Objective and Purpose

• Introducing some kinetic (nonlinear or linear) processes in plasma physics

• Emphasizing those topics relevant to solar-terrestrial physics rather than basic nonlinear theories

• No profound and sophisticated theories will be discussed.

My Suggestions

• Do not always agree with me.• I encourage you to form your own opinion

and have independent viewpoint.• During the course try to envision new

research task by yourself.• I would like to combine lectures with

discussions as much as possible.

Introductory Remarks

The scope of nonlinear plasma physics

Nonlinear versus Linear Plasma Physics

• Nonlinear and linear processes are generally not separable.

• Nonlinear processes are often studied with linear theories.

• Some nonlinear plasma processes we face everyday: heating and acceleration

Plasma Physics: Methodology & Approaches

• Single particle motion• Magneto-hydrodynamics approach• Kinetic and statistical theories• Wave kinetics • Wave-particle interactions• Wave-wave interactions• Linear and nonlinear approaches

Nonlinear Plasma Physics

• Nonlinear wave kinetics

mid 1960s to early 1970s

• Quasilinear kinetic theories

1960s~

• Nonlinear particle dynamices

1970s~

Topics to be lectured

• Introducing nonlinear plasma physics and dynamics

(4.7.09)• Perception of microscopic and macroscopic physics

(4.9.09)• Ion pickup (4.14.09)• Pitch-angle scattering (4.16.09)• Waves, fluctuations and turbulence (4.21.09)• Wave modes of general interest (4.23.09)• Proton heating (4.28.09)• Heating by Alfven waves (4.30.09)• Quasilinear theory and non-resonant interactions (5.5.09)• Quasilinear theory of Alfven waves (5.7.09)• Collisions and fluctuations (5.12.09)• Auroral kilometric radiation (5.14.09)• Solar radiophysics (O)(5.29.09)• Solar radiophysics (T) (6.2.09)• Alfvenic turbulence in various space regions (6.4.09)• Solar wind helium, cosmic ray modulation (6.9.09)

Sub-areas of nonlinear dynamics

• Nonlinear particle motion

• Approximate and special methods

• Nonlinear waves

• Particle motion in wave fields

Important References

• A. J. Lichtenberg and M. A. LiebermanRegular and Stochastic DynamicsSpringer-VerlagNew York (1992)

• R. Z. Sagdeev, D. A. Usikov, and G. M. ZaslavskyNonlinear PhysicsHarwood academic publishers, London(1988)

Nonlinear dynamics has a long history

• Weakly inhomogeneous system

Einstein (1911)

WKB method (1926)

• Nonlinear oscillations in three-body system in astronomy

Poincare (1892)

Von Zeipel (1916)

Historical Remarks

• Galileo (1564-1642)• I. Newton (1642-1727)• Lagrange (1736-1813)• W. R. Hamilton (1805-1865)• Jacobi (1804-1851)• Gauss (1777-1855)• S. Lie • H. Poincare

Lagrangian Equation of Motion

It is derived from the principle of least action

This work was done when he was about 19!

0, ( 1,..., )i i

d L Li n

dt q q

Lagrangian and Energy Constant

Since

we conclude

dL L L d L L d Lq q q q q

dt q q dt q q dt q

.L

q L H constq

The Hamilton’s equations

• The Hamiltonian equations of motion are

H Hq and p

p q

Hamiltonian formulation is widely used in nonlinear dynamics

• Perturbation methods play important roles in solving nonlinear problems

• Hamiltonian formulation enables theorists to develop systematic methods for a general class of problems.

• The Jacobian-Hamiltonian transformation makes the Hamiltonian approach even more powerful.

Why nonlinear dynamics is of interest

in plasma physics?

Some primary issues attracted attention:

• Stability of particle motion under nonlinear force or field

• The origin of stochasticity

• Evolution from local stochasticity to global stochasticity

Hamiltonian for waves

• In the early theories it is often considered that

• are conjugate canonical variables of the waves associated with wave number k while denotes frequency.

• Such an approach is no longer used. Instead the action-angle variables are preferred.

2 2 21

2w k k kk

H P Q

,k kP Q

k

Action-Angle Variables

• For a periodic system a pair of canonical variables are useful. These are the action and angle variables.

• The action is defined as

which is often a constant of motion.

1

2 cJ pdq

• The line integral is defined over a period of motion in the canonical coordinate q while p is the conjugate canonical momentum.

• The action J has a dimension of energy times time.

