Poincare’-Von Zeipel perturbation method

This article is based on a report entitled “A Pedestrian’s Guide to Lie Transforms: A New Approach to Perturbation Theory in Classical Mechanics” by Robert G. Littlejohn (1978). Before rushing  into Lie transform, understanding the method of average and Poincare’-Von Zeipel perturbation(PV) method would be helpful to understand advantage of Lie trasnform. I’m planning to update this article with more details. 

last modified : 6th Feb. 2010

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ES) How can we express old variables purely in terms of new variables?
Up to ,


ES) Note that on the account of near-identity transformation up to . What if we need to get higher order? We would face difficulties with disentangling mixed variables.
page 5.20 “In the Lie method, canonical transformations are generated without mixing old and new variables, therby completely bypassing the disentangling process.”
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page 5.13

“it is necessary to “disentangle” (5.27) to express old variables purely as a function of new variables.”

ES) Due to the generating function with mixed variables, the disentangling is inevitable in PV method. To the first order, it can be achieved easily since near-identity canonincal transformation is used.

page 5.15

“In (5.38) it is necessary to assume that the denominator does not vanish, which is equivalent to assuming that ther are no “first-order resonances” among the unperturbed oscillators.”

ES) What if first-order resononces exist among the unperturbed oscillators? What can we do?

page. 5.19

“The essence of the perturbation method of Poincare’ and Von Zeipel is the use of near-identity canoncical transformations, generated by mixed variable generating functions, to eliminate the dependence of a Hamiltonian on one or more variables or classes of terms.  ……….. in the case of time-dependent systems, it is possible to choose the transformation so that the new Hamiltonian K is independent of time. Or, with systems with some fast variables and some slow variables, it may be desirable to choose the transformatin so that the dependence on the fast generalized coordinates is eliminated.”

page 6.19
“A much more elegant derivation has been summarized by Cary, who centers his arguments around a certain differential equation in operator space. Cary’s formulas, including a Lie generator equivalent of the Hamilton-Jacobi equation, are expressed in closed form, i.e. not as a power seires in .”

Streamline, Streakline, and Pathline

From Wiki =)

Funny picture

Time dependent field is a key!

The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), previously released ink moves up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time.

Trace and pressure tensor

Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. (Symon (1971) Ch. 10) Thus:

where δ_ij is the Kronecker delta. The first term on the right is the constant tensor, also known as the pressure, and the second term is the traceless symmetric tensor, also known as the shear tensor.

Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7.

From wiki

Working directory

import os
print os.getcwd()

Data visualization tool

A quick look at six open source graphics utilities

M. Tim Jones (mtj@mtjones.com), Senior Principal Software Engineer, Emulex Corp.

http://www.ibm.com/developerworks/linux/library/l-datavistools/

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This is a good document in that the author explains, compares, and recommend data visualization tools.

Fourier series theorem

Fourier Series theorem is relevant to Ballooning representation.

Theorem 21.2 (Fourier Series Theorem)
.

1. Let  be a function which is piecewise continuous on  .Its Fourier series is given by

at each point  where the one sided derivatives  and  both exist.

2. If  is continuous the result is

Excerpt from http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node27.html#Fourier_series_theorem

Ref. A. Thyagaraja, J. Plasma Physics 59, 367 (1998)

Cygwin Setting only for X11

Following module should be installed :

X11>xorg-server (required, the Cygwin/X X Server)

X11>xinit (required, scripts for starting the X server: xinit, startx, startwin.sh, startxwin.bat (and a shortcut on the Start Menu to run it), startxdmcp.bat )

NET> openssh

Material Derivative(Lagrangian)

This comes from Wiki. I just liked this description for distinction between Eulerian and Lagrangian.

—————————————————————–

This static position derivative is called the Eulerian derivative.

An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.

If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature, by swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer’s changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.

That is, the path follows the fluid current described by the fluid’s velocity field v. So, the material derivative of the scalar φ is:

Method of characteristics (from wiki)

For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

For the sake of motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form

		( 1)

Suppose that a solution u is known, and consider the surface graph z = u(x,y) in R³. A normal vector to this surface is given by

As a result, equation (1 ) is equivalent to the geometrical statement that the vector field

is tangent to the surface z = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.

The equations of the characteristic curve may be expressed invariantly by the Charpit-Lagrange equations^[1]

or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):

These are the characteristic equations for the original system.

2D turbulence

2D turbulence – Inverse cascade, self-organization

ref)

http://www.fluid.tue.nl/WDY/2Dturb/2Dturb.html