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cds140b-2004-Lecture5B

# cds140b-2004-Lecture5B - ALTEC Dynamical S H C C Final...

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C D ynamical 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

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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 3-body 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

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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 (-μ,0) ( 1 -μ,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 ) - μ 1 r 1 - μ 2 r 2 where r 1 and r 2 are the distances of P from m 1 and m 2 5

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Example Problem and the only parameter of the system is μ = m 2 m 1 + m 2 where μ (0 , 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 , 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

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Equilibrium points Consider x -axis solutions ¯ U x = ¯ U y = 0 polynomial in x depends on parameter μ -2.0 - 1 . 5 - 1 .0 -0. 5 0 0. . 5 1 .0 1 . 5 2.0 - 4 . 5 - 4 .0 - 3 . 5 - 3 .0 -2. 5 -2.0 - 1 . 5 x U ( x , 0) _ L 3 L 1 L 2 The graph of ¯ U ( x, 0) for μ = 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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