What is Hamiltonian: Definition and 894 Discussions

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. quasar987

    Hamiltonian mechanics: canonical transformations

    Say I have a canonical transformation Q(q,p), P(q,p). In the {q,p} canonical coordinates, the Hamiltonian is H(q,p,t)=p\dot{q}-L(q,\dot{q},t) And the function K(Q,P,t)=H(q(Q,P),p(Q,P),t) plays the role of hamiltonian for the canonical coordinates Q and P in the sense that...
  2. R

    Can a system have the total energy conserved but the hamiltonian not conserved?

    Homework Statement Can a system have the total energy conserved but the hamiltonian not conserved? Homework EquationsIf the partial of the lagrangian w.r.t time is zero, energy is conserved. The hamiltonian is found by the usual method- get the generalized momentum from the lagrangian...
  3. L

    Hamiltonian Mechanics: Central Potential V(r) w/ Spherical Coordinates

    Homework Statement Using spherical coordinates (r, \theta, \phi), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p_{\phi}(0) = 0 and \phi (0)...
  4. L

    How Do You Derive a Lagrangian from a Time-Dependent Hamiltonian?

    Consider the time-dependent Hamiltonian H(q,p;t) = \frac{p^2}{2m \sin^2{(\omega t})} - \omega pq \cot{(\omega t)} - \frac{m}{2} \omega^2 \sin^2{(\omega t)} q^2 with constant m and \omega. Find a corresponding Lagrangian L = L(q,\dot{q};t) Ok, I know that the Hamiltonian is given by...
  5. R

    Finding the Hamiltonian for the Be Atom without Nuclear Terms

    I have a problem that uses the QM Hamiltonian for the berylium atom, but I am having trouble finding this Hamiltonian using the Born-Oppenheimer approximation (leaving out the nuclear-nucler and nucler-electron terms). Any know how to get this?
  6. K

    Zero Hamiltonian and its energies

    Let's suppose we have a Hamiltonian of the form: \bold H =0 then it's obivious that if you want to get its energies you would get E_{n}=0 for every n, this is a non-sense since you must have positive energies (and a ground state) ...then if we "cheat" :rolleyes: and make: \bold H...
  7. U

    Hamiltonian for a single particle

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  8. L

    Hamiltonian Math Problem: Finding the Time-Derivative of <p^2/2m>

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  9. J

    Hamiltonian (in Quantum Mechanics)

    The Hamiltonian is given: H=Aâ†â + B(â + â†) where â is annihilation operator and ↠is creation operator, and A and B are constants. How can I get the eigenenergy of this Hamiltonian? The given hint is "Use new operator b = câ + d, b† =c↠+ d (c and d are constants, too) But...
  10. J

    How can I calculate the eigenvalues of a Hamiltonian with spin 1/2 objects?

    Find the eigenvalues of the hamiltonian H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A) where S_A, S_B, S_C, S_D are spin 1/2 objects _________________________ I rewrite it as H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2] then i define...
  11. W

    Why do symmetry and Hamiltonian operators commute?

    Why do operators representing some symmetry commute with the Hamiltonian?
  12. J

    Angular Momentum and Hamiltonian Commutation

    I am working on a problem for homework and am supposed to show that the angular momentum operator squared commutes with H and that angular momentum and H also commute. This must be done in spherical coordinates and everything I see says "it's straightforward" but I don't see it. At least not...
  13. B

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  14. kakarukeys

    Conserved charge generates symmetry transformation in Hamiltonian Mechanics

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  15. E

    Obtaining the Hamiltonian for Einstein's Lagrangian: A Wheeler-De Witt Approach

    If we have the Einstein Lagrangian... L= \sqrt (-g)R my question is how do you get the Hamiltonian?..the approach by Wheeler-De Witt is to consider the line element: ds^2 = N(t)dt^^2 + g_ij dx^i dz^ j (Einstein sum convention) and then substitute it into the Lagrangian above and perform...
  16. K

    Helium Hamiltonian: Derive Ground State & Quantum Numbers

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  17. W

    Defination of eFz in Hamiltonian

    defination of "eFz" in Hamiltonian I want to know the defination of "eFz" in Hamiltonian for the electron and LO-phonon interaction in electric field, does "z" show the position of the electron?
  18. P

    How do you a find a Hamiltonian?

