# Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion

Save this PDF as:

Size: px
Start display at page:

Download "Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion"

## Transcription

1 Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Department of applied physics, Aalto University March 8, 2016

2 Hamiltonian versus Newtonian mechanics Newtonian mechanics: Time evolution is obtained from Newton s second law, the time-evolutions of both position and velocity are computed from the force balance: F = ma = q(e + v B) (1) Hamiltonian mechanics: Time evolution is obtained by computing the Hamiltonian of the system H(P, x, t) in the generalized momentum P and coordinate x and inserting it in the Hamiltonian equations: P = H x, ẋ = H P. (2) Hamiltonian mechanics is particular usefull when the system has more degrees of freedom.

3 Poincare Invariant In periodic motions the action integral I = P dx taken over a period is a constant of motion. Lecture assignment: Prove the Poincare invariant, i.e., show that di dt = 0

4 Poincare Invariant In periodic motions the action integral I = P dx taken over a period is a constant of motion. Lecture assignment: Prove the Poincare invariant, i.e., show that di dt = 0 Solution: di dt = = ( p x t ( H p s + p t p s + H x ) x s x s ( ds = p x s t + p t ) dh ds = ds ds = 0 ) x ds s

5 Adiabatic invariant Adiabatic invariants are first order approximations of the Poincare invariant : If a slow change is made to the system, so that the system is not quite periodic, and the constant of motion does not change it is called a adiabatic invariant. Adiabatic invariances play an important role in plasma physics as they allow us to obtain simple answers in many instances involving complicate motions. There are three adiabatic invariants, each corresponding to a different type of periodic motion in the plasma.

6 The first adiabatic invariant: Magnetic moment Periodic motion: Larmor gyration Guiding-centre frame-of-ref, Gyroangle θ = Ωt, velocity v = u(θ), position r = ρ(θ), momentum P = mv + qa, equation of motion dr dt = v Thus dr = u Ωdθ and expansion of A around R gives I = u [mu + q (ρ ) A] dθ + O(ɛ) Ω = 2πm u2 Ω + 2π q u (ρ )A + O(ɛ) Ω = πm u2 Ω + O(ɛ) = 2π m q µ + O(ɛ) (3) and µ = mu2 2B, the magnetic moment, is an adiabatic invariant

7 Bounce motion and µ Guiding-centre energy in the magnetic field E = 1 2 mv2 + µb Movement towards stronger magnetic field eventually reduces v to zero Reversion of the movement, or bounce point Magnetic mirror was one the first ideas to confine hot plasma

8 Loss cone in a mirror machine Lecture assignment: derive the velocity space condition for the particle trapping in a mirror machine

9 Loss cone in a mirror machine Lecture assignment: derive the velocity space condition for the particle trapping in a mirror machine Solution: E = 1 2 mv mv2 v2 v 2 = E µb 1 (4) for trapped particles E = µb mirror < µb max and the magnetic field B > B min v2 v 2 < B max B min 1 (5)

10 Loss cone in a mirror machine

11 Second adiabatic invariant: Longitudinal particle trapping J = m The guiding-centre trajectory is approximately closed adiabatic invariant is P ds ds is the arc-length along the field line v ds + q (A ˆb)ds = m v ds the contribution from the potential is zero, because no flux through the integration loop Van Allen radiation belts because of invariance of J.

12 Third adiabatic invariant The bounce center of the bounce motion between mirror points drifts in ϕ direction (grad-b and curvature) In mirror machine the configuration is cylindrically symmetric and drift orbit for the bounce center closes a loop J = P ϕ rdϕ = m v ϕ rdϕ + q A ϕ rdϕ = qφ

13 The three types of periodic motion

14 Guiding-centre Lagrangian Charged particle lagrangian L and Hamiltonian H L =(qa + mẋ) ẋ H, (6) H = 1 2 mẋ2 + qφ. (7) Guiding-centre lagrangian and Hamiltonian are derived with Lie perturbation theory. A first order theory gives L gc =(qa + mv ˆb) Ẋ + mµ e ζ H gc, (8) H gc = 1 2 mv2 + µb + qφ (9) with phase-space (X, v, µ, ζ) instead of (x, ẋ)

15 Equations of motion Minimization of the Lagrangian action integral L(q, q, t)dt leads to Euler equations: d L dt q = L q (10) One equation for each phase-space coordinate For a particle this will give the Lorentz force equation (Exercise).

