PhD and MPhil Thesis Classes - Université Libre de Bruxelles

3.2 Hamiltonian Mechanics0 = − d ∂Ldt ∂ ˙q i + ∂L∂q i (3.2)In general these are a set of second order differential equations for each configurationpoint; their solution requires that an initial condition (q(0), ˙q(0)) be given.Equations in Hamiltonian form are obtained by the Legendre transformation fromvelocity to canonical momentum:The definition of the canonical momenta isH(p, q) = p · ˙q − L(q, ˙q, t) (3.3)3.2.2 Maxwell-Lorentz Equationsp = ∂ L(q, ˙q, t) (3.4)∂ ˙qA plasma is a gas of charged and neutral particles viewed on scale lengths that arelarge compared to the Debye Length (see sec.1.8.2).Plasma usually contain several species of charged particles, typically electrons withcharge e e = −e and mass m e , and at least one species of ions with charge e i = Z i e andmass m i .Each (non-relativistic) particle can be described by the LagrangianL(q(t), ˙q(t), t) = m s2 | ˙q|2 − e s Φ(q, t) + e s ˙q · A(q, t) (3.5)cwhere Φ(q, t) is the electrostatic potential and A(q, t) is the electromagnetic potential.The electric, E, and magnetic, B, fields can be characterized by the electrostatic andelectromagnetic potentials asE = −∇Φ − 1 ∂Ac ∂tB = ∇ × A (3.6)27