Questions tagged [integrable-systems]

Question 1

I am currently taking a course in theoretical (classical) Mechanics, where I have learned about the Darboux theorem. My professor has also mentioned one can "reduce the system by symmetry", ...

Question 2

I am reading these lecture notes https://arxiv.org/abs/math/9908064 on the dynamical Yang-Baxter equation and have a question regarding the dynamical 2-cocycle condition. (See also the book "The ...

Question 3

I'm following notations/conventions of Srednicki, chapter 3. My question relates to some conserved currents and charges of the free real scalar field, that arise in addition to the translation and ...

Question 4

I read that the Kovalevskaya top has 4 invariants of motion: energy $H_K$ Kovalevskaya invariant $ K=\xi _{+}\xi$ angular momentum component in the $z$-direction $L_z$ magnitude of the $n$-vector: ${\...

Question 5

I'm trying to compute the propagators of the 4d Chern-Simons theory introduced in the paper "Gauge theory and Integrability, I" of Costello, Witten and Yamazaki. They pick a Lorenz like ...

Question 6

I am currently trying to explain why the free rigid body with a fixed point is an integrable system. My questions are the following: A $2n$-dimensional system is called integrable if there are $n$ ...

Question 7

In a (1+1)D integrable quantum field theory (e.g. the sine-Gordon model), there is a tower of infinitely many conserved currents with increasing spins. During a scattering, the existence of such a ...

Question 8

In many situations (e. g. elastic collisions) we have conservation of the quantities of momentum ($p=mv$) and kinetic energy ($K=\frac{1}{2}mv^{2}$). Does this extrapolate? Let's say in 1D for ...

Question 9

I am reading this paper on twist fields in 2D integrable QFT: https://arxiv.org/abs/0706.3384 In page 3, the authors offer the following definition of locality in QFT, relativistic as well as ...

Question 10

The general 3-body problem is well-known to be unsolvable in closed form due to its chaotic nature. However, I am curious about a specific, simplified scenario: If we consider three particles, each ...

Question 11

Integrable systems are usually defined as those with an extensive amount of conserved charges. Now take one of these systems, and make one parameter time-dependent for all times, with e.g. an ...

Question 12

From what I understand, the XY model is exactly mapped to free fermions tight-binding approximation. I follow this lecture notes to find the formula for ground state of the model to be $$ k = \frac{2\...

Question 13

Here is what I mean: In Lagrangian mechanics, we have the equation $$ \frac{\mathrm{d}}{\mathrm d t} \frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i},\quad i = 1,2, \cdots, n. $$ ...

Question 14

I am currently reading Chapter 2 of Theoretical Astrophysics Vol 1 by T. Padmanabhan. Here he discusses integrable systems in Hamiltonian mechanics. The idea of angle action variables are introduced ...

Question 15

Note: The origins of the Yang-Baxter equation is in physics, not mathematics. Let $V$ be a vector space. Let $R: V \otimes V \to V \otimes V$ be a linear operator. Consider a labelled (for notational ...

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Questions tagged [integrable-systems]

Using conserved quantities as variables in Mechanics

Dynamical 2-cocycle condition for $sl_2$

Additional conserved currents and charges of the free scalar QFT

Invariants of the Kovalevskaya top and Lie groups

Factor 4 in gauge fixing condition in Costello, Witten, Yamazaki paper "Gauge Theory and Integrability, I"

What are the conserved quantities for the Euler top? And how does the symplectic formalism differ from the Poisson?

Infinitely many conserved currents in an (1+1)D free QFT

Is mass times velocity cubed $mv^3$ also conserved?

Locality in QFT and energy density

Can the 3-body problem be solved in a 1D simplified case?

Integrable systems with explicit time dependence

What is the exact ground state energy of the 1D finite XY model in open boundary condition?

Is there a classical Lagrangian system with essentially no cyclic coordinates?

What is the need for angle-action variables in describing integrable systems?

How is the Yang-Baxter equation equivalent to the Braid equation?

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