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, 42, ..., 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 nonzero function. We denote (q'(t), p'(t)) the solutions for H' at fixed energy E' = f(E). Show that these trajectories are the same as the previous ones, but are governed by a different time dependence.

Figure 1 shows the 16‐APSK constellation with 3 different arrangements of transmitted carrier states on two rings with a ratio of 2.1756 between outer and inner rings radii. You are to write a MATLAB^® program (m‐file) to implement each of these M‐ary APSK transmissions (where M = 16) through an AWGN channel, including modulation (symbol generation) at the transmitter, addition of white noise in the channel, and symbol detection and determination of bit error ratio (BER) at the receiver. Your message signal will be a random sequence consisting of 403200 integers that have values between 0 and M-1 with equal probability. For each value of E_b/N_o in the range 4 dB to 16 dB in steps of 0.5 dB, where E_b is the average energy per bit and N_o is the noise power per unit bandwidth, run your simulation of the system repeatedly until at least 101-bit errors occur before calculating BER.