+ l2cos θ2(11)By using the Lagrange’s equations, we have¨θ1=g(sin θ2cos(4θ) − u sin θ1) − (l2˙θ22+ l1˙θ12cos(4θ)) sin(4θ)l1(u − cos2(4θ)¨θ2=gu(sin θ1cos(4θ) − sin θ2) + (ul1˙θ12+ l2˙θ22cos(4θ)) sin(4θ)l2(u − cos2(4θ),where 4θ = θ1− θ2and u = 1 + (m1/m2).9/16 Motion of double pendulum1. Periodic2. Quasiperiodic3. Chaotic10/16 PeriodicUse this input, we will find the periodic motion:M1 = 3M2 = 3L1 = 4L2 = 3AnV e1 = 0AnV e2 = 0Angle1 = pi/4Angle2 = pi/4tolerance = 1e − 006power = 1/311/16 QuasiperiodicUse this input, we will find the quasiperiodic motion:M1 = 3M2 = 3L1 = 4L2 = 3AnV e1 = 0AnV e2 = 10Angle1 = pi/2Angle2 = pi/2tolerance = 1e − 006power = 1/412/16 ChaoticUse this input, we will find the chaotic motion:M1 = 10M2 = 1L1 = 3L2 = 3AnV e1 = 2AnV e2 = 10Angle1 = piAngle2 = pitolerance = 1e − 006power = 1/313/16 References[1] David Arnold 2002 class notes[2] Robert L. Devaney Blowing Up Singularities inClassical Mechanical Systems, American Mathe-matical Monthly, Volume 89, Issue 8 (Oct,. 198 2),535-552[3] Robert L. Devaney The Exploding Exponential andOther Chaotic Bursts in Complex Dynamics, Amer-ican Mathematical Monthly, Volume 98, Issue 3(Mar., 1991), 217-233.[4] Peter M. Gent Pursuit Curves and Matlab[5] Franziska von Herrath and Scott Mandell14/16 http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp00/FranScott/finalpaper.pdf[6] Kenneth R. Me yer The Geometry of Harmonic Os-cillators, American Mathematical Monthly, Volume97, Issue 6 (Jun. - Jul., 1990), 457-465[7] Erik Neumannhttp://www.myphysicslab.com/dbl pendulum.html[8] A. Ohlhoff and P.H. Richter Forces in the DoublePendulum[9] Dave Petersen and Zachary Danielsonhttp://www.student.northpark.edu/petersend1/double pendulum.htm[10] John Pappas15/16 http://artemis1.physics.uoi.gr/∼rizos/diplomatikes/pappas j/pendulum/enpendindex.html[11] http://www.zarm.uni-bremen.de/2forschung/grenzph/ohlhoff/dynsys/pendel/index.htm[12] Doug Saucedo Latex experties[13] Gilbert Strang 1998 Introduction To Linear Algebra[14] Troy Shinbrot, Celso Grebogi, Jack Wisdom, andJames A. Yorke Chaos in a double pendulum, June1992 American of Physics Teachers[15] Eric W. Weissteinhttp://scienceworld.wolfram.com/physics/DoublePendulum.html[16] Jack Wisdom16/16 http://geosys.mit.edu/∼solar/text/node2.htmlIN SCIENCE, THERE IS ONLY PHYSICS; ALL THEREST IS STAMP COLLECTINGTHE
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