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Unformatted text preview: A LITTLE BIT ON SYMMETRIES HENRY JACOBS Consider a dynamical system on R n (1) ˙ x = f ( x ) Let x ( t ) be a trajectory that satisfies the ODE. Consider a transformation, Φ, from trajectories to trajectories. That is to say, if Γ is the set of curves on R n , then Φ : Γ → Γ. For example, if n = 2 and γ is the curve ( x 1 ( t ) ,x 2 ( t )) then Φ( γ ) could be the curve ( x 1 (- t ) ,- x 2 (- t )) or (- x 1 ( t ) ,x 2 ( t )) or anything involving a space transformation and/or time reparametrization. We say that the system is Φ symmetric if given a curve γ ∈ Γ that satisfies 1, the curve Φ( γ ) also satisfies 1. 1. Example, a conservative system with reflection symmetry Let ¨ x =- x 3 In first order form this is ˙ x = v (2) ˙ v =- x 3 (3) We observe the trajectories move along the level sets of the energy H ( x,v ) = 1 2 v 2 + x 4 . This gives a good idea of the symmetries. You’ve all proven that this system is “reversible” in homework. Reversible is a type of symmetry where if“reversible” in homework....
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
- Fall '09