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Extra info for USPAS - Classical Mechanics and Electromagnetism for Accelerators and Beams
Example: particle at rest use a quantity which is Lorentz invariant ! 4-Vector ? USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA 44 how do we formulate a Lagrange function? example: particle at rest use a quantity which is Lorentz invariant ! energy-momentum 4-vector for particle at rest: 1 , cp , cp , iE cp x y z c space-time 4-vector 0, 0, 0, imc dx , dy , dz , icdt product of both here, particle system is most convenient 0, 0, 0, imc dx , dy , dz , icdt mc 2 dt mc 2 1 2 dt and L mc 2 1 2 USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA 45 how do we formulate a Lagrange function?
Q f , H, p x , p y , p z , . . p f 0 Hamilton’s equations: dq l d H p l ; dp l d qH ; l for l 0, 1, 2, . . f USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA 64 Extended Hamiltonian – 2 for l0 dq 0 d and dp 0 d 1 H q 0 H H q0 C 1 t H q 0 dp 0 d H t dH dt p 0 H C 2 0 H const.
4 0. 56 relativistic 20 MeV electrons: A 1 highly 2 mc 0. 511 2 v c 1 0. 999 67c 2 20 USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA 33 summary of formulas relativistic factor total energy E 2 c 2p 2 A 2m 2 c 4 momentum cp E velocity 1 1 2 or 1 2 E Amc2 1 1 1 E 1 kin Amc 2 Ekin Amc2 E2kin 2Amc2 Ekin USPAS JAN 19-30, 2004, College of William & Mary, Williamsburg, VA Ekin Amc2 34 Emission of Radiation, spectral and spatial distribution consider EM wave in particle system phase of the wave is: t i E0e 1 c n x x n y y E n z z Phase is product of two 4-vectors!