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I was wondering if there is a known link between dynamical algebra and symmetry group. In particular:

Do all Hamiltonians belonging to certain dynamical algebra share the same symmetry group?

Do all Hamiltonians sharing the same symmetry group belong to the same dynamical algebra?

If one knows that two Hamiltonians belong to the same dynamical algebra, what Physical insight can be deduced? In other world, what Physics do they share?

Reminder: a set of operators forms a dynamical algebra if it is closed under commutation. In other words, the commutator of any two algebra elements must still be an algebra element, i.e. a linear combination of the generators. An Hamiltonian is said to a belong to a dynamical algebra if it can be written as linear combination of algebra’s generators.

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  • $\begingroup$ Comment to the post (v2): Would you call the trivial 1-dim Lie algebra ${\rm span}\{H\}$, generated by the Hamiltonian $H$, for a dynamical algebra?? Your definition (v2) of a dynamical algebra seems incomplete. $\endgroup$
    – Qmechanic
    Aug 5, 2018 at 6:17
  • $\begingroup$ Can you reformulate your comment? I don’t understand your point. $\endgroup$
    – AndreaPaco
    Aug 5, 2018 at 23:32
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    $\begingroup$ Please see my answer to the following question. I hope it will be helpful physics.stackexchange.com/q/105042 $\endgroup$ Aug 6, 2018 at 6:49

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In a complete general framework, probably as you know, you can define an hamiltonian system as a dynamical system whose vector field reads: $$[X_{H}(x)]_{i}= \{x_{i},H\} $$ where $$x=(q_{1},...,q_{n},p_{1},...,p_{n})=(q,p)$$ is a vector of the phase space $\Gamma$ (that is practically always an Hilbert space), $\{ ; \}$ is a Poisson Bracket that satisfies the usual properties and $H=H(x)$ is the Hamiltonian of the system, defined on the phase space. A really important object that one can define, is the so-called Poisson tensor, defined as: $$J_{jk} = \{x_{j},x_{k}\}$$ Note that from this definition is obvious that $J$ has got the same skew-symmetry property of the Poisson brackets: this is fundamental in order to redefine a formulation of any Hamiltonian system in terms of $J$. At this point, without enter a lot in details, one can develope what is written above redefining the Poisson brackets in terms of the Poisson tensor and in this way one can write a complete general form of the Hamilton equation of ANY Hamiltonian system: $$\dot{x} =J(x)\nabla_{x}H(x)$$ where $J$ is the Poisson tensor and the notation $\nabla_{x}$ means that the gradient acts on a vector of the phase space $x$. You can make ANY change of variable $x \rightarrow y=f(x)$ that you want, but you will NEVER lose the hamiltonian properties of your system (i.e. the properties of the Poisson brakets). This is really important and means that if a system is Hamiltonian, according to the definitions above, it will remains Hamiltonian INDIPENDENTLY of the coordinate chosen to describe it.

The concept of the Poisson tensor is also important in order to characterize the set of Casimir invariants (i.e the symmetry group) of a given Hamiltonian system: A Casimir invariant $C(x)$ is a function defined on the phase space, such that: $$J(x)\nabla C(x)=0$$ in other words, is a function which has got the gradient that is in the Kernel of the Poisson tensor. These functions describe the invariace of your system. The main point (that i hope can answers to your question) is easy to understand just from the definition of a Casimir invariant and is that ANY Hamiltonian system that is described by the same Poisson tensor, has got the same Casimir invariants (i.e the same symmetry)! Is not correct the way in which your are posing your question, beacause every Hamiltonian system has got a well defined algebra of functions $A(\Gamma)$ whose bilinear product is a Poisson bracket that satisfies the usual properties that you know: if is not possible to define a Poisson bracket you can't have any Hamiltonian system. It's not meaningful what you are asking for beacause the definition of the classical Hamiltonian algebra is really general and , i repeat, well-defined in any case. In other words, the Hamiltonian algebra defined on the phase space doesn't count if you are talking about invariance and symmetries of an Hamiltonian system but, as i wrote to you above, the main instrument in order to characterize this fact is the Poisson tensor of the system: the same Poisson tensor -> the same Casimir(symmetries)

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