fa_09lecture_set4 - 1 & We are focused on oscillators...

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Unformatted text preview: 1 & We are focused on oscillators (undamped) Then, is a first integral of motion , i.e., for all solutions. defines curves in the (x 1 ,x 2 ) phase plane along which H is a constant. Since H is a constant along a trajectory, the curves of constant H are a trajectory itself. Thus, level curves of H are the trajectories ! 1 2 2 x x V x in first order form : x V / x x = + = &- = - & && & 2 2 2 1 H x / 2 V(x) x / 2 V(x ) = + = + & H = & 2 2 1 H x / 2 V(x ) = + & Ex : Undamped pendulum The total energy function H is: Total energy and level curves for the undamped (nonlinear) pendulum n ( 1) = 2 2 1 1 pot. energy H x /2 cosx (x ) =- = 2 & Ex: Simple harmonic oscillator: The equation of motion, and the total energy function H are Total energy and level curves for the undamped simple (linear) harmonic oscillator 1 2 2 2 2 1 2 1 x x x x and H x / 2 x / 2 x x = & + = = + = - & && & Equilibria and Stability The equation of motion in first-order form is: The equilibrium points correspond to where the RHS is zero: (or, points where the potential energy has an extremum ) = = 2 1 x 0, V / x = = - & & 1 2 2 1 x x , x V / x x V x * x * x * x * 3 & Motions near a local minimum Consider a potential function, and the related dynamic system. The construction here shows that the solution curves are level curves for constant total energy h. The total energies satisfy h 1 <h * <h 2 . Around an equil. Pt. which is a local minimum, the orbits are closed curves - the motion is periodic. x V x * x 1 x 2 1 x 1 x h 2 h * h 1 level curve Lagrange - Dirichlet theorem: Equilibria corresponding to local minima of the potential energy are Lyapunov stable (nonlinearly stable); note that for an equilibrium point to be a local minima , Motions near a local maximum A local maximum is an unstable equilibrium ; for it one can check that 2 2 V/ x=0, and V/ x >0 x V x * x 1 x 2 1 x 1 x h 2 h * h 1 level curves 2 2 V/ x=0, and V/ x <0 4 & Motion near a Cusp or an inflexion point Clearly, the equilibrium is unstable One can think of this as the merging of a center on the right & saddle on the left. Note that at this equilibrium point, x V x * x 1 x 2 1 x 1 x h 2 h * h 1 level curves 2 2 3 3 V/ x=0, V/ x =0, and V/ x Example: Consider a system with potential function given by Potential function and phase portraits then are: = =- + + 2 4 5 6 x x V(x) mgy(x) mg( 0.01x 0.1x ) 2 2 V(x) x * 1 x * 2 x * 3 x * 4 x * 5 x 5 & A Homoclinic orbit : & - limit sets are the same saddle point. A Heteroclinic orbit has different saddle points as its & - limit sets....
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This note was uploaded on 12/28/2011 for the course ME 580 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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fa_09lecture_set4 - 1 & We are focused on oscillators...

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