Analytical Mechanics ( AM ) Olaf Scholten KVI, kamer v8; tel nr 6-55; email: scholten@kvinl Web page: http://wwwkvinl/ scholten Book: Classical Dynamics of Particles and Systems, Stephen T Thornton & Jerry B Marion ISBN: 95-556-6 ISBN: 978-95-556-7 Content: - Numerical methods for solving physics problems is stressed The subjects that are being addressed include: - Damped and driven oscillators; non-linear effects (higher harmonics, hysteresis and chaos); Fourier transforms and Greens - Variational calculus; Lagrange multipliers; Hamilton and Lagrange formalism, Lagrange density for continuous systems - Central potentials, Kepler problem, - Rotating rigid body, non-inertial systems, inertial tensors, Euler angles Generalizations to problems in Quantum Mechanics, Relativistic Mechanics, Field Theory,
November 9, 4 Advanced Analytical Mechanics (a)- - Prelim - Preliminaries website: //wwwkvinl/ scholten/aam/collegehtm Homeworks are an integral part of the course! Language English? Read through slides BEFORE lecture I may give flash exams (pop-quiz) during class Assumed prior knowledge: Mechanics- &, Complex analysis, Introduction to Programming and Numerical Methods
November 9, 4 Advanced Analytical Mechanics (a)- - Prelim - Mechanics- from website: Mechanics- Ocasys: Upon completion of Mechanics-, the student: - is able to describe the three-dimensional motion of a rigid body as a translation of and a rotation about the center of mass; - knows the definition of the moment of inertia tensor and is able to compute the principal values of this tensor for sets of discrete particles and for simple continuous bodies; - is able to apply conservation of angular momentum for bodies rotating in two and in three dimensions (including the effect of an external angular impulse); - is able to derive the complete equations of motion, ie involving translation and rotation, of rigid bodies subject to external forces and torques in two dimensions; - can explain precession of spinning objects and compute the precession frequency; - is able to identify the degrees of freedom of a mechanical system and to formulate its Lagrangian when subjected to conservative forces; - can apply constraint potentials in order to incorporate constraints in the Lagrangian; - is able to show that the conservation laws of energy, momentum and angular momentum are contained in the Lagrangian; - is able to derive the equations of motion for coupled linear oscillators, and determine eigenfrequency and eigenmodes for systems up to two degrees of freedom
November 9, 4 Advanced Analytical Mechanics (a) - Oscil - Harmonic Oscillators particle, mass m, subject to force F (x) In general: F (x) = F + x F + x F + Equilibrium point: F (x ) = choose here x = Good approximation near equilibrium: F (x) = x F = x k Equation of motion (eom) from F = m a Solution: k x = m ẍ x(t) = A sin(ωt δ) with ω = k/m and A and δ determined by initial conditions x(t = ) and ẋ(t = ) Alternative: x(t) = A + e iωt + A e iωt ω: Angular frequency (hoekfrequentie) [radians/second] ν: Frequency = ω/π oscillations per second τ : Oscillation time = /ν Example: Sphere (radius R, mass M) attached by point on its surface, making small oscillations Equation of motion: N = I θ Torque: N = R F = RMgθ moment of inertia: I = 5 MR + MR 7 5 MR θ = RMgθ ω = 5 7 g/r [prblm,,4,5,6,7,8,9]
November 9, 4 Advanced Analytical Mechanics (a)-5 - Oscil - D Harmonic Oscillator Motion in directions around equilibrium F x = k x and F y = k y Coupled equations of motion mẍ + k x = mÿ + k y = Solution, ω = k/m: ( integration constants each) x(t) = A cos(ωt α) y(t) = B cos(ωt β) special cases: α = β and α β = π/ More general F x = k x x and F y = k y y Coupled equations of motion mẍ + k x x = mÿ + k y y = [Fig ] Solution: x(t) = A cos(ω x t α) y(t) = B cos(ω y t β) Lissajous figures
