17

I think one must distinguish between chaotic "Hamiltonian" systems and chaotic dissipative systems. In the latter, the phase space volume is not conserved, so it is much more difficult to find "integrals of motion" because Liouville's theorem is broken. Remember, a quantity "A" is an integral of motion if $\frac{dA}{dt} = \frac{\partial A}{\partial t} + \{...


14

Explicitly proving non-integrability of an arbitrary Hamiltonian system is an open problem. For some classes of Hamiltonian systems (e.g systems on a plane) is possible to prove explicitly the non-integrability of the system, using theorems of Poincare, Burns, Ziglin and Yoshida (and generalizations). For example there is a theorem of Poincare: For a ...


14

Quantum integrability basically means that the model is Bethe Ansatz solvable. This means that we can, using the Yang-Baxter relation, get a so-called "transfer matrix" which can be used to generate an infinite set of conserved quantities, including the Hamiltonian of the system, which, in turn, commute with the Hamiltonian. In other words, if we can find a ...


14

Yes there is an analog of the Laplace-Runge-Lenz vector, even more so! The N-dimensional harmonic oscillator is one of a handful superintegrable systems where you have a maximal number (2 N - 1) of independent constants of the motion, leading to closed trajectories in classical mechanics, and spectrum degeneracies in QM. It's just that higher dimensional ...


11

I am not an expert on these issues, but if a further integral existed the orbit would be confined in a codimesion-1 embedded submanifold (for almost all the possible values of that function due to Sard's theorem). An embedded submanifold is a very regular subset, it cannot have self-intersections and cannot be dense in the space for example. Instead the ...


10

Let there be given a $2n$-dimenional real symplectic manifold $(M,\omega)$ with a globally defined real function $H:M\times[t_i,t_f] \to \mathbb{R}$, which we will call the Hamiltonian. The time evolution is governed by Hamilton's (or equivalently Liouville's) equations of motion. Here $t\in[t_i,t_f]$ is time. On one hand, there is the notion of complete ...


10

1) A constant of motion $f(z,t)$ is a (globally defined, smooth) function $f:M\times [t_i,t_f] \to \mathbb{R}$ of the dynamical variables $z\in M$ and time $t\in[t_i,t_f]$, such that the map $$[t_i,t_f]~\ni ~t~~\mapsto~~f(\gamma(t),t)~\in~ \mathbb{R}$$ doesn't depend on time for every solution curve $z=\gamma(t)$ to the equations of motion of the system. An ...


10

i will try this one. A Hamiltonian system is (fully) integrable, which means there are $n$ ($n=$ number of dimensions) independent integrals of motion (note that completely integrable hamiltonian systems are very rare, almost all hamiltonian systems are not completely integrable). What this states in essence (and intuitively) is that the hamiltonian system ...


9

There shouldn't be any more independent ones. This is a 4D system in phase space, so 3 independent phase-space surfaces (including the Hamiltonian) intersect on a line--a trajectory in phase space. Another independent constant would intersect that trajectory at a point and the system would freeze, so all PBs with the Hamiltonian would vanish. (Note $G=(p_2 - ...


8

The Yangian is a deformation of the universal enveloping algebra of a certain Lie Algebra, whose generators satisfy the Yang-Baxter relation. For certain systems (such as those you mentioned) the generators commute with the Hamiltonian and as such the entire Yangian Hopf algebra constitutes symmetries of the system. The physical significance of these ...


8

(1) In general, what is meant by non-linear system in classical mechanics? A linear system is described by a set of differential equations that are a linear combination of the dependent variable and its derivatives. Some examples of linear systems in classical mechanics: A damped harmonic oscillator, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k x(t) ...


8

General analytic methods to prove non-integrability are discussed in e.g. this Phys.SE post. In this answer, we will sketch how to apply the following Poincare corollary. Poincare corollary: If an autonomous Hamiltonian Liouville-integrable system has a periodic solution $z_0(t)=z_0(t+T)$, then the monodromy matrix for the linearized system along $z_0$ ...


8

integrability, chaos and non-linearity are intimately related to ergodicity. Is there any set of conditions relating these concepts with ergodicity? Yes. In order to exhibit chaos, a system must be nonlinear: either through an explicitly nonlinear term, or indirectly, such as when the nonlinearity arises from partial differentiation or a time-delay. Now, ...


