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 Download Pdf

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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

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Conservative Systems et ) be potential energy is conservative system with total energy as conservative quan- tity: iven ), ) is conservative quantity if dE/dt = 0. onservative system cannot have any attracting ﬁxed points. ne generally ﬁnds saddles and centers

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Particle Moving in Double-Well Potential onsider particle ( = 1) with ) = Vector ﬁeld y, ixed points: (0 0) ( saddle ), ( 0) ( centers ). rajectories are closed curves and contours of constant energy: eriodic orbits. Homoclinic orbits Energy surface Theorem of Nonlinear Centers for

Conservative Systems

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Reversible Systems echanical systems ) have time-reversal symmetry. ts vector ﬁeld y, stay the same if t,y f ( ,y )) is a solution, then so is ( )). very trajectory has a twin, they diﬀer only by time-reversal and a reﬂection in -axis. x,y x,y ) is reversible if invariant under t,y x, ) = x,y ,g x, ) = x,y )).

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Nonlinear Centers for Reversible Systems heorem: Suppose = 0 is a linear center for a reversible system, then all trajectories are closed curves near the origin. how that has a nonlinear center and plot phase

portrait. addles (( 1)). Heteroclinic connections

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General Deﬁnition of Reversibility ) is reversible if invariant under t, ) with ) = how that cos cos cos cos is reversible but not conservative . Plot phase portrait. Reversible with x,y ) = ( x, ). ixed point ( ) is attracting. Not conservative

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Pendulum Equation f no damping and external driving, motion of pendulum dt sin = 0 ith ωt (dimensionless time) where g/L (frequency), ν, sin θ. where is (dimensionless) angular velocity.

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Nonlinear Centers and Saddle Points ixed

points: ( kπ, 0). Nonlinar center at (0 0). Reversible τ, ). Nonlinear center at (0 0). Conservative cos ). Saddle at ( π, 0). 1 with = (1 1). traight down. Librations. Inverted pendulum. Rotations.

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Adding Linear Damping Equation becomes + sin = 0 Centers become stable spirals Saddles remain saddles hange in energy along a trajectory dE d + sin ) = hirl clockwise. Settle into small oscillation. Come to rest.

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