16 Summary Single particle motion: In space and Magnetic fusion, the particle path is governed by the magnetic field The fast gyrating motion can be averaged to reveal guiding-center motion. Collisions can be interpreted as transformations from one orbit to another Adiabatic invariants Help to categorize different time scales in the particle motion Make it possible to develop (Hamiltonian) theories for, e.g, guiding-center dynamics and bounce-center dynamics. These theories are out of the scope of this course.

### Single Particle Motion

Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic

### Motion of Charged Particles in Fields

Chapter Motion of Charged Particles in Fields Plasmas are complicated because motions of electrons and ions are determined by the electric and magnetic fields but also change the fields by the currents

### Single particle motion

Single particle motion Plasma is a collection of a very large number of charged particles moving in, and giving rise to, electromagnetic fields. Before going to the statistical descriptions, let us learn

### Charged particle motion in external fields

Chapter 2 Charged particle motion in external fields A (fully ionized) plasma contains a very large number of particles. In general, their motion can only be studied statistically, taking appropriate averages.

### Single Particle Motion in a Magnetized Plasma

Single Particle Motion in a Magnetized Plasma Aurora observed from the Space Shuttle Bounce Motion At Earth, pitch angles are defined by the velocity direction of particles at the magnetic equator, therefore:

### Single particle motion and trapped particles

Single particle motion and trapped particles Gyromotion of ions and electrons Drifts in electric fields Inhomogeneous magnetic fields Magnetic and general drift motions Trapped magnetospheric particles

### Single Particle Motion

Single Particle Motion Overview Electromagnetic fields, Lorentz-force, gyration and guiding center, drifts, adiabatic invariants. Pre-requisites: Energy density of the particle population smaller than

### Nonlinear Single-Particle Dynamics in High Energy Accelerators

Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 2: Basic tools and concepts Nonlinear Single-Particle Dynamics in High Energy Accelerators This course consists of eight lectures: 1.

### Lecture 16 March 29, 2010

Lecture 16 March 29, 2010 We know Maxwell s equations the Lorentz force. Why more theory? Newton = = Hamiltonian = Quantum Mechanics Elegance! Beauty! Gauge Fields = Non-Abelian Gauge Theory = Stard Model

### Neoclassical transport

Neoclassical transport Dr Ben Dudson Department of Physics, University of York Heslington, York YO10 5DD, UK 28 th January 2013 Dr Ben Dudson Magnetic Confinement Fusion (1 of 19) Last time Toroidal devices

### Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger

### cos 6 λ m sin 2 λ m Mirror Point latitude Equatorial Pitch Angle Figure 5.1: Mirror point latitude as function of equatorial pitch angle.

Chapter 5 The Inner Magnetosphere 5.1 Trapped Particles The motion of trapped particles in the inner magnetosphere is a combination of gyro motion, bounce motion, and gradient and curvature drifts. In

### Methods of plasma description Particle motion in external fields

Methods of plasma description (suggested reading D.R. Nicholson, chap., Chen chap., 8.4) Charged particle motion in eternal electromagnetic (elmg) fields Charged particle motion in self-consistent elmg

### Motion of a charged particle in an EM field

Department of physics Seminar - 4 th year Motion of a charged particle in an EM field Author: Lojze Gačnik Mentor: Assoc. Prof. Tomaž Gyergyek Ljubljana, November 2011 Abstract In this paper I will present

