Questions tagged [integrals-of-motion]
The integrals-of-motion tag has no usage guidance.
96 questions
3
votes
1
answer
222
views
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 ...
- 31
17
votes
2
answers
3k
views
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{...
- 331
5
votes
1
answer
203
views
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 ...
- 1,000
2
votes
0
answers
89
views
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/...
- 43
2
votes
1
answer
438
views
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. ...
- 938
1
vote
1
answer
53
views
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 ...
- 31
0
votes
0
answers
101
views
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 ...
- 799
1
vote
1
answer
199
views
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 ...
- 219
-1
votes
1
answer
66
views
RMS Speed derivation Integral [closed]
I am looking for a good table of integral to use for a class, but I also need help solving an integral that leads to RMS SPEED of a gas particle. The integral is of the following form:
$\int_{0}^{\...
1
vote
0
answers
119
views
Integrability of one-dimensional system of motion?
How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)?
It is clear that if $H$ does not depend explicitly on time then $H$ is indeed a constant ...
1
vote
1
answer
131
views
How to obtain the $2s-1$ integrals of motion for a mass with a spring?
In according with Landau's Mechanics the number of independent integrals of motion for a closed mechanical system with $s$ degrees of freedom is $2s-1$. We can express the $2s-1$ arbitrary constants $...
- 41
0
votes
0
answers
36
views
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 ...
- 41
2
votes
1
answer
57
views
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 ...
- 1,748
2
votes
3
answers
556
views
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 ...
- 5,821
0
votes
2
answers
61
views
Arguments of specific Hamiltonian, always conserved?
I'm studying an introductory course in theoretical physics, I stumbled upon something I really can't understand.
So, in my book there is written the following statment:
Consider a Hamiltonian system $...
- 77
1
vote
2
answers
201
views
Is 2D rectangular billiard an integrable system? What's the form of explicit solution?
Suppose the free particle moving inside 2D box.
$$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+V$$
where the potential is zero inside the box and infinite outside the box.
It's clear that $p_x,p_y$ are not ...
- 12.1k
3
votes
2
answers
569
views
When is the first integral equal to the total energy of the system?
For a function that is a solution of the Euler-Lagrange equation there is a constant known as the first integral which is given by:
$$E=\sum_i(\frac{\partial f}{\partial y'_i}y'_i)-L$$
I am trying to ...
- 187
1
vote
0
answers
25
views
ODEs with rational first integrals [closed]
I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$
...
- 11
1
vote
1
answer
1k
views
Calculate distance based on - Resistance - Acceleration - Time - Initial Velocity
I have been looking at equations that can represent how far something travels given these three variables. Initial Velocity, Resistance, Acceleration, and Time.
The main thing I saw which was similar ...
- 111
6
votes
2
answers
524
views
Global conserved quantities for point particle coupled to a Schrodinger field
We have a box (represented by a potential $V$) with a classical particle in it. If the box has a finite inertia and it's floating in space, then it shakes as the particle bumps on the walls. The total ...
- 5,244
4
votes
2
answers
717
views
Number of integrals of motion
In Landau-Lifshitz Classical mechanics textbook, it is said that there are generally $2s-1$ integrals of motion where $s$ is the number of degrees of freedom. Why is that? I couldn't find anywhere an ...
- 1,048
-1
votes
1
answer
51
views
Question of ball falling down.Difficulty in understanding the formula
A ball is thrown upward from the top of a tower 40m high. u = 10m/s.Find time for it reach AD.
g = $10m/s^2$.
Taking upwards direction as +ve and downward as -ve.
u = +10m/s.$g=-10m/s^2$.s=-40.
$-40 = ...
- 735
1
vote
1
answer
526
views
Example of time-dependent constant of motion in classical mechanics
In classical mechanics text, when learning about Poisson brackets, one gets
$\frac{df}{dt} = \{f,H\} +\frac{\partial f}{\partial t}$, where $H$ is the Hamiltonian of the system and for $\frac{df}{dt}=...
- 13
1
vote
1
answer
43
views
What term describes the trajectory space splitting behavior when parametrizing a pendulum?
So I was thinking about this post I made earlier: What is the second conserved Quantity of the Pendulum?
In which a pendulum appears two have significant properties. It's Kinetic Energy and its Phase.
...
- 3,009
1
vote
1
answer
352
views
2D harmonic oscillator trajectory
Consider the Hamiltonian for the classic planar harmonic oscillator:
$$H = H_x + H_y$$
where $$H_x~=~\frac{1}{2}(p_x^2+x^2), \qquad H_y~=~\frac{1}{2}(p_y^2+y^2).$$
So it is possible to obtain a set of ...
- 647
1
vote
0
answers
56
views
Constants of motion [duplicate]
For any system performing any kind of motion with $n$ degrees of freedom, are $2n-1$ integrals of motion and also $2n$ constants of motion always present? If yes, then is there always a symmetry for ...
