Monday, September 25, 2017

Week Unknown: "Open" Problems

The four problems on which I want to work when I have time:

On the circle, S1, with its standard Riemannian metric it inherits as a subspace of ℝ2, find a second-order ODE that is
  1. gradient but not Hamiltonian
  2. Hamiltonian but not gradient
  3. both gradient and Hamiltonian ("harmonic") (I think I have a heuristic proof there are no non-trivial ones)
  4. neither gradient nor Hamiltonian
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3 comments:

  1. Dr. Jeff RollandSeptember 25, 2017 at 5:50 PM

    Wash, rinse, and repeat with the closed surfaces

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  2. Dr. Jeff RollandSeptember 26, 2017 at 11:37 AM

    This is all examples of harmonic second-order ODEs on ℝ https://math.stackexchange.com/questions/2004616/vector-field-on-tangent-bundle-that-is-simultaneously-a-gradient-field-and-hamil

    ReplyDelete
  3. Dr. Jeff RollandNovember 8, 2019 at 10:10 AM

    Per a theorem of Marsden, a vector field is Hamiltonian if and only if, when viewed in syplectic coordinates, the Jacobian matrix is in the symplectic group at all points.

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