# An example of non-Hamiltonian systems

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. Thank you.

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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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In infinite case yes. What about finite numbers of particles. I think I should construct some system for wich we cant writ Hamiltonian. –  Daniel Aug 24 '12 at 9:26
what's your infinite case example? –  Yrogirg Aug 24 '12 at 9:28
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 –  Daniel Aug 24 '12 at 9:49

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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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. –  Ron Maimon Aug 24 '12 at 20:19
What is the subtlety with (ii)? Classically, a Hamiltonian formulation is possible, right?. –  drake Aug 24 '12 at 20:28
@ 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. –  Lupercus Aug 25 '12 at 11:09
@ 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). –  Lupercus Aug 25 '12 at 11:25

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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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. –  Ron Maimon Aug 24 '12 at 20:20
@Ron, I have added a some definitions and a reference that makes the point more clear. –  John Sidles Aug 25 '12 at 0:14
friction is irrelevant for rattleback?! –  Yrogirg Aug 25 '12 at 5:38
@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. –  John Sidles Aug 27 '12 at 3:22