33
votes
What exactly are the 12 conserved quantities in the Two-Body Problem?
There's not much difference in having two bodies rather than one. For two bodies, you can separate the center of mass position/velocity, and the relative position/velocity. The six center of mass ...
- 105k
17
votes
Accepted
Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential?
Yes there is an analog of the Laplace-Runge-Lenz vector, even more so! The N-dimensional harmonic oscillator is one of a handful superintegrable systems where you have a maximal number (2 N - 1) of ...
- 66.3k
17
votes
Why are we sure that integrals of motion don't exist in a chaotic system?
I think one must distinguish between chaotic "Hamiltonian" systems and chaotic dissipative systems. In the latter, the phase space volume is not conserved, so it is much more difficult to find "...
- 1,450
12
votes
Why are we sure that integrals of motion don't exist in a chaotic system?
I am not an expert on these issues, but if a further integral existed the orbit would be confined in a codimesion-1 embedded submanifold (for almost all the possible values of that function due to ...
- 78.1k
9
votes
Accepted
Poisson brackets and Hamiltonian Invariants
There shouldn't be any more independent ones. This is a 4D system in phase space, so 3 independent phase-space surfaces (including the Hamiltonian) intersect on a line--a trajectory in phase space. ...
- 66.3k
9
votes
Analytic proof of the non-integrability of the Henon-Heiles system?
General analytic methods to prove non-integrability are discussed in e.g. this Phys.SE post. In this answer, we will sketch how to apply the following Poincare corollary.
Poincare corollary: If an ...
- 213k
8
votes
Is a system Liouville integrable if and only if its Hamilton-Jacobi equation is separable?
In this answer, we elaborate on the various definitions of integrability, separability & AA-property, in order to expose their (slight) differences.
Let there be given a finite-dimensional ...
- 213k
6
votes
Accepted
Finding higher order first integrals of a Hamiltonian
The simple answer, overall, is that you always have to assume an integral of motion in a certain form and look if the conditions for its existence are even fulfilled. Generically, they are not, and if ...
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6
votes
Question about a 2D Harmonic Oscillator with incommensurate frequencies and Integrability
OP has already noted that the 2D harmonic oscillator is completely Liouville-integrable with 2 globally defined, Poisson-commuting, real integrals of motion $H_1$ and $H_2$.
Since the phase space has ...
- 213k
6
votes
Number of integrals of motion
Every autonomous dynamical system arising from mechanics can be described as a first-order ODE on a $2n$-dimensional manifold $M$ (the spacetime of kinetic states of Lagrangian mechanics with local ...
- 78.1k
5
votes
Why are we sure that integrals of motion don't exist in a chaotic system?
The Bunimovich stadium is well known to be ergodic. Here is a nice description by Terry Tao. This notion extends naturally to quantum (or wave) chaos where instead of the trajectories being ...
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5
votes
How to prove that a Hamiltonian system is *not* Liouville integrable?
User Nikos M has already given a good answer. Here we would like to mention the following Poincare theorem, which can be used to prove non-existence of integrals of motion, cf. e.g. this Phys.SE post.
...
- 213k
5
votes
Accepted
Is the motion of a particle in the surface of a torus always periodic?
The OP has different questions on the title and body:
Is the motion of a particle in the surface of a torus chaotic?
Generically, yes, mechanical systems are chaotic — though not always, and the ...
- 12.7k
4
votes
Integrals of Motion
One more easy way to understand it is as follows. Motion of a particle can be described as a trajectory in 2n dimensional phase space (composed of n positions and n conjugate momenta). Now a function ...
4
votes
Accepted
2D harmonic oscillator having 4 constants of the motion and superintegrability
The four invariants
$$
2E_x=p_x^2+x^2, \qquad 2E_y=p_y^2+y^2, \qquad L=yp_x-xp_y, \qquad K=xy+p_xp_y,
$$
are linearly independent, as per the requirement of that article's linear algebraic method; but ...
- 66.3k
4
votes
Why are we sure that integrals of motion don't exist in a chaotic system?
On one hand, a positive maximal Lyapunov exponent (MLE) is often taken as a de-facto definition of (deterministic) chaos. (Note that chaos also requires topological mixing.)
On the other hand, ...
