Hamiltonian Definition and 899 Threads

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.
Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs.
Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard, the knight's tour, had been studied in the 9th century in Indian mathematics by Rudrata, and around the same time in Islamic mathematics by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler.

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  1. J

    I Hamiltonian for mass on a smooth fixed hemisphere

    I am trying to figure out how to get the Hamiltonian for a mass on a fixed smooth hemisphere. Using Thorton from example 7.10 page 252 My main question is about the Potential energy= mgrcosineθ is the generalized momenta Pdotθ supposed to be equal to zero because θ is cyclic? Or is Pdotθ=...
  2. Konte

    I Hamiltonian matrix - Eigenvectors

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  3. redtree

    A Conjugate variables in the Fourier and Legendre transforms

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  4. P

    I Yes, thank you for explaining it to me. I have a much better understanding now.

    Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly) 1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be...
  5. T

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  6. L

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  7. binbagsss

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  8. G

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  9. P

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  10. P

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  11. S

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  12. H

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  13. 4

    I QFT for the Gifted Amateur Question (2)

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  14. Milsomonk

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  15. TheCapacitor

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  16. A

    I Finite difference Hamiltonian

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  17. Gopal Mailpalli

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  18. F

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  19. M

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  20. A

    I How to diagonalize Hamiltonian with Zeeman field

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  21. H

    I Meaning of the Hamiltonian when it is not energy

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  22. I

    I How to change the Hamiltonian in a change of basis

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  23. I

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  24. I

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  25. M

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  26. V

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  27. Arnd Obert

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  28. P

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  29. AwesomeTrains

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  30. B

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  31. U

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  32. H

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  33. Z

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  34. R

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  35. I

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  36. T

    I Schrödinger equation and interaction Hamiltonian

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  37. T

    A Interaction Hamiltonian coupling question

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  38. L

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  39. 1

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  40. A

    I Is the Hamiltonian always the total energy?

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  41. S

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  42. S

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  43. S

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  44. M

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  45. M

    A Hamiltonian and constant potential

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  46. S

    I Why is the KE operator negative in QM?

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  47. S

    I Symmetry of Hamiltonian and eigenstates

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  48. S

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  49. P

    I Do All Physical States Satisfy the Hamiltonian Equation Hψ = Eψ?

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  50. M

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