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Background

I was reading this article on the unviersal $SO(4)$ and $SU(3)$ symmetries in all central potential problem. Turns out every bounded planar motion in any smooth central potential will all have these two inherent symmetries. They are respectively generated by integrals of motion named generalized Laplace–Runge–Lenz vector and generalized Jauch-Hill-Fradkin tensor, which are almost trivially constructed based on the conserved angular momentum and an arbitrary constant vector perpendicular to it.

The implication being that the Kepler motion and the isotropic oscillator are not special in terms of symmetries among the 3-D central potential problems, but their integrability (or even super-integrability in these cases) of dynamics.

And since the action variables are the straightforward integrals of motion defined for classically integrable Hamiltonian systems, so it's easier to investigate the relations between the integrability and all kinds of symmetries through them:

Principal questions

What is the physical interpretation of symmetries generated by N action variables $\left( J_i \right)_{1\leq i\leq N}$ of a completely-integrable system? And how are they related to other general symmetries of the system, generated by $\{ G_k \}$ such that $[H,G_k]=0$ ?

I read that the connected abelian Lie group generated by these mutually commuting integrals of motion must be isomorphic to $ R^{N-k} \times (S^1)^k $, but even if the property of the symmetries generated by such special set of integrals of motion is just trivially derived from their commutation relation, is there any physical meaning behind them?

And what is the relation between these symmetries and other symmetries generated by a integral of motion which cannot be functionally independent on the former? $~$(Like, maybe other general symmetries preserve these N particular "abelian" symmtries representing the integrability, or we might be able to obtain other general symmetries from these N symmetries, I guess?)
$~$


Related definitions

Considering a classical mechanics system $(H,M,\omega)$ with $Q$ the $N$-dimensional configuration space manifold ($q_i,~i=1,\ldots,N$). We note its Hamiltonian:
$$H: ~~ M = T^{\ast}Q ~~~ \to ~~~~ \mathbb{R} ~~~~~~~~~~~~$$ $$~~~~~\eta_{\mu}=(q_i,p_i)\mapsto H(q_i,p_i)$$ where $M=T^{\ast}Q$ is the cotangent bundle of $Q$ as the symplectic manifold ($\eta_\mu,~\mu=1,\ldots,2N$).

  • Integral of motion $G$: $[H,G]=0$. It's the generator of infinitesimal canonical transformation $\eta \to \eta'=\eta+\epsilon[\eta,G]$, which is a symmetry satisfied by the system;
  • Completely integrable system: There exist and only exist $N$ functionnally independent & mutually Poisson-commuting globally defined integrals of motion $I_1=H, I_2, \ldots, I_N: M \to \mathbb{R}$. Then the system will have the following properties:
    1. Its level set $M_{f}= \{ \eta|{I_i}(\eta)={f_i} \}$ is a smooth manifold, invariant under the Hamiltonian flow (which means the globally defined function $I_i$ conserve their initial value $f_i$ during the time evolution), and if it's again compact and connected, then it is diffeomorphic to the N-dimensional tori $(S^1)^N$;
    2. When $M_{f}$ is compact and connected (physically the motion is bounded), we can transform to the canonical coordinates action-angle variables of the system which can significantly simplify its dynamics. Here we focus on the action variables (sometimes known as adiabatic invariants) whose definition (David Tong's lecture notes. Eqns(4.145), pp.111) is $J_i=\displaystyle\sum_{a=1}^N{\frac{1}{2\pi}\oint_{\gamma_i}p_a dq_a}$ , where each $\gamma_i$ are different periods of the invariant tori, physically corresponding to bounded movements like libration or rotation;
  • Mutually Poisson-commuting integrals of motion (or in involution): $[I_i,I_j]=0, \forall i, j $;
  • Functionally independent between them: the linear independence of their 1-forms $( dI_i )_{1\leq i\leq N}$, meaning that $\mathrm{d}I_1\wedge \ldots\wedge \mathrm{d}I_N\neq 0$ is nowhere vanishing on the level set $M_f$.

Secondary questions

For the same principal question concerning in the place of the AV, arbitrary I3M which are functions of AV, are the answers different, or there's no particularity of AV compared to other I3V?

Abbreviations:
Action Variables abbreviated as AV;
Independent & in Involution Integrals of Motion abbreviated as I3M.

Do non-integrables systems always have a set of I3M (with cardinality $k<N$ of course), and what symmetries can they generate?

These two secondary questions concerning uniqueness of AV in completely-integrable systems and existence in non-integrable ones may not be as interesting as these final questions:

The super-integrable systems (there exist $N+1<=r<=2N-1$ globally defined I3M) may be even more intriguing than all the previous example because of the degeneracy of fundamental frequencies and other more properties among them.

So what do their generated symmetries represent (giving that the $r-N$ dimensional sub-algebra don't generally commute anymore)? And how are they even more special than the symmetries generated from completely-integrable systems?

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My answer will be to try to be more intuitive than in-depth quantitative-wise. If the latter is what you seek, I would recommend Olver's Book "Applications of Lie Groups to Differential Equations" chapters 4-7.

The concept of action variables $(J_i)_{1 \leq i \leq N}$ in a completely integrable system and their relation to general symmetries of the system, represented by $\{G_k\}$ such that $[H, G_k] = 0$, is a deep and fundamental aspect of classical mechanics, particularly in the context of Hamiltonian systems.

A system is said to be completely integrable if it has as many independent constants of motion as degrees of freedom. In Hamiltonian mechanics, this translates to having $ N $ independent integrals of motion for a system with $ N $ degrees of freedom.

In the Hamilton-Jacobi theory, a completely integrable system can be transformed into action-angle variables. The action variables $ J_i $ are integrals over the closed orbits in phase space, representing the "size" of the orbits. They are constants of motion, serving as the integrals of motion for the system.

$$ J_i = \oint p_i dq_i $$

where $ p_i $ and $ q_i $ are the canonical momenta and coordinates, respectively.


The action variables correspond to conserved quantities in the system. For example, in a planetary orbit, one of the action variables might correspond to the total angular momentum of the orbit, a conserved quantity due to rotational symmetry.

Relation to General Symmetries $ \{G_k\} $

The quantities $ G_k $ represent general symmetries of the system. The condition $[H, G_k] = 0$ means that these are also constants of motion, as they commute with the Hamiltonian $ H $. (You can think of this as the Lie Derivative) Now while action variables are specific to completely integrable systems and are associated with the fundamental symmetries of the system, the $ G_k $ are more general and can exist in non-integrable systems as well.

Noether's Theorem: The existence of these symmetries and conserved quantities is deeply related to Noether's theorem, which states that every continuous symmetry of a physical system corresponds to a conservation law. The action variables $ J_i $ are specific manifestations of this principle in integrable systems.

In integrable systems, the action variables and other integrals of motion like $ G_k $ often form a set of mutually commuting quantities. This commutativity is a hallmark of integrable systems and is crucial for their solvability.


Action variables in a completely integrable system represent the fundamental symmetries and conserved quantities of that system. They are intimately connected to the general symmetries represented by $ G_k $, which are also conserved due to their commutation with the Hamiltonian. This relationship underscores the deep connection between symmetries and conservation laws in physics, a cornerstone of our understanding of fundamental interactions and dynamics.

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  • $\begingroup$ Thank you, MrDBrane, for the feedback and the recommended reference. I'll read it soon. I'm not familiar with symplectic geometry and dynamical system manifold, but is there some connections between the integrals of motion in involution and the Poincaré-Cartan invariant form or other invariants in these fields? $\endgroup$
    – TheoVereka
    Nov 22, 2023 at 11:34

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