Consider a particle of mass of m which is constrained to move on the surface of a sphere of radius R. There are no external forces of any kind on the particle. How many generalised coordinates is required to describe this system? Construct the Lagrangian and obtain Lagrange's equations of motion for this system

**Problem 3.23 Consider the following Hermitian matrix: 2 i 1 T = -i i 1 -i 2 (a) Calculate det(T) and Tr(T). (b) Find the eigenvalues of T. Check that their sum and product are consistent with (a), in the sense of Equation 3.82. Write down the diagonalized version of T. (c) Find the eigenvectors of T. Within the degenerate sector, construct two linearly independent eigenvectors (it is this step that is always possible for a Hermitian matrix, but not for an arbitrary matrix-contrast Problem 3.18). Orthogonalize them, and check that both are orthogonal to the third. Normalize all three eigenvectors. (d) Construct the unitary matrix S that diagonalizes T, and show explicitly that the similarity transformation using S reduces T to the appropriate diagonal form.

a) Define a conservative system and state how the lagrange's equations are modified b) State three advantages why Lagrangian approach to solving problems is more superior compared to Newtonian approach c) Use Lagrange's equation to obtain the equations of motion of a linear harmonic oscillator (Hint: the system is conservative)