### CHARGED PARTICLE MOTION IN CONSTANT AND UNIFORM ELECTROMAGNETIC FIELDS

CHARGED PARTICLE MOTION IN CONSTANT AND UNIFORM ELECTROMAGNETIC FIELDS In this and in the following two chapters we investigate the motion of charged particles in the presence of electric and magnetic

### the EL equation for the x coordinate is easily seen to be (exercise)

Physics 6010, Fall 2016 Relevant Sections in Text: 1.3 1.6 Examples After all this formalism it is a good idea to spend some time developing a number of illustrative examples. These examples represent

### Lecture 2: Plasma particles with E and B fields

Lecture 2: Plasma particles with E and B fields Today s Menu Magnetized plasma & Larmor radius Plasma s diamagnetism Charged particle in a multitude of EM fields: drift motion ExB drift, gradient drift,

Radiation belt particle dynamics Prepared by Kevin Graf Stanford University, Stanford, CA IHY Workshop on Advancing VLF through the Global AWESOME Network Basic Motion Motion of charged particle q in presence

### Physics of fusion power. Lecture 13 : Diffusion equation / transport

Physics of fusion power Lecture 13 : Diffusion equation / transport Many body problem The plasma has some 10 22 particles. No description is possible that allows for the determination of position and velocity

### Hamiltonian and Non-Hamiltonian Reductions of Charged Particle Dynamics: Diffusion and Self-Organization

NNP2017 11 th July 2017 Lawrence University Hamiltonian and Non-Hamiltonian Reductions of Charged Particle Dynamics: Diffusion and Self-Organization N. Sato and Z. Yoshida Graduate School of Frontier Sciences

### Plasma Physics Prof. V. K. Tripathi Department of Physics Indian Institute of Technology, Delhi

Plasma Physics Prof. V. K. Tripathi Department of Physics Indian Institute of Technology, Delhi Module No. # 01 Lecture No. # 22 Adiabatic Invariance of Magnetic Moment and Mirror Confinement Today, we

### Toroidal confinement devices

Toroidal confinement devices Dr Ben Dudson Department of Physics, University of York, Heslington, York YO10 5DD, UK 24 th January 2014 Dr Ben Dudson Magnetic Confinement Fusion (1 of 20) Last time... Power

### Variation Principle in Mechanics

Section 2 Variation Principle in Mechanics Hamilton s Principle: Every mechanical system is characterized by a Lagrangian, L(q i, q i, t) or L(q, q, t) in brief, and the motion of he system is such that

### Generalized Coordinates, Lagrangians

Generalized Coordinates, Lagrangians Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 August 10, 2012 Generalized coordinates Consider again the motion of a simple pendulum. Since it is one

### The Particle-Field Hamiltonian

The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and

### FYS 3120: Classical Mechanics and Electrodynamics

FYS 3120: Classical Mechanics and Electrodynamics Formula Collection Spring semester 2014 1 Analytical Mechanics The Lagrangian L = L(q, q, t), (1) is a function of the generalized coordinates q = {q i

### APPENDIX Z. USEFUL FORMULAS 1. Appendix Z. Useful Formulas. DRAFT 13:41 June 30, 2006 c J.D Callen, Fundamentals of Plasma Physics

APPENDIX Z. USEFUL FORMULAS 1 Appendix Z Useful Formulas APPENDIX Z. USEFUL FORMULAS 2 Key Vector Relations A B = B A, A B = B A, A A = 0, A B C) = A B) C A B C) = B A C) C A B), bac-cab rule A B) C D)

### Gyrokinetic simulations of magnetic fusion plasmas

Gyrokinetic simulations of magnetic fusion plasmas Tutorial 2 Virginie Grandgirard CEA/DSM/IRFM, Association Euratom-CEA, Cadarache, 13108 St Paul-lez-Durance, France. email: virginie.grandgirard@cea.fr

