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 ; i = 1, 2,..., d} of the physical system, and their time derivatives q = { q i ; i = 1, 2,..., d}. The Lagrangian may also have an explicit dependence of time t. Lagrange s equations There is one equation for each generalized coordinate. Generalized momentum d L L = 0, i = 1, 2,.., d. (2) dt q i q i p i = L q i, i = 1, 2,.., d. (3) is also referred to as canonical or conjugate momentum. There is one generalized momentum p i conjugate to each generalized coordinate q i. The Hamiltonian H(p, q) = d q i p i L (4) is usually considered as a function of the generalized coordinates q i and momenta p i. i=1 Hamilton s equations q i = H p i, ṗ i = H q i, i = 1, 2,.., d (5) (6) Standard expressions for L og H L = T V H = T + V (7) with T as kinetic energy and V as potential energy. There are cases where H is not the total energy. 1
Charged particle in electromagnetic field (non-relativistic) 2 Relativity Space-time coordinates General Lorentz transformation L = L(r, v) = 1 2 mv2 eφ + ev A H = H(r, p) = 1 2m (p ea)2 + eφ (8) (x 0, x 1, x 2, x 3 ) = (ct, x, y, z) = (ct, r) (9) x µ x µ = L µ νx ν + a µ (10) Special Lorentz transformation with velocity v in the x direction x 0 = γ(x 0 βx 1 ) with β = v/c and γ = 1/ 1 β 2, and x 2 og x 3 are unchanged. Condition satisfied by Lorentz transformation matrices Invariant line element x 1 = γ(x 1 βx 0 ) (11) g µν L µ ρl ν σ = g ρσ (12) s 2 = r 2 c 2 t 2 = g µν x µ x ν = x µ x µ (13) Metric tensor 0, µ ν g µν = 1, µ = ν = 0 1, µ = ν 0 Upper and lower index x µ = g µν x ν, (x µ ) = (ct, r), (x µ ) = ( ct, r) x µ = g µν x ν, g µρ g ρν = δ ν µ (14) 2
Proper time - time dilatation dτ = 1 ds c 2 = 1 dt, (15) γ dτ: proper time interval = time measured in an (instantaneous) rest frame of a moving body (by a co-moving clock) ds 2 : invariant line element of an infinitesimal section of the object s world line dt: coordinate time interval = time interval measured in arbitrarily chosen inertial system Length contraction L = 1 γ L 0 (16) Lengths of a moving body measured in the direction of motion. L 0 : length measured in the rest frame of a moving body L: length measured (at simultaneity) in an arbitrarily chosen inertial frame. Four velocity Four acceleration U µ = dxµ dτ = γ (c, v), U µ U µ = c 2 (17) A µ = du µ dτ = d2 x µ dτ 2, Aµ U µ = 0 (18) Proper acceleration a 0 Acceleration measured in instantaneous rest frame, A µ A µ = a 0 2 (19) Four momentum p µ = m U µ = mγ(c, v) = ( E, p) (20) c with m as the (rest) mass of a moving body. Relativistic energy E = γmc 2 (21) γm is sometimes referred to as the relativistic mass of the moving body. 3
3 Electrodynamics Maxwell s equations Maxwell s equations in covariant form Electromagnetic field tensor E = ρ ɛ 0 B 1 c 2 t E = µ 0j B = 0 E + t B = 0 (22) ν F µν = µ 0 j µ, ν x ν ν F µν = 0, F µν 1 2 ɛµνρσ F ρσ (23) F 0k = 1 c E k, F ij = ɛ ijk B k F 0k = B k, F ij = 1 c ɛ ijke k (24) Four-current density (j µ ) = (cρ, j) (25) Charge conservation Electromagnetic potentials µ j µ = 0, t ρ + j = 0 (26) E = φ t A, B = A (27) Four potential F µν = µ A ν ν A µ, (A µ ) = ( 1 φ, A) (28) c Lorentz force Force from the electromagnetic field on a point particle with charge q Potentials from charge and current distributions in Lorentz gauge, µ A µ = 0: F = q(e + v B) (29) φ(r, t) = 1 ρ(r, t ) 4πɛ 0 r r dv A(r, t) = µ 0 4π 4 j(r, t ) r r dv (30)
Retarded time t = t 1 c r r (31) Electric dipole moment p = rρ(r)dv (32) Electric dipole potential (dipole in origin) Force and torque (about the origin) φ = n p 4πɛ 0 r 2, n = r r (33) F = (p )E, M = p E (34) Magnetic dipole moment m = 1 2 r j(r) dv (35) Magnetic dipole potential (dipole in the origin) Force and torque (about the origin) A = µ 0 m n 4πr 2, n = r r (36) F = (m B) (current loop), M = m B (37) Lorentz transformation of the electromagnetic field Lorentz invariants Special Lorentz transformations F µν = L µ ρl ν σf ρσ (38) E 2 c 2 B 2 = c2 2 F µνf µν E B = c 4 F µν F µν (39) E = E, E = γ(e + v B) B = B, B = γ(b v E/c 2 ) (40) The fields are decomposed in a parallel component ( ), along the direction of transformation velocity v, and a perpendicular component ( ), orthogonal to v. Electromagnetic field energy density u = 1 2 (ɛ 0E 2 + 1 µ 0 B 2 ) = ɛ 0 2 (E2 + c 2 B 2 ) (41) 5
Electromagnetic energy current density (Poynting s vector) S = 1 E B (42) µ 0 Monochromatic plane waves, plane polarized E(r, t) = E 0 cos(k r ωt), E 0 = E 0 e 1 B(r, t) = B 0 cos(k r ωt) ; B 0 = B 0 e 2 E 0 k = B 0 k = 0, B 0 = 1 c n E 0, n = k k (43) Monochromatic plane waves, circular polarized Polarization vectors Four-wave vector E(r, t) = Re (E 0 exp[i(k r ωt)]), E 0 = E 0 1 2 (e 1 ± ie 2 ) B(r, t) = Re (B 0 exp[i(k r ωt)]), B 0 = B 0 1 2 (e 2 ie 1 ) (44) e 1 k = e 2 k = 0, e 1 e 2 = 0, e 2 1 = e 2 2 = 1 (45) (k µ ) = ( ω, k), ω = ck (46) c Radiation fields, in the wave zone (r >> r, λ) B(r, t) = µ 0 n 4πc r d dt j(r, t )dv, n = r r Electric dipole radiation E(r, t) = cb(r, t) n (47) B(r, t) = µ 0 n p(t r/c), E(r, t) = cb(r, t) n (48) 4πc r Radiation from accelerated, charged particle B(r, t) = µ 0q 4πcr [a n] ret, with r(t) as the particle s position vector. Radiated power, Larmor s formula E(r, t) = cb(r, t) n ret n = R/R, R(t) = r r(t) (49) P = µ 0q 2 6πc a2 (50) 6