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

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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.