### Radiation Damping. 1 Introduction to the Abraham-Lorentz equation

Radiation Damping Lecture 18 1 Introduction to the Abraham-Lorentz equation Classically, a charged particle radiates energy if it is accelerated. We have previously obtained the Larmor expression for the

### Geometric Gyrokinetic Theory and its Applications to Large-Scale Simulations of Magnetized Plasmas

Geometric Gyrokinetic Theory and its Applications to Large-Scale Simulations of Magnetized Plasmas Hong Qin Princeton Plasma Physics Laboratory, Princeton University CEA-EDF-INRIA School -- Numerical models

### Space Physics. An Introduction to Plasmas and Particles in the Heliosphere and Magnetospheres. May-Britt Kallenrode. Springer

May-Britt Kallenrode Space Physics An Introduction to Plasmas and Particles in the Heliosphere and Magnetospheres With 170 Figures, 9 Tables, Numerous Exercises and Problems Springer Contents 1. Introduction

### Physical Dynamics (SPA5304) Lecture Plan 2018

Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle

### Dynamics of charged particles in spatially chaotic magnetic fields

PSFC/JA-1-38 Dynamics of charged particles in spatially chaotic magnetic fields Abhay K. Ram and Brahmananda Dasgupta a October 21 Plasma Science and Fusion Center, Massachusetts Institute of Technology

### Lectures on basic plasma physics: Introduction

Lectures on basic plasma physics: Introduction Department of applied physics, Aalto University Compiled: January 13, 2016 Definition of a plasma Layout 1 Definition of a plasma 2 Basic plasma parameters

### Magnetism II. Physics 2415 Lecture 15. Michael Fowler, UVa

Magnetism II Physics 2415 Lecture 15 Michael Fowler, UVa Today s Topics Force on a charged particle moving in a magnetic field Path of a charged particle moving in a magnetic field Torque on a current

### Today s lecture: Motion in a Uniform Magnetic Field continued Force on a Current Carrying Conductor Introduction to the Biot-Savart Law

PHYSICS 1B Today s lecture: Motion in a Uniform Magnetic Field continued Force on a Current Carrying Conductor Introduction to the Biot-Savart Law Electricity & Magnetism A Charged Particle in a Magnetic

### Curves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,

Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal

### Theory for Neoclassical Toroidal Plasma Viscosity in a Toroidally Symmetric Torus. K. C. Shaing

Theory for Neoclassical Toroidal Plasma Viscosity in a Toroidally Symmetric Torus K. C. Shaing Plasma and Space Science Center, and ISAPS, National Cheng Kung University, Tainan, Taiwan 70101, Republic

### Final Review Prof. WAN, Xin

General Physics I Final Review Prof. WAN, Xin xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ About the Final Exam Total 6 questions. 40% mechanics, 30% wave and relativity, 30% thermal physics. Pick

### Fundamental Concepts of Particle Accelerators III : High-Energy Beam Dynamics (2) Koji TAKATA KEK. Accelerator Course, Sokendai. Second Term, JFY2012

.... Fundamental Concepts of Particle Accelerators III : High-Energy Beam Dynamics (2) Koji TAKATA KEK koji.takata@kek.jp http://research.kek.jp/people/takata/home.html Accelerator Course, Sokendai Second

### 0 Magnetically Confined Plasma

0 Magnetically Confined Plasma 0.1 Particle Motion in Prescribed Fields The equation of motion for species s (= e, i) is written as d v ( s m s dt = q s E + vs B). The motion in a constant magnetic field

### Summer College on Plasma Physics August Introduction to Nonlinear Gyrokinetic Theory

2052-24 Summer College on Plasma Physics 10-28 August 2009 Introduction to Nonlinear Gyrokinetic Theory T.S. Hahm Princeton Plasma Physics Laboratory Princeton University USA Introduction to Nonlinear

