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Questions tagged [integrals-of-motion]

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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
239 views

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 ...
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 ...
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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 $a < \... 1answer 251 views Determining constants of motion for this Hamiltonian? [closed] Let$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 + ... 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 ... 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-...
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 ...