Adiabatic Invariant

It seems adiabatic invariant is important to analyze and understand physics of nature heuristically like magnetic moment which is usually assumed as a conserved quantity in plasma physics field. To understand adiabatic invariant in the structure of classical mechanics, we need to deal with some theory.

This article is written to understand more deeply where adiabatic quantities come from. When I solve some problems of a particle motion in slowly varing field( Electric or Magnetic field), I did a mistake that I thought energy of particle is a conserved quantity, and the problem required adiabatic moment conserving flux of mangtic field in the circle of drift motion of the particle. To exactly figure out this kind of conserved quantities is a final object of this article. Following is a very similar example of above problem,

<Excerpt from Classical Mechanics by Goldstein>

“At the first Solvay Conerence 1911, which grappled with the problems of introducing quantum notions into physics, a deceptively simple problem in classical mechanics was raised. Consider a bob on a string oscillating as a plane pendulum, with the string pssing through a small hole in the ceiling. Now imagine that the string is either pulled up or let down slowly, so slowly that there is little change in the length of the pendulum during one period of oscillation. What happens to the frequency of oscillation during this process?”

 

To figure out how it works, I’ll cover following subjects.

1. Hamilton-Jacobi equation

2. Action-Angle Principle

3 Canonincal Perturbation Theory.

 

Example )

1. A counterintuitive problem by Greg Hammet

http://w3.pppl.gov/~hammett/courses/gpp1/counter-intuitive.pdf

 

MHD and Turbulence

http://www.damtp.cam.ac.uk/user/as629/PartIIIL05/course_blogL05.html

 

Energy Principle Ref.

The original famous paper on the MHD energy principle is I. B. Bernstein, E. A. Frieman, M. D. Kruskal & R. M. Kulsrud, Proc. Roy. Soc. LondonA244, 17 (1958)
Lagrangian formulation of MHD and the action principle are discussed in the excellent original paper by Newcomb:
W. A. Newcomb, Nucl. Fusion: 1962 Supplement, Part 2, p. 451 (distributed in class)
A more recent useful reference is D. Pfirsch & R. N. Sudan, Phys. Fluids B 5, 2052 (1993)

My favorite songs

Pullback

<Wiki>

The notion of pullback in mathematics is a fundamental one. It refers to two different, but related processes: precomposition and fiber-product.

Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y(x).

Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ^*f on M defined by (φ^*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ^-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)

More generally, if f:N→A is a smooth map from N to any other manifold A, then φ^*f(x)=f(φ(x)) is a smooth map from M to A.

<Frankel> p.57

The presence of the Poincare’ 1-form field on T*M and the capability of pulling back 1-form fields under mappings endow T*M with a powerful tool that is not available on TM.

싸이월드에 대한 단상

모든 홈페이지나 블로그 등이 여러 역할이 있겠지만, 무엇인가를 누군가에게 보여주는 역할이 강하다는 것에 대해서는 부인할 수 없을 것이다. (물론, 이 블로그는 자료의 저장과 내 기억을 위해서 만들어지긴 했다만 이 글은 누군가를 위해서 쓰는지도 모르겠다.) 자신을 다른 사람에게 알리고 싶은 욕구에 대한 표출일 수 있겠다. 달리 말하면 사랑받고자 하는 욕망인가.

싸이월드를 했었고, 그리고 마지막으로 탈퇴를 하면서 느낀 바는, 내가 항상 일기를 남기고 있다는 것이다. 일기는 관념적으로 남에게 보여주기 보다는 주로 사적인 영역으로 감추어지는 경향이 크다.  그럼에도 불구하고 계속적으로 일기를 남기고자 했었던 것이, 어떻게 보면 내가 다른 사람에게 말하고 싶었지만 하지 못했던 것들을 해결하려고 했었던 것이 아닌가 싶다.  또한 내가 다른 사람들의 여러 일기들을 들여다보면서, 그들의 생활에 관심을 가지게 되고 궁금증을 가지게 되었었다. 이런 반응들을 목적으로 적혀져 있는 일기들이 있음을 부인하지는 못할 것이다.

