LORENTZ FORCE DECOUPLING A particle of mass m carrying the electric charge q is mobile in a region of space where there is a uniform electric field E=Ezand a uniform magnetic fieldB=By . The particle has zero initial velocity and leaves the origin when t=0. We will note?=mqB? and R=(B?)E?. The frame of reference(O,x,y,z) is assumed to be Galilean and orthonormal. In this repository, the coordinates of the particle are noted (x,y,z). At each moment the particle experiences the force of LorentzqvB+qE when its speed is v. 1. Show that Newton's second law, qvB+qE=m???xy?z????, is translated here by the equation matrix, dtdX?=AX+C, where we noted, X=???x?y??z?????with A=???00??000???00????andC=???00R?2????

2. Show thatA has three eigenvalues ???1?=0,?2?=i? and ?3?=?i?, withi2=?1 3. Show that we can associate the three eigenvalues ??of with the eigenvectors,u1?=???010????,u2?=???i01????,u3?=???10i???? We then define the passage matrixP by putting the eigenvectors in a column (the order is arbitrary), and we give its inverse (which you are not asked to calculate):P????i01?10i?00???? P?1????2?i?21?0?000?21?2?i?0????. In addition, we defineZ=???z1?z2?z3?????=P?1X 4. Calculate the matrix productP?1AP of which we will note the result D. What do you notice?