Is there any relation between the automorphism of the group and group of permutation? If exists, give example.

1. This first question contains a number of unrelated problems that should each be given a short answer. (a) Write down the Euler-Lagrange equations for a system with generalised coordinates (41, 92, ..., qN). (b) What is a symmetry? State the general definition (expressed just in words), and not an example such as translational or rotational symmetry. (c) Give the general definition of the Poisson bracket for a Hamiltonian system described by the scalar parameters (q, p). Write down the three fundamental Poisson brackets and their values. (d) Demonstrate that the first integral of a time-independent Lagrangian is a constant of motion. (e) Starting from the conserved energy E = ² + V(q) for a scalar q, derive the integral formula for the time t = t(q) as a function of q. (f) Let H(q, p) be a Hamiltonian and (q(t), p(t)) a solution of its equations of motion for a fixed energy E. We consider now a new Hamiltonian H'(q,p) obtained through H'(q, p) = f(H(q, p)), where f is an arbitrary…

In Poincare transformation if scalar field is invariant under translation, then prove that generator of translation is momentum 4 vector.

The three matrix operators for spin one satisfy sz Sy – Sy 8z = isz and cyclic permutations. Show that s = sz, (sz tisy )³ = 0. For the sane in, m, can have the values im, m – 1, .., -m, while A12 has eigenvalue m(m + 1). Thus M? = m(m + 1) ±2 5 x 1 = 5 times each once 6 15 4 x 8 = 32 times +3/2 ±1/2 3/2 each 8 times 4 ±1 3 x 27 = 81 times 1 each 27 times 3 2 x 18 = 96 times 1/2 ±1/2 each 48 times 1 × 12 = 42 times 0, each 42 times Total 256 eigenvalues A certain state | 4) is an eigenstate of L? and L,: L'|v) = 1(l+ 1) h² |»), mh|v) . For this state calculate (La) and (L²).