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  1. In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system?

  2. Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed dynamical systems that can't be described by a time-invariant Hamiltonian?

  3. Are those that can't be described by a time-invariant Hamiltonian "unphysical"?

Edward Hughes
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Ricardo
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  • Essentially a duplicate of http://physics.stackexchange.com/q/175021/2451 – Qmechanic Aug 13 '15 at 10:24
  • Hmm - I'm not sure it's quite a duplicate. The OP in this case probably wants some broader physical intuition rather than the mathematics. I think it's worth having both questions to complement each other! – Edward Hughes Aug 13 '15 at 10:34

1 Answers1

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You are confusing two definitions - closed system and conservation of energy. I'll clear them up for you.

In classical dynamics a closed system is one where no force external to the system acts. In a closed system, the total energy, total momentum and total angular momentum must be conserved. This follows from Noether's theorem. If a has no interaction with the outside world, then we expect it to obey spacetime symmetries. Noether's theorem then guarantees the conserved quantities above.

A2. Yes, every closed dynamical system can be represented as a time-invariant Hamiltonian.

Note that this definition of closed system is not the same as the definition of closed system in thermodynamics. In fact, a closed system in classical dynamics is the same as an isolated system in thermodynamics. It's rather irritating that there's this abuse of nomenclature - I assume it's historical!

Our best fundamental theories assume that we can describe the universe as a closed system. Therefore in a sense, all of physics reduces to this case. However, just describing things in terms of closed systems is quite impractical. After all most experiments we perform are very much not in a closed system. At the moment, we're all sitting in the Earth's gravitational field, for instance. Therefore it's convenient (and completely physical) to describe the world in terms of more general non-closed systems.

A3. No. Fundamentally we believe all physics is described by closed systems, which would have time-invariant Hamiltonians. But it is sometimes convenient to describe parts of the system on their own. That's a completely physical thing to do! There's no reason why parts of the system should have time-invariant Hamiltonians, or indeed obey any symmetries at all.

Finally we come to conservation of energy. Total energy is conserved if and only if the Hamiltonian is time-invariant. But notice that this is a less stringent condition than having a classical dynamics closed system. For example for a particle moving in a central field, energy is conserved but momentum is not. We can't describe this situation as a closed system (because there's an external force) but we can say its Hamiltonian is time invariant.

A1. No, there are time-invariant Hamiltonians which don't obey other spacetime symmetries, so don't describe closed systems!

Edward Hughes
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  • Thanks for your answer, it indeed cleared things up :) Trying to further understand what having a (time-invariant) Hamiltonian means, I found this phys.SE where the second answer mentions two energy-conserving systems that are non-Hamiltonian. However you said energy is conserved only if the system has a time-invariant Hamiltonian. What am I missing? – Ricardo Aug 13 '15 at 14:44
  • Excellent question. There's a subtle difference between "having a Hamiltonian" and "being a Hamiltonian system". The latter usually means that you can describe things in terms of Poisson brackets. All the unconstrained closed systems I've ever come across can be described as Hamiltonian systems. I can't find a proof of this however. When you have constraints (specifically non-holonomic ones) then closed systems don't have to be Hamiltonian, as this presentation points out. – Edward Hughes Aug 14 '15 at 10:28
  • Note that both the sleigh and the rattleback fall into the category of closed systems with non-holonomic constraints. This accounts for the examples in your linked question. There may be other circumstances in which closed systems are non-Hamiltonian, but you'd probably need a dynamical systems expert to answer that question! – Edward Hughes Aug 14 '15 at 10:31
  • So we could say that a system has a time-invariant Hamiltonian if and only if it is holonomic and energy-conserving. That's quite a restricted class I'd say, I don't understand then what's the fuzz with Hamiltonian mechanics. Do you know if quantum systems with a time-invariant Hamiltonian also have these restrictions? – Ricardo Aug 14 '15 at 11:52
  • The point is that having a time-invariant Hamiltonian doesn't guarantee you can describe the dynamics using Poisson brackets! That's why the issue is a little subtle. Quantum systems must have a Hamiltonian and commutation relations by definition, which automatically bypasses such issues. – Edward Hughes Aug 14 '15 at 12:03
  • I'm not sure why one would want to describe the dynamics using Poisson brackets (conciseness?). Do you know of a reference where I can learn about the difference between "having a Hamiltonian" and "being a Hamiltonian system"? How does this difference manifest in Newtonian mechanics? – Ricardo Aug 15 '15 at 00:42