• If we consider J to be a canonical variable its conjugate pair is the phase angle of the periodic motion. The reason is that the pair must satisfy the Hamiltonian equation of motion.

The equations of motion are:

• If the Hamiltonian is not a function of phase angle, the action J is not time dependent. Then for a system that is autonomous, Hamiltonian may be written a very simple form

H

J

HJ

H J

System with more than one degree of freedom

• So far, for simplicity, we have only consider one degree of freedom. If the system has several degrees of freedom, the Hamiltonian may be written as

• In the following we give an example.

1 1 2 2 3 3 ....H J J J

An Example

• Particle moving in a magnetic field with an axially symmetrical configuration. In such a field the Hamiltonian takes the form

• We have used cylindrical coordinates in which the local magnetic field is in the z direction. There is a vector potential .

2

2 21

2 r z

pH qA p p

m r

• The azimuthal component of canonical momentum is

• In the following we study several actions.• The first one is

which is the magnetic moment.

2p mr qA r

22

1

1 1

2 2 2rC T

mvJ p dr mv dt

Bounce Motion• We consider that the magnetic field forms an

adiabatic trap so that a particle can be trapped due to mirror effect. The action of interest is

• The integral is defined over a bounce period. is an averaged kinetic energy .

2

22

' 2z z

z zb bC T

m v WJ p dz mv dt

zW

Azimuthal drift motion

• We suppose that the magnetic field has a mild gradient in the radial direction so that particles can execute drift motion in the azimuthal direction. The corresponding action is

• The integral is defined over the azimuthal angle.

23

1( )

2 C

J r mr qA d mr qrA qrA

• The above discussions show that there are three adiabatic invariants.

• However, one should always keep in mind that these invariants are only in the approximate sense.

• It is assumed implicitly that the magnetic field varies very mildly.

Adiabatic Invariants

• In general adiabatic invariants are discussed in either asymptotic or approximate sense.

• The system is considered to be either slightly time dependent or spatially weakly inhomogeneous.

• Adiabatic invariant breaks down in the presence of resonance.

Particle Motion

Trajectories in phase space

Liouville’s theorem

Poincare’s integral invariants

Resonance

Action-angle variables

Poincare’s mappings

Singularity and resonance

• Singularity is usually of mathematical origin while resonance is of physical nature.

• In some cases singularity is created unnecessarily and may not be physically important.

Generic Phase Space Orbits

Kol’mogorov (1954), Arnol’d (1963) and Moser (1962) (KAM) show: when an integrable Hamiltonian system is perturbed there are in general three types of orbits in phase space.

Four Primary Types of Orbits

• Stable periodic (regular) orbits

• Stable quasi-periodic orbits

(KAM tori)

• Chaotic (stochastic) orbits

• Arnol’d diffusion

Particles in wave fields

• Nonlinear effects of wave field may result in regular or stochastic particle orbits in a wave field.

• Stochastic orbits are due to high order nonlinear resonance.

Methods

Approximate methodsperturbation theoriesaveraging methodsSpecial methodsKAM theorynonlinear resonancephase space trajectories

Three Cases of General Interest

• An oscillating system driven by an external force which is a function of the amplitude.

• The system is driven by a periodic force which tends to modulate the oscillation.

• An oscillating system which is slowly varying.

Perturbation Methods

• In the first case nonlinear effect can invalidate the simple-minded method of perturbation due to secular terms in the perturbation solution.

• In the second case resonance can lead to singularity.

• In the third case the problem of so-called “singular perturbation” appears.

Three Perturbation Methods

• The Poincare’s method

• Method of removing resonance

• Slowly varying system

Expansions for small perturbation and slow perturbation

• For small perturbation

• For slow perturbation

0 0 1( ) ( , , ) ...H H H t J J θ

0 0 1( , , ) ( , , , ) ...H H J y t H J y t

Numerical Method

• A useful numerical method by Poincare is often adopted.

Poincare’s section

Early KAM theorem

Different chaotic regions are often isolated by KAM tori

More recent theories show

• In high-dimensional cases due to “Arnold diffusion” , now a well known phenomenon, the chaotic orbits can reach every where in phase space.

• When the perturbation increases the KAM tori destabilize and become discrete sets

Issues

• Single particle dynamics versus kinetic theory

• Mathematics versus physics

• Numerical versus analytic methods

• Common sense versus usefulness

Comments

• The stochastic particle motion found in nonlinear dynamics may be relevant to plasma heating.

• But heating is a process involving many particles. Motion of single particle is not conclusive.

• Relative motion among particles may be chaotic while the motion of each particle is still regular.