Lesson plan
Arnold diffusion
In applied mathematics, Arnold diffusion is the phenomenon of instability of nearly-integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly-integrable Hamiltonian systems that exhibit a significant change in the action variables. Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (i.e. unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems. It occurs in systems with more than N=2 degrees of freedom, since the N-dimensional invariant tori do not separate the 2N-1 dimensional phase space any more. Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori.
Article summary
In applied mathematics, Arnold diffusion is the phenomenon of instability of nearly-integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly-integrable Hamiltonian systems that exhibit a significant change in the action variables. Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (i.e. unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems. It occurs in systems with more than N=2 degrees of freedom, since the N-dimensional invariant tori do not separate the 2N-1 dimensional phase space any more. Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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