onservative system cannot have any attracting 64257xed points ne generally 64257nds saddles and centers brPage 2br Particle Moving in DoubleWell Potential onsider particle 1 with Vector 64257eld y ixed points 0 0 saddle 0 centers rajectorie ID: 27060
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ConservativeSystems I L etV(x)be potentialenergy ,mx=�Vxisconservativesystemwith totalenergy asconservativequan-tity:E=1 2m_x2+V(x): I G iven_x=f(x),E(x)isconservativequantityifdE=dt=0. I C onservativesystemcannothaveany attracting xedpoints. I O negenerallynds saddles and centers . ParticleMovinginDouble-WellPotential I C onsiderparticle(m=1)withV(x)=�1 2x2+1 4x4. I Vectoreld _x=y;_y=x�x3: I F ixedpoints:(0;0)( saddle ),(1;0)( centers ). I T rajectoriesareclosedcurvesand contours ofconstantenergy:E=1 2y2�1 2x2+1 4x4: I P eriodicorbits. Homoclinicorbits . Energysurface . I TheoremofNonlinearCentersforConservativeSystems . NonlinearCentersforReversibleSystems I T heorem:Supposex=0isalinearcenterforareversiblesystem,thenalltrajectoriesare closedcurves neartheorigin. I S howthat_x=y�y3;_y=�x�y2hasa nonlinearcenter andplotphaseportrait. I S addles((�1;1)). Heteroclinicconnections . GeneralDenitionofReversibility I _x=f(x)is reversible if invariant undert!�t;x!R(x)withR2(x)=x. I S howthat_x=�2cos(x)�cos(y);_y=�2cos(y)�cos(x)is reversible but notconservative .Plotphaseportrait. I Reversible withR(x;y)=(�x;�y). I F ixedpoint(� 2;� 2)isattracting. Notconservative . PendulumEquation I I fnodampingandexternaldriving,motionofpendulumd2 dt2+g Lsin=0: I W ith =!t(dimensionlesstime)where!=p g=L(frequency),_=;_=�sin:whereis(dimensionless)angularvelocity. AddingLinearDamping I Equation becomes+b_+sin=0: I Centers become stablespirals . Saddles remain saddles . I C hangein energy alongatrajectorydE d =_(+sin)=�b_20: I W hirlclockwise.Settleintosmalloscillation.Cometorest.