November 9, 4 Advanced Analytical Mechanics (a)-6 - Oscil - One dimensional oscillator x(t) = A cos(ωt δ) Phase Diagram Here we consider position as function of time Initial conditions: value of x and ẋ at a particular time Consider x and ẋ two coordinates (in phase space) specifying position and velocity (motion) of particle at a particular time Curve given by functional p(x, ẋ) = Question: at which points (x, ẋ) in phase space can the particle be found? Answer (directly from the solution): x /A + ẋ /(ω A ) = Moves with time in clock-wise direction on circle Manipulating the eq of motion ẍ + ω x = gives dtẋ d = ω x thus dẋ dx = dtẋ d dt dx = ω ẋ x giving ẋ dẋ = ω x dx, or ẋ = ω x + C Poincaré section: Point in phase diagram at times t n ω = n π To Do: - write computer program to numerically solve eom of oscillator - make plots of x(t) and x(ẋ) - check different initial conditions
November 9, 4 Advanced Analytical Mechanics (a)-7 - Oscil4 - Computer simulations- Solving ẍ = ω x numerically: ẋ(t) = [x(t ( + δ/) x(t δ/)]/δ and thus ) ẍ(t) = [x(t + δ) x(t)]/δ [x(t) x(t δ)]/δ /δ or ( ) ẍ(t) = x(t + δ) x(t) + x(t δ) /δ and thus giving x(t + δ) x(t) + x(t δ) = δ ω x(t) x(t + δ) }{{} New point = ( δ ω ) x(t) x(t δ) }{{} previous points Initial conditions (take grid, t = n δ): x() = a and ẋ() ẋ(δ/) = [x(δ) x()]/δ = ȧ giving x(δ) = a + ȧ δ
November 9, 4 Advanced Analytical Mechanics (a)-8 - Oscil5 - Solving ẍ = ω x numerically: x(t + δ) }{{} New point Computer simulations-b = ( δ ω ) x(t) x(t δ) }{{} previous points Initial conditions (take grid, t = n δ): x() = a; x() = a Use http://wwwonlinecompilernet/fortranhtml or GFORTRAN or whatever program ho! Solving a Harmonic oscillator integer i real x(:),del,omga print *,"Hello World, give me delta and omega" read *, del,omga write(9,*) del,omga x()= ; x()= do i=,99 x(i+)=( - del** * omga**) * x(i) - x(i-) write(9,*) i+, x(i+) enddo print *,"end ho",del,omga end program ho
November 9, 4 Advanced Analytical Mechanics (a)-9 - Oscil6 - Computer simulations- Simple harmonic oscillator ẍ + βẋ + ω rx = parameters ω r =, β = (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 NL-pendulum Sun Mar 9 9 ::5 time [ sec] Initial conditions at t = : x = and ẋ = Q: take Initial conditions at t = : x = and ẋ = ω
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil7 - Damped Oscillator Eq of motion (eom): [chapter 5] ẍ + βẋ + ω rx = trial solution: x(t) = Ae γt gives γ + βγ + ω r = Solutions: γ = β ± β ω r when ω r β define ω = ω r β Solution now: γ = β ± i ω x(t) = e βt[ A e iω t + B e iω t ] Numerically: ẋ(t) = [x(t + δ) x(t δ)]/( δ) and thus giving x(t + δ) x(t) + x(t δ) = δ ω x(t) δ β [x(t + δ) x(t δ)] x(t + δ) = ( δ ω ) x(t) + ( + δ β) x(t δ) + δ β To Do: [prblm,,,,,4,6]
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil8 - Computer simulations-a Damped harmonic oscillator ẍ + βẋ + ω rx = parameters ω r =, β = 5 (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 NL-pendulum Sun Mar 9 9 :5:4 time [ sec] Initial conditions at t = : x = and ẋ = Q: Take different initial conditions [prblm 7,8,,,]
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil9 - Computer simulations-b Damped harmonic oscillator ẍ + βẋ + ω rx = parameters ω r =, β = (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 NL-pendulum Sun Mar 9 9 :6:45 time [ sec] Initial conditions at t = : x = and ẋ = Q: Change friction force
