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Questions tagged [integrals-of-motion]

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41
votes
5answers
3k views

Why are we sure that integrals of motion don't exist in a chaotic system?

The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$. Why ...
18
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4answers
3k views

If all conserved quantities of a system are known, can they be explained by symmetries?

If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1 conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver ...
15
votes
1answer
8k views

Constants of motion vs. integrals of motion vs. first integrals

Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
15
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3answers
780 views

Is a system Liouville integrable if and only if its Hamilton-Jacobi equation is separable?

I am asked to show that, a system is completely integrable Liouville if and only if its Hamilton-Jacobi equation is completely separable. I get the idea and understand that is very related to the ...
13
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2answers
974 views

How to prove that a Hamiltonian system is *not* Liouville integrable?

To show that a system is Liouville integrable, we just need to find $n$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For ...
12
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3answers
733 views

What could cause an asymmetric orbit in a symmetric potential?

My question can be summarized as: Given a potential with a symmetry (e.g. $z\rightarrow-z$), should I expect orbits in that potential to exhibit the same symmetry? Below is the full motivation for ...
12
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1answer
884 views

Is there an analog to the Runge-Lenz vector for a harmonic potential?

The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force ...
10
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4answers
4k views

Integrable vs. Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each ...
10
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2answers
895 views

Why are there only 3 Additive Integrals of Motion?

1. I was reading Landau & Lifschitz's book on Mechanics, and came across this sentence on p.19: "There are no other additive integrals of the motion. Thus every closed system has seven such ...
7
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1answer
462 views

Poisson brackets and Hamiltonian Invariants

Consider this Hamiltonian of two degrees of freedom, $$ H=q_1p_1-q_2p_2-aq_1^2+bq_2^2 \, . $$ Define $$A\equiv\frac{p_1-aq_1}{q_2} \hspace{10mm} B\equiv q_1q_2 \, .$$ $A$, $B$, and $C$ are constants ...
7
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2answers
2k views

Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, or ...
6
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1answer
239 views

Analytic proof of the non-integrability of the Henon-Heiles system?

The Henon-Heiles potential is $$ U(x,y ) = \frac{1}{2} (x^2 + y^2 + 2 x^2 y - \frac{2}{3} y^3) .$$ This is a two degree-of-freedom system. The full Hamiltonian is $$ H = p_x^2 + p_y^2 + U(x,y ) ....
6
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1answer
162 views

Finding higher order first integrals of a Hamiltonian

Suppose I have some general Hamiltonian of the form, $$ H(q,p) = \frac{1}{2} g^{\mu \nu} p_{\mu} p_{\nu} + V(q)$$ It is simple to find basic first integrals $F$ such that the Poisson bracket is zero,...
5
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3answers
3k views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
4
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3answers
4k views

Integrals of Motion

Landau & Lifshitz write on the first page of chapter 2 of their Mechanics book (p.13) The number of independent integrals of motion for a closed mechanical system with $s$ degrees of freedom is ...
4
votes
2answers
445 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 ...
4
votes
1answer
93 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 ...
4
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1answer
554 views

2D harmonic oscillator having 4 constants of the motion and superintegrability

A 2D harmonic oscillator \begin{align} H=p_x^2+p_y^2+x^2+y^2 \end{align} has 4 constants of the motion: $E$ the total energy, $D$ the energy difference between coordinates, $L$ the angular momentum ...
4
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1answer
85 views

Constancy of Coefficients of Additive Integrals Throughout Subsystems of a Closed System

I'm studying Landau and Lifshitz's Statistical Physics, Part 1, 3rd edition and am looking for clarification on the following statement, which appears on page 11 in the section on The Significance of ...
4
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1answer
275 views

Integrals of motion for a free particle

I'm struggling to understand the argument on p. 13 in Landau and Lifshitz that for a system with $N$ degrees of freedom there must be $2N-1$ integrals of motion. In particular, I can't understand ...
4
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0answers
63 views

Why and how almost periodic series constitute the algebra of observable of integrable systems?

