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Equimass n-Body Problem

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New Orbits for the Equimass n Body Problem Robert J Vanderbei The LOQO Guy Home Page Title Page Contents May 14 2004 Workshop on Large Scale Nonlinear and Semidefinite Programming Waterloo Ontario JJ II J I Page 1 of 9 Go Back Full Screen Close Operations Research and Financial Engineering Princeton University http www princeton edu rvdb Quit Optimization minimize f x subject to b h x b r l x u Linear Programming LP f and h are linear Home Page Title Page Contents Convex Optimization f is convex each hi is concave and r JJ II Nonlinear Optimization f and each hi is assumed to be twice differentiable J I Page 2 of 9 Go Back Generally we seek a local solution in the vicinity of a given starting point If problem is convex which includes LP any local solution is automatically a global solution Full Screen Close Quit Least Action Principle Home Page Given n bodies Title Page Let mj denote the mass and zj t denote the position in R2 C of body j at time t Contents JJ II J I Page 3 of 9 Action Functional Z A 2 X mj 0 j 2 2 kz j k mj mk dt kz z k j k j k k j Go Back X Full Screen Close Quit Equation of Motion First Variation X X X zj zk zj zk dt mj mk A mj z j z j 3 kzj zk k 0 j j k k j Z 2 X X X zj zk mj z j zj dt mj mk 3 kzj zk k 0 j k k6 j Z 2 Home Page Title Page Contents JJ II J I Setting first variation to zero we get Page 4 of 9 mj z j zj zk mj mk 3 kz j zk k k k6 j X j 1 2 n 1 2 Go Back Full Screen Note If mj 0 for some j then the first order optimality condition reduces to 0 0 which is not the equation of motion for a massless body Close Quit Periodic Solutions We assume solutions can be expressed in the form X zj t k eikt k C Home Page k Title Page Writing with components zj t xj t yj t and k k k we get x t a0 y t b0 X k 1 X ack bck cos kt cos kt ask bsk sin kt sin kt Contents JJ II J I Page 5 of 9 k 1 Go Back where Full Screen a0 0 b0 0 ack bck k k k k ask k k bsk k k The variables a0 ack ask b0 bck and bsk are the decision variables in the optimization model

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