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil - Driven & Damped Oscillator Driving force F (t) = m A cos(ωt) Eq of motion (eom): [chapter 6] ẍ + βẋ + ω rx = A cos(ωt) Same as real part of ẍ + βẋ + ω rx = A e iωt trial solution: x(t) = Ae iωt (why same frequency?) gives A ω + i Aβω + Aω r = A Special solution: A = A (ω r ω ) + iβω = A e iδ with tan δ = Im(A)/Re(A) = βω/(ωr ω ) A = A / (ωr ω ) + 4β ω Special solution now: ( x(t) = A R e iδ e iωt) = A cos(ωt δ) Full sol = Special sol + sol homogeneous eq To Do: - check importance real & imaginary parts [prblm 4,5]
November 9, 4 Advanced Analytical Mechanics (a) - Oscil - Computer simulations- Damped & Driven harmonic oscillator ẍ + βẋ + ω rx = A cos(ω d t) parameters ω r =, β =, A =, ω d = (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 NL-pendulum Sun Mar 9 9 :: time [ sec] Initial conditions at t = : x = and ẋ = Q: Take different initial conditions; Change driving frequency; Change friction force
November 9, 4 Advanced Analytical Mechanics (a)-5 - Oscil - Non-linear Oscillator Spring constant k, unextended length l mass m then [chapter 4] F x = k(s l ) sin θ with s = l + x and sin θ = x/s Problem 4: for d = l l we have F x = k d x l k l x + l eom including driving force F cos ωt: mẍ + k d x l + k l x F cos ωt = l L L S x S Substitute y = x/l, ω = k d m l, ɛ = k l m l, g = F m l giving ÿ = ω y ɛy + g cos ωt
November 9, 4 Advanced Analytical Mechanics (a)-6 - Oscil - ẍ = ω x ɛx + g cos ωt Interested in the effect of the x term on the motion Assume that this correction is small and that we can write x(ɛ; t) = x () (t) + ɛx () (t) + ɛ x () (t) + explain! Substitute in eom Collect terms proportional to ɛ ẍ () = ω x () + g cos ωt gives x () (t) = g/(ω ω ) cos ωt A () cos ωt Collect terms proportional to ɛ ẍ () = ω x () [x () ] use cos ωt = 4 cos ωt + 4 cos ωt gives x () (t) = A () cos ωt + B () cos ωt with A () = 4 [A() ] /(ω ω ) and B () = 4 [A() ] /(ω 9ω ) Conclusion: Due to x term solution has frequencies ω and higher [prblm 4,,,4,5,6,7,8]
November 9, 4 Advanced Analytical Mechanics (a)-7 - Oscil4 - Computer simulations Damped & Driven Anharmonic Oscillator ẍ + βẋ + ω rx + ω rc x = A cos(ω d t) parameters ω r =, β =, A =, ω d = 8, C = (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 NL-pendulum Thu Apr 9 8:59: time [ sec] Initial conditions at t = : x = and ẋ = Q: what happens if sign C is reversed and if sign ωr is reversed?
November 9, 4 Advanced Analytical Mechanics (a)-8 - Oscil5 - [45] Hysteresis Simple estimate by assuming linear regimes with different spring constants [prblm 49,] Computer simulations-5 Oscillator with memory effect ẍ + βẋ + ω rx + ω rc x = A cos(ω d t) parameters ω r =, β =, A = 8, ω d = 8, C = Max Ampl Phase - - 4 pendulum Tue Mar 9 :9: 4 5 6 Tune up Tune down x oscil 4 5 6 Frequency Initial conditions at slowly tune up or down frequency Q: try different non-linearities and strength friction force
November 9, 4 Advanced Analytical Mechanics (a)-9 - Oscil6 - Chaos Computer simulations-6 Driven and damped non-linear Oscillator inverted linear part ẍ + βẋ + ωrx + ωrc x = A cos(ω d t) parameters ωr = 4, β =, A = 4, ω d = 4, C = (dx/dt)/ - - amplitude Phase - - - x Poincaré - - - 4 - - - - - 4 5 6 7 8 time [ sec] NL-pendulum Wed Apr 9 :: Q: try different non-linearities and strength friction force
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil7 - In the coming lectures: Fourier transform Solve damped & driven oscillator for a general periodic force [8] [prblm 7,8,9,9,4] Greens function Solve damped & driven oscillator for an impulsive force [9] [prblm,,,4,5,8] other: [prblm 4,4,44,45]
November 9, 4 Advanced Analytical Mechanics (a)- - Oscil8 - Useful sin α = eiα e iα i cos α = eiα + e iα sin α + β = cos α sin β + sin α cos β cos α + β = cos α cos β sin α sin β sin α α cos α α