    So when I first learned about Hamiltonians, my teacher presented them as something which you derived based Newton's second law, with the intermediate step of the Lagrangian. Doing it this way gets a Hamiltonian of the form H = T + V. Now I have a new teacher, who says that that these sorts of...
  19. S

    The Hamiltonian: Solving dH/dt Formally

    This is a question, not a homework problem, as i am currently studying for my test on classical mechanics suppose H = \sum_{i} \dot{q_{i}}(p,q,t) p_{i} - L(p,q,t) also i can prove that dH = \sum_{i} (\dot{q_{i}}dp_{i} - \dot{p_{i}} dq_{i}) - \frac{\partial L}{\partial t} dt suppose...
  20. P

    Hamiltonian formulation of general relativity

    I am currently studying loop quantum gravity, and therefore GR in Ashtekar variables (A,E). I see the vector constraint E^a_iF^i_{ab}=0 is said to generate spatial diffeomorphisms (where F is the Yang-Mills field strength in terms of A), but how can I show this? How do spatial diffeomorphisms...
  21. R

    .Solving [X,H] for the SHO Hamiltonian

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  22. X

    What can this Hamiltonian can do?

    please see the following Hamiltonian mode. two two-level atoms(or two qubits) interact with photon. á、à are the creation and annihilation operators of photon please rewriting H in the way of pauli operators , will looks very simple H = E1 ( |e1><e1| －|g1><g1| ) + E2 ( |e2><e2| －|g2><g2|...
  23. quasar987

    Hamiltonian of a spin 1/2 particle in a constant mag. field

    Why is there no kinetic energy term in said hamiltonian? Suppose I take a magnetic dipole in my hand, and I throw it in the field. Then surely its classical energy is E = p²/2m - \vec{\mu} \cdot \vec{B}. Then why is the p²/2m term absent in the hamiltonian?
  24. Q

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  25. S

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    Hi. Can anyone tell me what exactly the "Heisenberg Hamiltonian" is? I found it in an article related to: the renormalization group. Thanks in advance. Somy:smile:
  26. F

    Angular momentum commutes with Hamiltonian

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  27. marcus

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    http://www.arxiv.org/abs/gr-qc/0510011 "Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the...
  28. Y

    Lagrange and Hamiltonian question

    I can't understand what the question is asking~ hope somebody can help me~ A particle of mass m moves in a plane under the influence of Newtonian gravitation force, described by the potential V(r) = - GmM/r (symbol in conventional meaning) Now introduce a new variable u(theta) = 1/r(theta)...
  29. H

    Poisson brackets and EM Hamiltonian

    Consider the following general Hamiltonian for the electromagnetic field: H = \int dx^3 \frac{1}{2} E_i E_i + \frac{1}{4}F_{ij}F_{ij} + E_i \partial_i A_0 + \lambda E_0 where \lambda is a free parameter and E_0 is the canonical momentum associated to A_0, which defines a constraint (E_0 =...
  30. E

    Is the Potential in Quantum Hamiltonian II Always Real?

    Let be a Hamiltonian in the form H=T+V we don,t know if V is real or complex..all we know is that if E_n is an energy also E*_n=E_k will be another energy, my question is if this would imply V is real... my proof is taking normalized Eigenfunctions we would have that...
  31. E

    Hamiltonian with complex potential

    let,s suppose we have the Hamiltonian H=T+V but V is V=F(z) being z=a+ix then would the energies be real?..thanks.
  32. N

    Is the Hamiltonian Cycle an NP Problem?

    sup, just realized there is not graphtheory/combinatorial subforums. is the hamiltonian cycle still considered and NP problem?
  33. L

    What are Hamiltonian Mechanics?

