Autonomous Mechanical systems of 1 degree of freedom

Transcription

1 Autonomous Mechanical systems of degree of freedom = f (, ), = y = y = f (, y) = f (, ) : position y : velocity f : force function Solution : = ( t; c, c), y = y( t; c, c ), ci = ci(, y) Integral of motion : F(, y) = c, c = c(, y ) Analytic solutions / integrals?

2 Autonomous Mechanical systems of degree of freedom Eample. = η = y, y = y d d d dy = = = dt d dt d y y dy y = y dy = d y = + c d We consider c > ( ) d d y = = ( + c ) = dt tan ( / c ) = c t + c dt c + d c ( t) = c tan( c t + c ) and y( t) = = ( c ) dt cos ( ct + c ) F(, y) = y = c Integral of motion or. c =, c = sin tan c y

3 Autonomous Mechanical systems of degree of freedom Eample. = ( + ) = ydy / d ydy = + = + + y d y ln y c y ln + y = c Integral of motion d = y =...? dt

4 Autonomous Mechanical systems of degree of freedom General features = f (, ), or y = y = f (, y) Dimensions : (dynamical variable and y) Phase Space : The plane position-velocity (,y) Autonomous system : phase space trajectories do not intersect on the plane (,y) Area Preservation div f f f y f (, y) f (, y) y y = + = + = div f f = = y conservative systems

5 Part A. Conservative Systems = f( ) or = y, y = f ( ) where U=U() the potential function U ( ) = f ( ) d E = y + U( ) = E σταθ., E = E y The energy integral = (, ) Time Evolution analytic solution () =, E = E d inv t = t = t( ; ) = ( t; ) ( E U( )) = (initial conditions) Equilibrium solutions (points) ( t) =, y( t) = y t critical points of vector field y =, f ( ) = du f( ) = f( ) = = see math3.nb d = etrema of potential function

6 Part A. Conservative Systems Phase space trajectories f(,y)=c are given analytically by the energy integral E = y + U( ) = E Each trajectory corresponds to a constant value of E. An energy value E may corresponds to more than one trajectories (different orbits with the same energy) Phase space trajectories are symmetric with respect to ais y= y = ( E U ( ) Flow is directed from left to right for y> and the opposite for y< If a trajectory intersects the ais y=, this intersection should be perpendicular i.e. the vector field on the y= ais is vertical to it, f=(, f() ) Phase space diagram or Phase Portrait : a sufficient collection of trajectories for identifying all possible trajectories see math3.nb

7 Part A. Conservative Systems Restriction of range of motion (valid -interval) E U( ) min min ma ma unbounded motion to both directions unbounded motion on the right unbounded motion on the left bounded motion * At = min and = ma is y= bounded motion closed phase space trajectory periodic motion (oscillation) period of oscillation T = ma min d ( E U ( )) see math33.nb

8 Conservative Linear Systems. Harmonic oscillator (elliptic system) k =, k or + =, = k solution : Energy : ( t) = cos( t) + y sin( t), y( t) = y cos( t) sin( t) U =, E = y + U,.8.6 trajectories : y + = E E/ Range : E E period : T = independent of the initial conditions! see math35harmonic.nb

9 Conservative Linear Systems. Repulsive force proportional to the distance (hyperbolic system) = k, k o r a = solution : ( t) = cosh( at) + y sinh( at), y( t) = y cosh( at) + a sinh( at) a Energy : trajectories : Range :,, U = a E = y + U y = E E / a if E E E if E or a a Asymptotic orbits : E = y = a - see math35hyperbolic.nb

10 Part A. Conservative Systems Linear stability of equilibrium points ODEs : = y, y = f ( ) equilibrium solution (EQP): =, y= solution near the equilibrium solution : = + Δ, y= + Δy ODEs = y, y = f( +) equations of deviations from EQP (after st order Taylor epansion of f ) k : linear stability inde,, df = y y = k k = = d d V, k> harmonic oscillations linearly stable EQP k< hyperbolic motion linearly unstable EQP k= critical case d or = k Linear ODE of deviations

11 Part A. Conservative Systems Linear stability of equilibrium points Phase space topological structure near Equilibria Linear stability Linear instability

12 Part A. Conservative Systems General study of Dynamics Find EQPs and their linear stability Plot The phase space portrait including sufficient orbits (Energy levels : below, between and above EQPs). Determine the trajectories of asymptotic solutions, which have the same energy level with the unstable EQPs. The set of the above asymptotic trajectories (for each EQP) is called the separatri curve Separatri = stable + unstable manifold The separatri may form close loops containing closed-periodic trajectories Determine the periods of the periodic trajectories. Close to the stable EQP the period is approimated given by the linear system. As the separatri is approimated then T. Eample : Strongly nonlinear oscillator Eample : 3-mode oscillations in polynomial potential Eample 3 : Oscillations in the Yukawa potential math37eample.nb math37eample.nb math37eample3.nb

13 The dynamics of the simple pendulum mr = F, F = B + T, B = mgk in polar coordinates r = e + + e ( r r ) r ( r r ) B = mg cose mg sine T= Te r r ml = mgcos T ( a) ml = mgsin ( b), cos cos ( ) * T = ml V = mgz = mgl ml mgl = E const ( b) = sin, = g / l f = sin V = f d = cos cos = E (Energy integral * ) ( E = E / ml )

14 The dynamics of the simple pendulum Equilibrium points : sinθ= =,,4,... V '' = ευσταθεια =,3,5,... V '' = σταθεια Remark: We observe that the ODE of motion is invariant under the translation + k, kz i.e. the system can be studied in the interval -π θ π

15 The dynamics of the simple pendulum Energy of Separatri = Energy of unstable equilibrium point Width of Separatri Types of motion E<E S : Librations E>E S : Rotations E=E S : Asymptotic motion E = = arccos, E ES Amplitude of libration ma min Α θ = θ ma -θ min = θ ma T Period of librations ma d = 4 ( E + cos ) Period of Rotation T = d ( E + cos ) PND.NB

16 Εξάσκηση : Ενδεικτικές Ασκήσεις του Βιβλίου που δεν απαιτούν Mathematica 3.3., 3.3., (χωρίς το Τ), 3.4.