# Questions tagged [integrals-of-motion]

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### Particularity of symmetries generated by the action variables of a classically integrable system

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 ...
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### What exactly are the 12 conserved quantities in the Two-Body Problem?

The Two-Body problem consists of 6 2nd-order differential equations \begin{equation} \ddot{\mathbf{r}}_1 = \frac{1}{m_1}\ \mathbf{F_g} \\ \ddot{\mathbf{r}}_2 = -\ \frac{1}{m_2}\ \mathbf{F_g} \end{...
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### Relationship between symmetries and additive integrals of motion

I'm currently reading Landau and Lifshitz's Statistical Physics. In it, they attempt to justify that the density function only depends on the energy by arguing that the logarithm of this function is ...
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### Explicit construction of integrals of motion in 1d XXZ model for few sites

I was studying the algebraic Bethe ansatz for the spin-1/2 XXZ model. In the end one ends up with $2^L$ integrals of motion $Q_k$ that commute with the Hamiltonian, (https://doi.org/10.1103/...
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### Conserved Quantities in Kepler Problem?

In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. ...
1 vote
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### Integrals of motion for a rotational symmetric 3D Hamiltonian $H=\frac{{\bf p}^2}{2m}+V(r)$ [closed]

A particle of mass $m$ moves in three dimensions under the action of the conservative force with potential energy $V(r)$. Using the spherical coordinates $r, \theta, \phi$ find the Hamiltonian of the ...
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### Solubility of integrable systems and the classical XXZ model

I've been learning about integrability in the Hamiltonian sense, and trying to wrap my mind around the analytic power afforded by integrability, both in quantum and classical systems. My goal with ...
1 vote
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### Why are there $2s -1$ independent integrals of motion?

I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and can be considered an additive ...
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### Which is the integrals of the motion deriving from the isotropy of time? [duplicate]

In Landau's Mechanics it is written that there are some integrals of the motion deriving from the fundamental homogeneity and isotropy of space and time. Momentum is related to the homogeneity of ...
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### Symmetry associated to a part of a separable Hamiltonian

The harmonic oscillator in 3D is: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+ \frac{k}{2} (x^2+y^2+z^2) = H_x + H_y + H_z,$$ where $H_x$, $H_y$ and $H_z$ are all constants of motion (alongside $\vec{L}$). Time ...
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### Is the motion of a particle in the surface of a torus always periodic?

I am trying to see if there are ballistic trajectories in the surface of the torus that are not periodic and to what extent. Maybe it is not only quasiperiodic but chaotic. I guess there are ...
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### How to know the number of constants of a free particle?

Landau-Lifshitz Mechanics says that there are $2s-1$ constants of a system with $s$ degrees of freedom (beginning of the second chapter on Conservation Laws). If this is true, for a single free ...
362 views

### Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. 'Remark 11.12' on pg 443 of Fasano-Marmi's 'Analytical Mechanics' suggest ...
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### Hamiltonian with one constant of motion (besides the Hamiltonian itself)

The background of my question is a well known fact: a Hamiltonian system with $n$ degrees of freedom with $n$ constants of motion is integrable. My question is about the case in which there are only ...
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### Different action-angle variables for a 2D harmonic oscillator

Consider a bidimensional harmonic oscillator. Ref. 1 says that, when the frequencies are commensurable, separating the variables in cartesian or polar coordinates leads to different action-angle ...
1 vote
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I want to obtain some clarification on the concept of Noether charge. Given conserved current $J^\mu$ e.g. in free scalar field theory in $(n+1)$ dimensional Minkowski spacetime $M$, i.e. $\partial_\... 2 votes 1 answer 222 views ### Apparent emergence of conserved quantities in non-integrable systems This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable? Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities.... 0 votes 1 answer 146 views ### Inferring the conservation of angular momentum from linear momentum [duplicate] Working in 3-dimensions, if we are given a Lagrangian containing$N$particles. Say, through Noether's theorem, we know that the sum of the linear momentum of all$N$particles in each direction are ... 3 votes 2 answers 673 views ### Determining if constants of motion are independent Say, in Hamiltonian mechanics, we know two constants of motion,$A$and$B$. It could be proven that the quantity$[A,B]$is also a constant of motion, where$[A,B]$denotes the Poisson brackets of$... 1 vote
I have had this confusion for a while now. We solve the Hamilton Jacobi equation, $$H+\frac{\partial S}{\partial t}=0$$ Say we get a solution $S(q,\alpha,t)$ where $\alpha$ is a constant of ...