Questions tagged [integrals-of-motion]
100 questions
45
votes
5 answers
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 are we sure that no other, independent on $E$,…
Alexey Sokolik
- 2,453
29
votes
1 answer
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 them, $2N-1$ are initial positions and velocity.…
user17116
22
votes
4 answers
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 these conserved quantities by means of Noether's…
Tobias Kienzler
- 6,950
20
votes
4 answers
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 other are zero.
The way I understand it, these…
Ronak M Soni
- 820
20
votes
1 answer
Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric 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 potentials.
I suppose it is not a coincidence…
user2640461
- 821
18
votes
2 answers
Why are there only $1+3+3=7$ 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 integrals: energy, three components of momentum, and three…
Chill2Macht
- 554
18
votes
2 answers
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 example, how to do this for the three-body problem?
Jiang-min Zhang
- 4,262
17
votes
2 answers
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{equation}
where $\mathbf{F_g}$ is the gravitational…
Matías Cerioni
- 341
- 2
- 6
16
votes
3 answers
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 Action-Angle coordinates and one of Liouville's…
César González
- 161
12
votes
3 answers
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 this question.
A while back I came across an…
Kyle Oman
- 18,883
- 9
- 68
- 125
8
votes
1 answer
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 of motion (i.e $\{A,H\} =\{B,H\}=0$), but $C=\{A,…
Sergi
- 297
8
votes
1 answer
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
$$ H = p_x^2 + p_y^2 + U(x,y ) . $$
It is shown by numerics that it is…
kaiser
- 1,227
8
votes
2 answers
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 independent coordinates of $q$ and generalized…
user929304
- 4,910
7
votes
3 answers
What does it mean, when one says that system has $N$ constants of motion?
For example for an isolated system the energy $E$ is conserved. But then any function of energy, (like $E^2,\sin E,\frac{ln|E|}{E^{42}}$ e.t.c.)
is conserved too. Therefore one can make up infinitely many conserved quantities just by using the…
Kostya
- 20,288
7
votes
3 answers
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 on an arbitrary function; such a solution is called…
a06e
- 3,840
- 4
- 41
- 75