Questions tagged [integrals-of-motion]

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How many constants of motion are there in a 2D two body problem?

A system consists of two masses interacting with gravitational force, rotating around their centre of mass. If we only consider the $xy$ plane where the masses rotate, the system has 8 degrees of ...
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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 $...
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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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How many independent constants of motion are there?

Any mechanical system with $f$ degrees of freedom can be described by a set of $2f$ first-order Hamilton's equations. Solving this set of equations leads to $2f$ constants of integration. Equivalently,...
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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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3 answers
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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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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 $...
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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 ...
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3 votes
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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 ...
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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 $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ ...
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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 ...
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2 answers
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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 ...
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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 ...
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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 = ...
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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}=...
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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. ...
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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 ...
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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 ...
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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 $x_i$ ...
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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. $$...
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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 ...
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2 votes
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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 $...
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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} ...
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1 answer
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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{...
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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,...
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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 ...
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1 vote
1 answer
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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 ...
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4 votes
1 answer
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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 ...
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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 ...
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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\...
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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 < \...
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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 ...
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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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4 votes
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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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2 votes
1 answer
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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 ...
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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_\...
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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....
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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 ...
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2 answers
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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 $...
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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 ...
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4 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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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 ...
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What do I need an integral of motion for and why should I care?

I'm fine with Lagrangian mechanics without an integral of motion, but for some reason I keep seeing these integrals mentioned within the context of the Lagrangian with no real explanation as to their ...
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1 answer
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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?
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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 ...
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2 votes
2 answers
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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 ...
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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 ...
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4 votes
1 answer
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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 ...
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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,...
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