### The Accelerator Hamiltonian in a Straight Coordinate System

Hamiltoninan Dynamics for Particle Accelerators, Lecture 2 The Accelerator Hamiltonian in a Straight Coordinate System Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. Given

### Chapter 2 Single-Particle Motions

Chapter 2 Single-Particle Motions 2.1 Introduction What makes plasmas particularly difficult to analyze is the fact that the densities fall in an intermediate range. Fluids like water are so dense that

### Physics 2D Lecture Slides Jan 15. Vivek Sharma UCSD Physics

Physics D Lecture Slides Jan 15 Vivek Sharma UCSD Physics Relativistic Momentum and Revised Newton s Laws and the Special theory of relativity: Example : p= mu Need to generalize the laws of Mechanics

### Mechanics IV: Oscillations

Mechanics IV: Oscillations Chapter 4 of Morin covers oscillations, including damped and driven oscillators in detail. Also see chapter 10 of Kleppner and Kolenkow. For more on normal modes, see any book

### M2A2 Problem Sheet 3 - Hamiltonian Mechanics

MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,

### Part II. Classical Dynamics. Year

Part II Year 28 27 26 25 24 23 22 21 20 2009 2008 2007 2006 2005 28 Paper 1, Section I 8B Derive Hamilton s equations from an action principle. 22 Consider a two-dimensional phase space with the Hamiltonian

### Longitudinal Beam Dynamics

Longitudinal Beam Dynamics Shahin Sanaye Hajari School of Particles and Accelerators, Institute For Research in Fundamental Science (IPM), Tehran, Iran IPM Linac workshop, Bahman 28-30, 1396 Contents 1.

### Modelling of Frequency Sweeping with the HAGIS code

Modelling of Frequency Sweeping with the HAGIS code S.D.Pinches 1 H.L.Berk 2, S.E.Sharapov 3, M.Gryaznavich 3 1 Max-Planck-Institut für Plasmaphysik, EURATOM Assoziation, Garching, Germany 2 Institute

### Chapter 4. Symmetries and Conservation Laws

Chapter 4 Symmetries and Conservation Laws xx Section 1 Hamiltonian Legendre transform A system is described by a function f with x, y as independent variables. f = f (x, y) df = f x d x + f dy = ud x

### Overview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities

Overview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities G. W. Hammett Princeton Plasma Physics Lab (PPPL) http://w3.pppl.gov/~hammett AST559: Plasma & Fluid Turbulence Dec. 5, 2011 (based

### Space Physics (I) [AP-3044] Lecture 5 by Ling-Hsiao Lyu Oct. 2011

Lecture 5. The Inner Magnetosphere 5.1. Co-rotating E-field A magnetohydodynamic (MHD) plasma is a simplified plasma model at low-frequency and long-wavelength limit. Consider time scale much longer than

### Introduction to Plasma Physics

Introduction to Plasma Physics Hartmut Zohm Max-Planck-Institut für Plasmaphysik 85748 Garching DPG Advanced Physics School The Physics of ITER Bad Honnef, 22.09.2014 A simplistic view on a Fusion Power

### Physics 106a, Caltech 13 November, Lecture 13: Action, Hamilton-Jacobi Theory. Action-Angle Variables

Physics 06a, Caltech 3 November, 08 Lecture 3: Action, Hamilton-Jacobi Theory Starred sections are advanced topics for interest and future reference. The unstarred material will not be tested on the final

### Planetary Magnetospheres: Homework Problems

Planetary Magnetospheres: Homework Problems s will be posted online at http://www.ucl.ac.uk/ ucapnac 1. In classical electromagnetic theory, the magnetic moment µ L associated with a circular current loop

### PROBLEM SET. Heliophysics Summer School. July, 2013

PROBLEM SET Heliophysics Summer School July, 2013 Problem Set for Shocks and Particle Acceleration There is probably only time to attempt one or two of these questions. In the tutorial session discussion