In the introduction of his book Noncommutative Geometry, p. 42, Connes explains that when a classical dynamical system has enough constants of motions, the motion of the system is almost periodic, ...
3
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2answers
232 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 $...
3
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1answer
384 views

Help finding equations of motion from Hamiltonian with integral of motion

I've given this Hamiltonian with one degree of freedom: $$ H(x,p)=\frac{p^2}{2}+\frac{\omega_0^2x^2}{2}+\lambda\left(\frac{p^2}{2}+\frac{\omega_{0}^2x^2}{2}\right)^2 $$ I need to find the general ...
3
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2answers
73 views

Maximal number of conserved quantities (classical integrability)

In these notes on page 4 the author says that if a $2d$-dimensional phase space has $d$ conserved quantities $F_{\mu}$ that Poisson commute, then $H$ can be written as a function of the $F_{\mu}$. Why ...
3
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0answers
81 views

How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
3
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0answers
585 views

What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
2
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1answer
51 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
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2answers
568 views

Cyclic coordinates don't actually seem to have constant generalised momentum?

Say I am working on a problem where I have two generalized coordinates $q_1,q_2$ and my Lagrangian is independent of $q_1$, i.e. I have $L(\dot{q}_1,q_2,\dot{q}_2)$. Then we say that $q_1$ is a cyclic ...
2
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2answers
98 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 ...
2
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1answer
2k views

Applying velocity Verlet algorithm

I want to implement a simple particules system using the velocity form of the Verlet algorithm as integrator. Initial conditions at $t=0$ for a given particule $p$: mass: $ m $ position: $\...
2
votes
1answer
124 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 ...
2
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1answer
903 views

Angular momentum components as independent integrals of motion

I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so ...
2
votes
1answer
287 views

Action-angle coordinates

I'm studying action-angle coordinates and I've come across two distinct, yet correct, definitions for the new generalized momenta, $\text{J}_k$: $$\text{J}_k \equiv \oint p_k dq_k $$ and $$\text{J}...
1
vote
1answer
104 views

Derivative of line integral with respect to initial conditions

In a fluid mechanics application I compute the following integral to find position given a known velocity field that varies in both time and space: $$ \mathbf{x}(t) = \mathbf{x}_0 + \int_0^t \mathbf{v}...
1
vote
1answer
187 views

What does it mean for a system to be integrable?

I am reading this paper: http://aleph.physik.uni-kl.de/~korsch/papers/JPA_24_45.pdf and in section 3 they talk about the integrability of the system. What does that mean?
1
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2answers
278 views

Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian which doesn't depend on the new coordinates $w_k$, but only in the ...
1
vote
1answer
277 views

Integrals of Motion for s Degrees of Freedom

From Landau & Lifshitz, Classical Mechanics, the number of integrals of independent integrals of motion for a system of $s$ degrees of freedom is $2s-1$. I am considering a spherical pendulum in ...
1
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1answer
78 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
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0answers
45 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
vote
1answer
54 views

Question about a system with all bounded orbits closed and maximal integrable

Given Hamiltonian system with $2n$-dim phase space, if there exist $k\ge n$ independent integrals of motions then we call it integrable Hamiltonian system. The largest number of independent integrals ...
1
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0answers
114 views

Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
1
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0answers
329 views

Finding conserved quantities from Hamiltonian when Symmetry is not evident [closed]

A particle is moving in 3D space, under a potential $$V = -\frac{\alpha}{r}-\frac{\vec{r} \cdot \vec{\mu}}{r^3 } $$ where $\vec{\mu}$ is some constant vector. I need to show there are three ...
1
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0answers
187 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
1
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0answers
885 views

Do symmetries increase the number of conserved quantities? [closed]

Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential ...
0
votes
2answers
71 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_\...
0
votes
2answers
40 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 < \...
0
votes
1answer
251 views

Determining constants of motion for this Hamiltonian? [closed]

Let $S$ be a Hamiltonian system with three degrees of freedom, let $q_1, q_2, q_3$ be the generalized coordinates. Suppose the Hamiltonian of the system is $$ H = \alpha \sum_{i=1}^3 p_i + V(q_1^2 + ...
0
votes
1answer
69 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 ...
0
votes
0answers
18 views

Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below: In free space, an electron, initially at rest at $z=0$, is subjected to an intense laser field $\vec E=\hat x A \cos(\omega t-...
0
votes
1answer
107 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 ...