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Unformatted text preview: C Dynamical S C A L T E C H Final Project Shane D. Ross Control and Dynamical Systems, Caltech www.cds.caltech.edu/ shane/cds140b CDS 140b, February 5, 2004 Final Project Issues to address in project Equilibrium points Periodic orbits low order analytical approximations More accurate p.o.s Higher order numerical approximations of p.o.s using differential correction and continuation To be covered later Invariant manifolds of p.o.s Poincar e section 2 Example Problem Planar, circular, restricted 3body problem (3BP) From Chapter 2 of KLMR book (on class website) P in field of two bodies, m 1 and m 2 x y frame rotates w.r.t. X Y inertial frame Y X x y t P m 2 m 1 3 Example Problem Equations of motion describe P moving in an effective potential U ( x.y ) in a rotating frame x m 1 m 2 P (,0) ( 1,0) ( x ,y ) y U ( x,y ) _ L 4 L 5 L 3 L 1 L 2 Position Space Effective Potential 4 Example Problem Point in phase space: q = ( x y v x v y ) T R 4 Equations of motion, q = f ( q ) , are x = v x , y = v y , v x = 2 v y U x , v y = 2 v x U y , where U ( x, y ) = 1 2 ( x 2 + y 2 ) 1 r 1 2 r 2 where r 1 and r 2 are the distances of P from m 1 and m 2 5 Example Problem and the only parameter of the system is = m 2 m 1 + m 2 where (0 , . 5) 6 Equilibrium points Find q = ( x y v x v y ) T s.t. q = f ( q ) = 0 Have form ( x, y, , 0) where ( x, y ) are critical points of U ( x, y ) , i.e., U x = U y = 0 , where U a = U a U ( x,y ) _ L 4 L 5 L 3 L 1 L 2 Critical Points of U ( x, y ) 7 Equilibrium points Consider xaxis solutions U x = U y = 0 polynomial in x depends on parameter 2.0 1 . 5 1 .00. 5 0. 5 1 .0 1 . 5 2.0 4 . 5 4 .0 3 . 5 3 .02. 52.0 1 . 5 x U ( x , 0) _ L 3 L 1 L 2 The graph of U ( x, 0) for = 0 . 1 8 Equilibrium points Phase space near equilibrium points Transform coordinates, placing q at origin, q = q + u Linearize vector field about q q = q + u = f ( q ) + Df ( q ) u + O (  u  2 ) ....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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