    Can anyone give me a basic definition of Hamiltonian Mechanics without all the fancy mathematics, and perhaps could supply a few examples as to this? I am trying to make sense of this, but everywhere I go, I run into very large mathematical equations and no defintions I can understand...
  34. Z

    Understanding the Relationship between the Hamiltonian and Magnetic Field

    I have a question regarding the Hamiltonian in a magnetic field. First Hamiltonian with potential V is given by Ho = (1/2m)*p^2 + V but if a vector potential A is also present then H1 = (1/2m)*(p+eA)^2 + V there is a way to write H1 interm of Ho H1 = exp(-ier.A) Ho exp(ier.A) where r...
  35. H

    P&S Chapter 2: Derivation of the KG Hamiltonian

    I am not familiar with calculations such as the following one, and I want to be sure I do the right steps before going on with Peskin & Schröder. I want to derive step by step the Hamiltonian of the real scalar field. I have no problem to arrive at the first part of (2.31) in page 21: H =...
  36. H

    Symplectic runge kutta for hamiltonian system

    Hi ! I'm trying to solve the restricted problem of three bodies, where a negligeable mass particule is moving in the gravitationnal field of two heavy objects which are in circular orbit around their common center of mass. this is a plane problem... I describe the mouvment in the mobile...
  37. S

    Hamiltonian of flyball governor

    Hamilton equations of flyball governor I'm trying to find 1. The Hamiltonian 2. The Hamilton equation of motion for the flyball governor shown in problem 2 here http://www.srl.caltech.edu/phys106/1999/Homework3.pdf This is what i have. Can someone tell me if I'm right...
  38. C

    Angular Momentum vs Hamiltonian in Dirac Field Theory (Canonical)

    I need some suggestions and/or corrections if I understand this correct? My questions are based on the book by Mandl and Shaw. Conserved currents are based on Noethers theorem and directly connected to spacetime and field transformations (rotations, translations, phase, ...). One can...
  39. R

    Quantum Mechanics and General Relativity: Understanding H Y = E Y and T Y = G Y

    Generally we write H Y = E Y in quantum mechanics. Would it make any sense to use T ( stress-energy tensor) from general relativity and G (curvature) to write : T Y = G Y
  40. I

    Relativistic Lagrangian -> Hamiltonian

    I came cross this problem when solving some older tests from Classical Mechanics, so I was hoping anybody can help me. I have expression for relativistic lagrangian of a particle in some potential that is function only of coordinates and not velocities: L = - mc^2* sqrt( 1 - (x'^2 + y'^2 +...
  41. T

    Can Identical Quantum Particles Have Different Energies?

    Let say I have prepared two identical particle, both describable by a wavefunction Psi, whereby, Psi = a*1 + b*2, where, 1 and 2 are two stationary wavefunctions. If I perform an experiment to find out the systems' energy, this is equivalent to operating a Hamiltonian on Psi. Operating...
  42. E

    Hamiltonian is commutable with momentum operator

    [H,P]=0 , where P is momentum operator. Hamiltonian is commutable with momentum operator. so H and p have wave function simultaniously, but in 1-dimensional potential well degeneracy not exist. what is the reason?
  43. suyver

    Standard Model: Lagrangian vs. Hamiltonian

    I was wondering: why is the SM always written with a Lagrangian? Couldn't you just as well write it with a Hamiltonian? The way I understand, the Lagrangian gives me the kinetic energy minus the potential energy (basically a measure for the "free energy", though not in the thermodynamical...
  44. E

    How Do You Quantize a Classical Hamiltonian of the Form H=f(x)P^n?

    given a classic hamiltonian of the form H=f(x)P**n what would be it quantum version of it?..how do you quantizy this?... (n is an integer) Should you take all the possible permutation of it?..i have this problem...thanks.
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