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I am preparing for the exam. And I need to know the answer to one question which I can't understand.

"Give an example of non-Hamiltonian systems: in case of infinite number of particles; for a finite number of particles".

I hope somebody can help me.

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That's easy. Hamiltonian mechanics describes reversible dynamics. Just introduce irreversibility in your system. like friction, dissipation, viscosity etc.

Can you answer the question now?

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  • $\begingroup$ In infinite case yes. What about finite numbers of particles. I think I should construct some system for wich we cant writ Hamiltonian. $\endgroup$
    – Daniel
    Commented Aug 24, 2012 at 9:26
  • $\begingroup$ what's your infinite case example? $\endgroup$
    – Yrogirg
    Commented Aug 24, 2012 at 9:28
  • $\begingroup$ System where total energy is not conserved. I think it may be some motion of particles which loses theire energy due to radiation, may be? Sorry I cant upvote because of low reputation $\endgroup$
    – Daniel
    Commented Aug 24, 2012 at 9:49
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    $\begingroup$ Ehhh, I dunno. You can actually put dissipation into Hamiltonian language pretty easily. $\endgroup$
    – DanielSank
    Commented Apr 13, 2016 at 19:51
  • $\begingroup$ Friction can be described by the Hamiltonian method, using finite number of degrees of freedom, see e.g. physics.stackexchange.com/questions/11905/… $\endgroup$
    – Ján Lalinský
    Commented Feb 9 at 20:21
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We define a Hamiltonian system to be the triad $(H,\mathcal{M},\omega)$ of a Hamiltonian function $H$ on a state-space manifold $\mathcal{M}$ that that is equipped with a (closed) symplectic form $\omega$.

Two long-known and much-studied yet (relatively) simple examples of energy-conserving yet non-Hamiltonian dynamical systems are the (1) the Chaplygin Sleigh, and (2) the rattleback.

Note Added  In particular, the reason that the dynamics of the Chaplygin Sleigh are not Hamiltonian is geometrically elementary: the state-space manifold of a Chaplygin Sleigh is odd-dimensional — namely, the x and y spatial coordinates of the sleigh, the angular orientation of the sleigh, its linear momentum, and its angular momentum — whereas symplectic forms exist only on even-dimensional manifolds.

Viewed as a flow on $\mathcal{M}$, the dynamics of these systems is energy-preserving but not a symplectomorphism. In thermodynamic terms, the First Law holds, but the Second Law need not.

For example, in we read in Advances in the Theory of Control, Signals and Systems with Physical Modeling:

One of the striking features of non-holonomic systems is that while they conserve energy they need not conserve volume in the state space.

The study of the thermodynamic properties of ensembles of these systems (and other non-symplectomorphic systems like them), and their quantum generalizations, are active areas of research.

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  • $\begingroup$ This is not right--- just because some constraints are not holonomic doesn't mean that the system is not described by a Hamiltonian. You can't have nondissipative motion that isn't Hamiltonian, it would require information to leave without energy leaving. $\endgroup$
    – Ron Maimon
    Commented Aug 24, 2012 at 20:20
  • $\begingroup$ @Ron, I have added a some definitions and a reference that makes the point more clear. $\endgroup$
    – John Sidles
    Commented Aug 25, 2012 at 0:14
  • $\begingroup$ friction is irrelevant for rattleback?! $\endgroup$
    – Yrogirg
    Commented Aug 25, 2012 at 5:38
  • $\begingroup$ @Yrogirg, in the macroscopic equations, there is no friction. At the microscopic level, you have asked a deep question! Namely, can quantum mechanics encompass rolling/sliding mechanical constraints that have zero entropy gain? I do not know the answer to this question, and I suspect that no one does. $\endgroup$
    – John Sidles
    Commented Aug 27, 2012 at 3:22
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    $\begingroup$ A comment: simply because a system has three state variables one cannot conclude that it is not Hamiltonian. A counter-example often cited by Arnold is provided by the Euler equations for a free rigid body (cf. web2.ph.utexas.edu/~morrison/94IFSR640_morrison.pdf, p.42). They are an example of a non-canonical Hamiltonian system. Another example of a non-canonical Hamiltonian system is fluid Flow when expressed in Eulerian coordinates. $\endgroup$
    – Cyclone
    Commented Jan 9, 2018 at 8:21
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Yrogirg is right: include friction and your hamiltonian dynamics is away.

Example for single/few particles: a bead on o wire frame with friction.

Example for an infinite number of particles:

i) fluid dynamics or the dynamics of charged systems (excluding the radiation hamiltonian)

ii) a quantized field in an expanding universe with Friedman metrics: ds^2= dt^2- R(t)^2 (dx^2+dy^2+dz^2) where R(t) would be the "radius" of the universe.

to name just two.

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  • $\begingroup$ The "finite number" of degrees of freedom is misleading, becuase friction is automatically infinite number of degrees of freedom, really, and there are no examples of friction in finite number of DOFs. $\endgroup$
    – Ron Maimon
    Commented Aug 24, 2012 at 20:19
  • $\begingroup$ What is the subtlety with (ii)? Classically, a Hamiltonian formulation is possible, right?. $\endgroup$
    – Diego Mazón
    Commented Aug 24, 2012 at 20:28
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    $\begingroup$ @ Ron: yes, friction implies an infinity of DOF, these are however not considered part of the system under scrutiny, but rather of the medium. Averaging out these DOFs results in friction, but this only at the level of the (open) subsystem considered. When discussing the sliding of metal plates upon each other, the quantized EM field between them is not always mentioned. $\endgroup$
    – Lupercus
    Commented Aug 25, 2012 at 11:09
  • $\begingroup$ @ Ron: you are right in what concerns the case of only few DOFs: there can be no friction, rigorously speaking. However, you do not need many particles to end up with chaotic dynamics (of which I admit to not know much). Hereby a single trajectory could fill an entire area of phase space. Statistically, this is akin to scattering a beam of representative points over some distributed scatterer medium. This (approximate, f.a.p.p.) statistical description of the few body system would be then markovian (not hamiltonian any more). $\endgroup$
    – Lupercus
    Commented Aug 25, 2012 at 11:25
  • $\begingroup$ Friction can be described by the Hamiltonian method, using finite number of degrees of freedom, see e.g. physics.stackexchange.com/questions/11905/… $\endgroup$
    – Ján Lalinský
    Commented Feb 9 at 19:40

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