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.