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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 ...
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### 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 ...
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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 ...
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 ...