### 2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson

2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasma physics Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas Transverse waves

### Classical Field Theory

1 Classical Field Theory Oscar Loaiza-Brito 1 Physics Department División de Ciencias e Ingeniería, Campus León, Universidad de Guanajuato January-June 2014 January-June 2018. Series of lectures on classical

### TURBULENT TRANSPORT THEORY

ASDEX Upgrade Max-Planck-Institut für Plasmaphysik TURBULENT TRANSPORT THEORY C. Angioni GYRO, J. Candy and R.E. Waltz, GA The problem of Transport Transport is the physics subject which studies the physical

### PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT. Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford

1 PLASMA: WHAT IT IS, HOW TO MAKE IT AND HOW TO HOLD IT Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford 2 Overview Why do we need plasmas? For fusion, among other things

### Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.

### P321(b), Assignement 1

P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates

### Hamilton-Jacobi theory

Hamilton-Jacobi theory November 9, 04 We conclude with the crowning theorem of Hamiltonian dynamics: a proof that for any Hamiltonian dynamical system there exists a canonical transformation to a set of

### Topics in Fluid Dynamics: Classical physics and recent mathematics

Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:

### Physical Dynamics (PHY-304)

Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.

### J10M.1 - Rod on a Rail (M93M.2)

Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight

### Particle orbits in the multidipole device

4/27/212 Lorentz Eqns. of Motion Multidipole confinement 1 Particle orbits in the multidipole device A simple way to make plasma is to place a heated emissive filament inside of a vacuum chamber with about

### CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017

CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and

### Gyrokinetic Field Theory without Lie Transforms

ASDEX Upgrade Gyrokinetic Field Theory without Lie Transforms for realisable tokamak codes B. Scott Max Planck Institut für Plasmaphysik Euratom Association D-85748 Garching, Germany Lectures in Gyrokinetic

### Problem 1, Lorentz transformations of electric and magnetic

Problem 1, Lorentz transformations of electric and magnetic fields We have that where, F µν = F µ ν = L µ µ Lν ν F µν, 0 B 3 B 2 ie 1 B 3 0 B 1 ie 2 B 2 B 1 0 ie 3 ie 2 ie 2 ie 3 0. Note that we use the

### Reminders: Show your work! As appropriate, include references on your submitted version. Write legibly!

Phys 782 - Computer Simulation of Plasmas Homework # 4 (Project #1) Due Wednesday, October 22, 2014 Reminders: Show your work! As appropriate, include references on your submitted version. Write legibly!

### Massachusetts Institute of Technology Department of Physics. Final Examination December 17, 2004

Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 004 Final Examination December 17, 004 Instructions Do not start until you are told to do so. Solve

### General classifications:

General classifications: Physics is perceived as fundamental basis for study of the universe Chemistry is perceived as fundamental basis for study of life Physics consists of concepts, principles and notions,

### Calculation of alpha particle redistribution in sawteeth using experimentally reconstructed displacement eigenfunctions

Calculation of alpha particle redistribution in sawteeth using experimentally reconstructed displacement eigenfunctions R. Farengo, H. E. Ferrari,2, M.-C. Firpo 3, P. L. Garcia-Martinez 2,3, A. F. Lifschitz

### PHYSICS OF HOT DENSE PLASMAS

Chapter 6 PHYSICS OF HOT DENSE PLASMAS 10 26 10 24 Solar Center Electron density (e/cm 3 ) 10 22 10 20 10 18 10 16 10 14 10 12 High pressure arcs Chromosphere Discharge plasmas Solar interior Nd (nω) laserproduced

Numerical Models for NNP Confinement Dr. Martin Droba Darmstadt 11.2.2008 Contents l NNP (Non-neutral Plasma) l Motivation l High current ring l Codes l Diocotron Instabillity and Diagnostic l Toroidal