 어쨌든, 과거의 싸이월드가 아닌 오프라인에서 적던 일기가 누군가에게 보여주기 위해서 적던 일기가 있었는지 궁금하다.  과거의 일기는 비밀 일기에 가깝겠다. 하지만 싸이월드에서의 일기들은 비밀 일기라기 보다는 읽고 상대의 반응을 바라는, 비밀을 가장한 목적 일기에 가까운 것이 아닌가라는 생각이 든다.

이러한 알게 모르게 인간 심리를 움직이는 것이 나를 열어 놓음으로 상대와 더욱 가까워 질 수 있다는 순기능도 있겠지만, 상대의 마음을 이용하여 원하는 바를 얻으려 하는 역기능이 될 수도 있겠다. 점점 사람의 마음에 교묘함을 심는 일기가 순수하게 하루를 회고하고 반성하던 일기의 역할을 대체하게 됨을 느껴 싸이월드를 그만 둘 수 밖에 없었다.

Geometry Reference Books

I’ve been tried to figure out differentiail geometry to use for gyrokientics. However, it was hard to find a book easily and readably describing all contents for non-mathematician. In addition, sometimes each book has their own notations or describes a same thing with other words or definition, which makes me more hard to understand geometry. 

It’s just reference books I’ve read. The description is not objective. It’s just my feeling.

THE GEOMETRY of PHYSICS (An Introduction) , Theodore Frankel

- My ex-officemate recommended this book. The book is thick, but it’s worthwhile spending time to read this book. Easy description and several figures.  Currently I’m reading it.

Geometry, Topology and Physics , Nakahara

- Readable book. Not too hard. This book pertains to quantum mechanics. It’s one of well-known books for Geometry.

Mathematical Methods of Classical Mechanics , V.I. Arnold

- One of well-known books for advanced classical mechanics. Because of focusing on classical mechanics, I felt the part introducing geometry is somewhat weak. If someone doesn’t have previous knowledge on Geometry, it can be hard to read after Chapter 7.

Flux Coordinates and Magnetic Field Structure, W.D. D’haeseleer, W.N.G.Hitchon, J.D.Callen, J.L.Shohet

- I studied geometry-related things for the first time via this book. It has intro. of geometry and focuses on practical calculation with respect to fusion plasma. More practical rather than abstract.

Plasma Confinement, R.D. Hazeltine, J.D. Meiss

- One of my favorite books. Plasma Physics. Lots of manipulations of differential calculation in it.  But, hard to understand contents in some points becuase the contents is too hard to catch for the first time.

2007 Student in PPPL

Symplectic manifolds

<Wiki>

In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g. in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field: The set of all possible configurations of a system is modelled as a manifold, and this manifold’s cotangent bundle describes the phase space of the system.

Any real-valued differentiable function H on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to the Hamilton–Jacobi equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville’s theorem, Hamiltonian flows preserve the volume form on the phase space.

*nondegenerate :  A nondegenerate bilinear form on a vector space V is one such that the map from V to V*( the dual space of V) is an isomorphism

 Definition :

A symplectic form on a manifold M is a nondegenerate closed two-form ω. Explicitly, nondegeneracy of the form means that, relative to any given basis X_i of the tangent space of M at a point, the matrix

is nonsingular (meaning that its determinant is non-zero). Note that Ω, being a skew-symmetric non-singular matrix, must have an even number of rows and columns. Thus the dimension of M is necessarily an even number 2n. In intrinsic terms, ω is nondegenerate if and only if its n-th exterior power is non-zero:

Furthermore, ω is required to be closed, meaning that

dω = 0

where d is the exterior derivative.

* The exterior power

The k-th exterior power of V, denoted Λ^k(V), is the vector subspace of Λ(V) spanned by elements of the form

한글, 그림, 수식 테스트

한글도 등록이 잘 되는지 테스트!!! =)

Intro. of blog

Even if I have a public homepage in PPPL, I make this blog in that I need some webpages supporting convenient functions like LATEX, easily editing and uploading graphics, and so on.

I plan to use this blog to summarize private results of calculation and other private stuffs in my life.