Conservative Systems et  be potential energy is conservative system with total energy as conservative quan tity iven   is conservative quantity if dEdt
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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 dEdt

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 dEdt




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Presentation on theme: "Conservative Systems et be potential energy is conservative system with total energy as conservative quan tity iven is conservative quantity if dEdt"— Presentation transcript:


Page 1
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.