7

This question is answered by Nima Arkani-Hamed in his Simons Center talk, at about 112 minutes in. His answer is that the structure of the amplituhedron itself does not directly use integrability of the theory in any way. It is only when you come to do the integrals themselves that integrability makes it possible. The amplituhedron itself is more linked to ...


7

In this answer, we elaborate on the various definitions of integrability, separability & AA-property, in order to expose their (slight) differences. Let there be given a finite-dimensional autonomous Hamiltonian system, defined on a connected $2n$-dimensional symplectic manifold $({\cal M},\{\cdot ,\cdot \})$. Definition. The system is (completely) ...


7

I'm not sure what intuition you are seeking in similarities of mathematical modeling... It's like intuition about the similar beat of two very different pieces of music? I fear it is all in the math. That is, the KdV being a solvable equation with the prototypical "magical" soliton solution $v(x,t)=-2c \operatorname{sech} ^2 (\sqrt{c}(x-ct))$, this shape ...


6

1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces. It was initially intended as a useful simplification to try to ...


6

As requested by the OP, I gather my points in an answer. Linear systems Linear systems are systems which are linear with respect to a physical quantity. Mathematically, their evolution can be written as a (possibly differential) equation. Examples: A linear spring is linear in the sense that is produces a force proportional to the displacement it ...


6

In local coordinates the canonical transformation to action angle coordinates $(q,p)\rightarrow (Q,P)$ can be related by, \begin{equation} \boxed{P_i=\frac{1}{2\pi}\oint p_idq^i \ \ \ \ \ \text{and}\ \ \ \ \ Q^i=\frac{\partial }{\partial P_i}\int p_idq^i} \end{equation} For Example: Consider the one dimensional harmonic oscillator with the following ...


6

OP's actual question follows directly from Theorem 5.3 in Ref. 2, but that leaves the obvious question: How to prove Theorem 5.3? It seems the only really satisfying answer would be to outline a complete proof of the Liouville-Arnold Theorem. This is what we intend to do in this answer. Let there be given a finite-dimensional autonomous Hamiltonian system, ...


5

Complete integrability is far stronger than solvability of the initial value problem. Complete integrability implies the absence of chaotic orbits. More precisely, all bounded orbits are quasiperiodic, lying on invariant tori. Perturbations of a completely integrable system preserve only some of these tori; this is the KAM theorem. http://en.wikipedia.org/...


5

In the general theory of partial differential equations and specifically for First-Order Partial Differential Equations one defines the general solution(Landau's general integral) and the complete integral as follows: For a two-dimensional first order partial differential equation $$f(x,y,z,z_x,z_y)=0. \tag{1}$$ Complete Integral: A two parameter ...


5

I found this, it may be of some help to answering your question: An Introduction to Yangian Symmetries. Denis Bernard. Int. J. Mod. Phys. B 7, pp. 3517-3530 (1993). arXiv:hep-th/9211133.


5

A linear system is one whose dynamics obeys linear differential equations, in contrast with those that are non-linear whose dynamics obeys non-linear differential equations. So if the dyanmics of the variable $x(t)$ obeys a a differential equation $$f\left(x(t),\frac{d}{dt}x(t),\dots,\frac{d^n}{dt^n}x(t),t\right)=0,$$ if $x_1(t)$ and $x_2(t)$ are differente ...


5

The key point here is that, any dynamical system that is not completely integrable will exhibit chaotic regimes1. In other words not all orbits will lie on an invariant torus (Liouville's torus is the topological structure of a fully integrable system), in principle a chaotic system can even have closed stable periodic orbits (typical for regular/integrable ...


5

The simple answer, overall, is that you always have to assume an integral of motion in a certain form and look if the conditions for its existence are even fulfilled. Generically, they are not, and if no explicit symmetry indicates full integrability, one cannot expect that the motion will be fully integrable; on the contrary, parts of the phase space will ...


5

The methods which one uses to understand or even construct Bertrand's theorem do indeed belong to the wide class of methods which are used also in chaos theory, but the connection is rather loose. I will describe how you can understand even this loose connection. When we have a dynamical system in classical mechanics, we need to analyze whether it is ...


5

The thing is that you perceive the situation upside down. The main interest in this topic is not because integrable models are useful for string theory. It's integrable models themselves that get much interest and it "just happens" that string theory is intimately related to some important QFTs. In most quantum field theories one can rarely go beyond the ...


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