Answered: Define a conservative system and state…

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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)

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**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 massless spring with equilibrium length d and spring constant k connects two particles. The system is flat and horizontal, yet it may spin and vibrate (ccompress\stretch).1- Determine the system's Lagrangian.2- Determine the system's Hamiltonian.3- Calculate Hamilton's equations of motion. It should be noted that the generalized momenta can be omitted. -It is worth noting that as the mass spins, it begins to expand. Hint: make your coordinate system's origin the center of the unstretched spring. also In generalized coordinates of (r_i) and (theta_i) , express your equations.
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Theoretical Mechanics Topic: Lagrangian and Hamiltonian Dynamics >Generate the necessary equations to this system. > Use the equations of motion >Generate equations for (x,y), (Vx,Vy), V², T > L = T-U --- For study purposes. Thank you!