### Plasmas as fluids. S.M.Lea. January 2007

Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider

### 2.1 The Ether and the Michelson-Morley Experiment

Chapter. Special Relativity Notes: Some material presented in this chapter is taken The Feynman Lectures on Physics, Vol. I by R. P. Feynman, R. B. Leighton, and M. Sands, Chap. 15 (1963, Addison-Wesley)..1

### Dispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson

2017-02-14 Dispersive Media, Lecture 7 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasmas as a coupled system Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas

### F1.9AB2 1. r 2 θ2 + sin 2 α. and. p θ = mr 2 θ. p2 θ. (d) In light of the information in part (c) above, we can express the Hamiltonian in the form

F1.9AB2 1 Question 1 (20 Marks) A cone of semi-angle α has its axis vertical and vertex downwards, as in Figure 1 (overleaf). A point mass m slides without friction on the inside of the cone under the

### Lagrangian and Hamiltonian Mechanics (Symon Chapter Nine)

Lagrangian and Hamiltonian Mechanics (Symon Chapter Nine Physics A301 Spring 2005 Contents 1 Lagrangian Mechanics 3 1.1 Derivation of the Lagrange Equations...................... 3 1.1.1 Newton s Second

### Physics 11b Lecture #10

Physics 11b Lecture #10 Magnetic Fields S&J Chapter 29 What We Did Last Time Electromotive forces (emfs) atteries are made of an emf and an internal resistance Resistor arithmetic R = R + R + R + + R series

### Homework 3. 1 Goldstein Part (a) Theoretical Dynamics September 24, The Hamiltonian is given by

Theoretical Dynamics September 4, 010 Instructor: Dr. Thomas Cohen Homework 3 Submitted by: Vivek Saxena 1 Goldstein 8.1 1.1 Part (a) The Hamiltonian is given by H(q i, p i, t) = p i q i L(q i, q i, t)

### Lecture 41: Highlights

Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final

### Relativistic Dynamics

Chapter 4 Relativistic Dynamics The most important example of a relativistic particle moving in a potential is a charged particle, say an electron, moving in an electromagnetic field, which might be that

### Gyrokinetics from variational averaging: existence and error bounds 1, a)

Gyrokinetics from variational averaging: existence and error bounds 1, a Stefan Possanner Technical University of Munich, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany Dated: 6

### 1.2 Coordinate Systems

1.2 Coordinate Systems 1.2.1 Introduction One of the critical factors in the development of the AE9/AP9/SPM model was the selection of coordinate systems for mapping particle flux measurements and the

### Physics 607 Exam 2. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2

Physics 607 Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all your

### Guiding Center Orbit Studies in a Tokamak Edge Geometry Employing Boozer and Cartesian Coordinates

Contrib. Plasma Phys. 48, No. -3, 4 8 (8) / DOI./ctpp.839 Guiding Center Orbit Studies in a Tokamak Edge Geometry Employing Boozer and Cartesian Coordinates Y. Nishimura,Y.Xiao,and Z. Lin Department of

### Lecture: Lorentz Invariant Dynamics

Chapter 5 Lecture: Lorentz Invariant Dynamics In the preceding chapter we introduced the Minkowski metric and covariance with respect to Lorentz transformations between inertial systems. This was shown

### 27. Impact Mechanics of Manipulation

27. Impact Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 27. Mechanics of Manipulation p.1 Lecture 27. Impact Chapter 1 Manipulation 1

### Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

### Quantum Black Hole and Information. Lecture (1): Acceleration, Horizon, Black Hole

Quantum Black Hole and Information Soo-Jong Rey @ copyright Lecture (1): Acceleration, Horizon, Black Hole [Convention: c = 1. This can always be reinstated from dimensional analysis.] Today, we shall

### Canonical transformations (Lecture 4)

Canonical transformations (Lecture 4) January 26, 2016 61/441 Lecture outline We will introduce and discuss canonical transformations that conserve the Hamiltonian structure of equations of motion. Poisson