- 213k
4
votes
Accepted
Constants of Integration In Hamilton-Jacobi theory
Well, the logic is as follows:
The HJ equation is a first-order non-linear PDE in $n+1$ variables $(q^1,\ldots q^n,t)$, which may in principle be solved using e.g. the method of characteristics. A ...
- 213k
4
votes
What is the need for angle-action variables in describing integrable systems?
There are several issues, e.g.
OP's functions $(F_1,\ldots,F_n)$ are presumably only locally defined in some open neighborhood. To be integrals of motion they should be globally defined, cf. e.g. ...
- 213k
3
votes
Conserved charge: partial or total derivative?
Hamilton's equation for a quantity $Q$ reads
$$ \frac{dQ}{dt}~=~\{Q,H\}+\frac{\partial Q}{\partial t} .$$
Therefore a constant of motion $Q$ satisfies
$$ \frac{dQ}{dt}~=~0 \qquad \Leftrightarrow\...
- 213k
3
votes
Determining if constants of motion are independent
Picking up on the check mentioned in J.G.'s answer:
For a $2n$ dimensional phase space, there are at most $2n$ independent constants of motion.
Similarly, there are at most $2n-1$ independent ...
- 213k
3
votes
Why are we sure that integrals of motion don't exist in a chaotic system?
This depends in ways on the system. Yet integrable domains can exist. The logistics map $x_{n+1}~=~rx_n(1~-~x_n)$ has for the parameter $r$ zones of stability. The image
illustrates bifurcation ...
- 12.9k
3
votes
Procedure for finding integrals of motion
A single autonomous (possibly nonlinear) 2nd-order ODE
$$F(x,\dot{x},\ddot{x})~=~0. \tag{1}$$
can in principle be written as a couple of autonomous (possibly nonlinear) 1st-order ODEs of the form
$$\...
- 213k
3
votes
Accepted
Reducing the degrees of freedom of a Lagrangian in a spherical potential by using integrals of motion
The main point is that one is not allowed to use EOM in the Lagrangian. For such problems one should instead form (minus) the Routhian
$$-R(r,\dot{r};\theta,\dot{\theta};\phi,p_{\phi})~=~L - p_{\phi}\...
- 213k
3
votes
Accepted
Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$
The 3D spherically symmetric harmonic oscillator $$ H~=~\frac{p_x^2+p_y^2+p_z^2}{2m}+ \alpha (x^2+y^2+z^2) ~=~ H_x + H_y + H_z $$ is a separable, Liouville integrable, and in fact a maximally ...
- 213k
3
votes
Global conserved quantities for point particle coupled to a Schrodinger field
Let us consider two quantum-mechanical particles, with coordinates $x$ and $X$ and momenta $p$ and $P$, interacting cia a central potential. Their joint Hamiltonian can be written as
$$
H=\frac{p^2}{...
- 65k
3
votes
Conserved Quantities in Kepler Problem?
He could (and should) count the mechanical energy as a conserved quantity. He also didn't mention that the total momentum of the 2 objects is 3 integrals of motion. However, not all quantities are ...
- 213k
3
votes
Particularity of symmetries generated by the action variables of a classically integrable system
My answer will be to try to be more intuitive than in-depth quantitative-wise. If the latter is what you seek, I would recommend Olver's Book "Applications of Lie Groups to Differential Equations&...
- 941
2
votes
Is the free 3D rigid body/Euler top an integrable system?
Ref. 1 argues that the 3D free rigid body/Euler top (which is well-known from e.g. the intermediate axis theorem/tennis racket theorem/Dzhanibekov effect) is an integrable system$^1$ as follows. Let ...
- 213k
2
votes
Accepted
Constancy of Coefficients of Additive Integrals Throughout Subsystems of a Closed System
For the $a$ subsystem:
$$ \log \rho_a = \alpha_a + \beta_a E_a + \mathbf{\gamma}_a \cdot \mathbf{P}_a + \delta_a \cdot \mathbf{M}_a . $$
Writing down this equation assumes that Liouville's theorem ...
- 46
2
votes
Accepted
Help finding equations of motion from Hamiltonian with integral of motion
I want simplification, so I'll take $\omega_0 = 1$. You can rescale $x$ and always get this kind of simplification, if $\omega_0$ is a constant. The Hamiltonian then becomes
$$H(x,p) = \frac{p^2}{2} +...
- 4,113
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