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Level Curves of a Hamiltonian System

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Various problems in my textbook ask me to sketch the level curves for a Hamiltonian system, but they don't suggest how to go about it. I know that I need to determine the eigenvalues for each equilibrium point in the given system, and these values hint at the behavior of solutions near each respective equilibrium point. Do I need to sketch a direction field to help me see how the curves behave? What else might I be missing?

Also, do the Hamiltonian level curves only suggest what the solutions to non-linear systems look like, or can it somehow lead explicit solutions?

Any help is appreciated.
 

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