Particles and fields: theoretical physics

2) Hamilton Formalism: Kepler Problem a) Set up the Lagrangian Function in Spherical Coordinates (r, 0 , (). b) Momentum Inputs: Specify the canonical pulses (Pr, PO, P Q). c) Set up the corresponding Hamiltonian function in spherical coordinates. d) Specify the cyclic coordinates.a) Construct the Lagrange function in spherical coordinates (r, 0, Q). b) Specify the canonical momenta (p_r, p_0, p_Q). c) Write down the corresponding Hamiltonian in spherical coordinates. d) State the cyclic coordinates. L(q,4, t) =T - V Lagrange function 0 = 7 al d OL Lagrange's motion equation aq. OL P. = Canonical Impulse: al - 0 agi + p; = const Cartesian coordinates: H(p. q.t) = _pi. - L) Hamiltonian function 4=4(9.P.) Pi = - und 9: = Motion equations: (.9) = ) (of ag _of ag) Poisson-brackets (f.9) = -(9. f) Antisymmetric: (f, 191 + 0292) = ci(f.91) + Calf.92). 1, 2 konstant linear (figh) = g(f,h) + (f,9)h Product rule: (f, (9, h)) + (9. (h, f)) + (h, (f.9)) = 0 Jacobi Identity: Movement in the Central Field E = -m(+ 83) + V(r) = -mr mi? + Van(r) Energy: 6 = - Angular velocity: mr? =1\\-(E - Va(r)) Radial Motion Equation " = (ct, z, y, =) Special Theory of Relativity Lorentztransformation: I'M = A" ,x" mit A". = VI - 37 und 8 = Contravariant Tensor Bad = dra ard By Covariant tensor. Energy-Impulse Relationship: = - Electrodynamics V . E = P V X E = - at Maxwell's equations: V . B=0 V x B = Hoj + Hoco at E = -Vo - OA B = V xA Potentials: at Eichtransformation: ax A + A = A+Vx at 6 = 17, B=0 Point charge at the origin Cylinder coordinates E. = sing Es - co Ex - ) Unit vectors: F = rey + ze: Location vector: Gradient