The dynamics of the simple pendulum

Transcription

1 .,, 9 G. Voyatzis, ept. of Physics, University of hessaloniki he ynamics of the simple penulum Analytic methos of Mechanics + Computations with Mathematica Outline. he mathematical escription of the moel. he qualitative escription of the ynamics 3. Approximate solutions 4. he analytic solution

2 .,, 9 he mathematical escription of the moel z m r F, F B, B mgk r e e ( r r ) r ( r r ) B mg e mg sine e r r ml mg ( a) ml mg sin ( b) () l () x e e r Also, an ml V mgzmgl V ml mgl (3) quation ()a gives the tension applie by the rob 3 ml mg mgl (4) l > or < > < Penulum with rob Penulum with fiber We can show that 3, if mgl quation ()b gives the equation of motion or g l sin si n, g / l (5)

3 .,, 9 he integral of the O.D.. his is a n orer OD, autonomous, nonlinear an its integral is obtaine as (5) sin t const (6) he potential V V ( ) of (5) is V sin V sin V ( ) (7) V() /t -ð -ð ð ð quilibrium points : sin=,,4,... V '',3,5,... V '' Remark: We observe that the OD of motion is invariant uner the translation k, k Z, consequently, the system can be stuie in the interval or.

4 .,, 9 he phase space portrait è. è ṁax - - s S With of Separatrix 4 he three types of motion. S (v) (Librations) (Rotations) S (v) è Amplitue of libration For (), () we get or min arc, S (Separatrix) (v) 3 4 t S Perio of librations From the energy integral we get t (8) t ( ) Consiering the motion from ( t) to ( t), the integration of (8) gives t t (9) ( ) O Subsequently, ue to the symmetry of motion, we get 4 ( ) = min = = PND.NB

5 .,, 9 Approximations sin, V ( ) sin sin ( ) sin ( ) ( ) O( ) = (stable equilibrium) 6 3 sin... he st orer ( linear) approximation, V ( ) ( t) ( t) p sin( t), (), p () /, / he approximation up to 3 r orer , V ( ) V() he potential of the original system, the 3 r orer approximation an the linear approximation.

6 .,, 9 Approximate solution using aylor series We expan the solution (t), with initial conitions ()= an () p, as series aroun t= up to 3 r orer (3) () () 3 4 ( t) pt t t O( t ) Replace the above solution to the 3 r orer approximation of the OD 3 (3) () ( p p ) t O( t ) 6 We keep terms up to st orer (with respect to t) because they are sufficient to etermine the quantities () an () namely Put on the same orer terms 3 () 6 (3) p p (3) p () ( 6), () ( ) 6 We replace () an (3) () in the solution an we 3 4 ( t) pt ( 6) t p( ) t O( t ) Remark. he above solution is a vali approximation only in a small time interval tt, t. In orer to construct an approximate solution in an interval ( t, t ) we procee step by step applying the series solution for a small time step. PND.NB

7 .,, 9 he analytic solution. he moern way eq=''[t]+^ Sin[[t]]; sol=dsolve[eq,,t] Solve::ifun: Inverse functions are being use by Solve, so some solutions may not be foun; use Reuce for complete solution information. More Functiont,JacobiAmplitue Functiont,JacobiAmplitue C tc 4, C, C tc 4, C We obtain that the solution is given by the special function which takes two arguments, u an k : u JacobiAmplitue(u,m), CtC m 4 C C[] an C[] are the arbitrary constants of integrations which are relate to initial conitions. But if we solve the initial value problem DSolve[{eq, [], '[]},, t] we get {}. he analytic proceure (Librations) he solution is base on the elliptic integral of st kin * (llipticf[,k]) u F(, k) k: moulo of u k=mou z : amplitue of u =am(u,k) : JacobiAmplitue[u,k ] lliptic trigonometric functions are efine through the amplitue : sn(u,k)=sin(), cn(u,k)=() z * Harol Davis, Introuction to nonlinear ifferential an Integral equations, Dover publ., 96.

8 .,, 9 Consiering a libration of amplitue, i.e. an orbit of energy energy integral gives, the t () We procee to transformations of the left part of () in orer to integrate it. a) u u thus sin, b) : sin, Note that sin sin () sin( / ) an, / sin( / ) hus () sin sin k k By ifferentiating the transformation formula : k k k ( / ) sin ( / ) k (3) Substituting () an (3) in (), we get t ( t t), ( t) (4) a) Integration of (4) by using the expansions : x x x x... ( x ) sin ( sin ), sin ( 8sin sin 4 ), etc. / sin n 3 5 (n) 4 6 n

9 .,, 9 b) use of elliptic integrals Let consier initial conitions () i.e. (), () t F(, k) t am( t, k) or sin ( t) ArcSink sin (5) ( t) ArcSin k sn( t, k) or where k sin Perio of librations We recall that F(, k) t t F(, k), /, thus 4 4 F(, k) K( k) K(k)=F(/,k) : he complete elliptic integral of st kin (lliptick[k ]) PNDan.NB PNDnum.NB xercise: Show that the solution of the separatrix trajectory is given by the formula 4 t 4arctan e tan, ()