From: Harald Hanche-Olsen
Subject: Re: Non-linear differential equation
Date: 08 Jun 2000 12:43:34 -0400
Newsgroups: sci.math.research
Summary: [missing]
+ attilam@libero.it (Attila):
| Solving a problem of rational mechanic, I got this differential
| equation:
|
| y'' + ay - by^3 = 0
|
| Does exist an analythic solution?
Multiply the equation by y' and integrate once to get
(1/2)(y')^2 + (a/2)y^2 - (b/4)y^4 = constant
which is a separable equations whose solution can be expressed in
terms of some elliptic integral.
| Considering that y is a positione and y'' un'acceleration, does
| exist a method to state whether the orbit in the phase plane
| (i.e. in the plane y, y') is periodical?
From the above you can see that the orbit lies on a level curve of the
Hamiltonian H(p,y) = (1/2)p^2 + (a/2)y^2 - (b/4)y^4 [where p=y'].
Of course, this is true of all Hamiltonian systems, of which this is
an example; i.e., those on the form y'=-dH/dp, p'=dH/dy [where you
should read the d's as partials].
Anyway, if b>0 you will find that some orbits are unbounded while
others are indeed closed curves.
--
* Harald Hanche-Olsen
- "There arises from a bad and unapt formation of words
a wonderful obstruction to the mind." - Francis Bacon
==============================================================================
From: parendt@nmt.edu (Paul Arendt)
Subject: Re: Non-Hamiltonian Flow in Physical Systems?
Date: 5 Sep 2000 17:34:21 GMT
Newsgroups: sci.physics.research
Summary: [missing]
In article <8p19p5$nqi$1@Urvile.MSUS.EDU>,
John Baez wrote:
>In article <8omij9$h10$1@nnrp1.deja.com>, wrote:
>>Are there any physical systems whose time evolution is described by a
>>non-Hamiltonian symplectic flow in the phase-space?
If so, you should expect that energy isn't conserved...
>First of all, a symplectic flow can only fail to be generated
>by a Hamiltonian if the phase space isn't simply connected.
... so here's an example! Suppose we have a pendulum that can
swing all the way around in a circle (and gravity is absent).
Then phase space is a cylinder, with nice coordinates p along
the cylinder axis and (locally) q around the circle of it.
The symplectic form can be written w = dp ^ dq, as usual.
Subject the pendulum to a constant force F in the clockwise
direction, say. Then time evolution is given by (partials)
d/dt = p/m d/dq + F d/dp.
We can solve for dH easily:
w(d/dt, . ) + dH = 0
==> dH = p/m dp - F dq
... but H = p^2/(2m) - F q doesn't cut it, since q isn't
defined globally.
>After all, a flow gives a vector field that we can convert into a
>1-form using the symplectic structure; the flow will be symplectic
>(i.e. preserve the symplectic structure) iff this 1-form is closed.
>In this case, we can always *locally* write this 1-form as dH for some
>Hamiltonian H - which will then be the Hamiltonian generating our
>symplectic flow. But *globally* we will run into trouble if the
>1-form fails to be exact.
And here's an ultra-cool theorem: the commutator of two locally
Hamiltonian vector fields is a globally Hamiltonian vector field.
The construction relies on the fact that the Poisson bracket
of F and G really only depends upon dF and dG. So we can define
{F,G} globally if dF and dG are known, even when F and G don't
exist globally. Then {F,G} is a Hamiltonian for the commutator
of the vector fields generated by dF and dG.
==============================================================================
From: parendt@nmt.edu (Paul Arendt)
Subject: Re: Non-Hamiltonian Flow in Physical Systems?
Date: 5 Sep 2000 23:10:10 GMT
Newsgroups: sci.physics.research
Paul Arendt wrote:
>Subject the pendulum to a constant force F in the clockwise
>direction, say. Then time evolution is given by (partials)
>
> d/dt = p/m d/dq + F d/dp.
Whoops -- this is a COUNTERclockwise force with the usual
convention for angle. Oh, bother.