2
votes
1
answer
118
views
Integral of Motion in the 1D Calogero Model
In section 4 of this article, suppose we have $N$ particles of the same mass $m$ moving in one dimension and interacting with each other via the potential $V_{ij}\equiv V(x_i - x_j)$, where $x_i$ ...
- 21
1
vote
2
answers
227
views
Are partial derivatives in the context of Action-Angle variables different from partial derivatives of functions?
Let's say I have a system with two degrees of freedom and I can find two independent action variables. One action variable is total energy expression, such as is often used in classical mechanics.
$$...
- 13
2
votes
1
answer
532
views
Conserved Quantities and Integrability in the $N$-Body Problem
Under my understanding of integrability, a system with $2n$-dimensional phase space is integrable when there are at least $n$ constants of motion satisfying some conditions (e.g., they are in ...
- 109
2
votes
0
answers
117
views
Grand-canonical partition function for an integrable system
When speaking about the grand-canonical ensemble of a statistical system, one usually works with a case, when there are several conserved quantities - total number of particles $N$, angular momentum $...
- 4,893
1
vote
0
answers
91
views
Problem with symmetry and integral of motion in Classical Mechanics [closed]
I am currently working through the problem 4.10 in Kotkin and Serbo's book "Collection of Problems in Classical Mechanics". The problem consists in showing that the quantity
\begin{equation}
...
- 39
6
votes
1
answer
629
views
Question about a 2D Harmonic Oscillator with incommensurate frequencies and Integrability
In Classical Dynamics by José & Saletan [section 4.2.2] they give the example of a 2D Harmonic Oscillator whose equations of motion are
\begin{equation}
\ddot{x}_i+\omega_i^2x_i=0 \ \ \ \ \ \text{...
- 4,737
2
votes
0
answers
44
views
Does the Hamilton-Jacobi equation imply that there are always $N$ conserved quantities for any system with $N$ degrees of freedom? [duplicate]
I'm reviewing the Hamilton-Jacobi equation because I'm working on a research project about Kerr black holes and the geodesics of particles gravitating them (This is not really relevant to the question,...
- 4,737
2
votes
0
answers
51
views
Destruction of integrals of motion in chaotic systems: Fermi-Pasta-Ulam (FPU) paradox
I am trying to understand behavior of system studied by Fermi, Pasta and Ulam i.e. chain of oscillators interacting via nonlinear forces. I am generally not very familiar with chaos theory and ...
- 41
1
vote
1
answer
840
views
Action-angle variables for anharmonic oscillator
I have an equation of potential given:
$$U = U_0\tan^2( \alpha(t)q)$$
I need to find a motion rules for that potential in terms of action-angle variables. Using the fact that Hamiltonian is equal to ...
- 19
4
votes
1
answer
518
views
Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$ [duplicate]
I know that in the case of the Kepler problem there is a quantity (other than energy, momentum,...) conserved which is the Runge-Lenz vector.
Is there also an "exotic" quantity conserved for a 2-Body ...
- 1,748
1
vote
0
answers
131
views
How to construct quasi-local integrals of motion in many-body localized (MBL) phase?
How does a set of quasi-local operators behave as integrals of motion in the presence of disorder to induce an effective integrability into the system?
Which principle should we use to construct local ...
- 11
3
votes
1
answer
168
views
Reducing the degrees of freedom of a Lagrangian in a spherical potential by using integrals of motion [duplicate]
I'm sure I've made a silly mistake here, so I would be very grateful if someone could help me clear it up! Here is my reasoning:
The Lagrangian in a spherical potential is
$$
\mathcal{L}=\frac{m\...
- 2,168
2
votes
1
answer
165
views
Why must the logarithm of the distribution function depend only upon additive integrals of motion (Landau & Lifshitz)?
Denote by $\rho(p,q)$ (the $p$ and $q$ are being used as shorthand for several degrees of freedom), the phase space probability distribution function, (so $\rho\,\text{d}p\text{d}q$ is the probability ...
- 553
0
votes
2
answers
131
views
Period of Small Oscillations for Perturbation on SHO
I am trying to find the period of small oscillations of the potential
$$
V(x) = \frac{1}{2}m\omega_0^2(x^2-bx^4)
$$
It is given that the particle oscillates between $-a$ and $a$ for some $a < \...
- 733
0
votes
2
answers
279
views
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 ...
- 108
3
votes
3
answers
472
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 ...
- 847
4
votes
2
answers
1k
views
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 ...
- 851
2
votes
1
answer
1k
views
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 ...
- 647
1
vote
2
answers
317
views
Conserved charge: partial or total derivative?
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
242
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....
- 1,252
0
votes
1
answer
150
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 ...
user148792
3
votes
2
answers
742
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 $...
user148792
1
vote
0
answers
82
views
How is BRST symmetry related to local integrals of motion?
I'm hoping someone can confirm or check my reasoning below:
In this wiki, they describe caos in a classical system as the spontaneous symmetry breaking of a BRST.
In this stackexchange, they clarify ...
- 1,444
4
votes
1
answer
578
views
Constants of Integration In Hamilton-Jacobi theory
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 ...
- 588