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Using the symplectic condition, show that the transformation is canonical
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- A ladder of length L and mass M is leaning against a wall. Assuming thewall and the floor are friction-less, the ladder will slide down the wall andalong the floor until the left end loses contact with the wall. Before the ladder loses contact with the wall there is one degree of freedom (θ). Using the formula for Euler-Lagrange EOM to find the Lagrangian and then the EOM for the ladder. Assume that gravity is present.Consider a system with l = 1. Use a three-component basis, |m>, which are the simultaneous eigenvectors of L2 and Lz, where |1> = (1, 0, 0), |0> = (0, 1, 0), and |-1> = (0, 0, 1). In this basis, find the matrix representations for Lx, Ly, Lz, L+, L-, and L2.Consider the functions f(x) = x and g(x) = sin x on the interval (0, ). (a) Complete the table and make a conjecture about which is the greater function on the interval (0, ). (b) Use a graphing utility to graph the functions and use the graphs to make a conjecture about which is the greater function on the interval (0, ). (c) Prove that f(x) > g(x) on the interval (0, ). [Hint: Show that h′(x) > 0, where h = f − g.]
- Evaluate the commutator è = [x², Pe** =?suppose S and T are infinite sets and Φ : S→T is a function that is not onto have S and T the same cardinality? prove (use mapping)Consider the question of finding the points on the curve xy² = 2 closest to the origin. (a) State what function is being minimized for this problem and what the constraint is. Label each. (b) Use Lagrange multipliers to find a system of equations for finding the closest point. Write this system of equations without any vectors. Include the constraint as one of the equations. (c) Solve the system of equations from part (b) to find the points closest on the curve xy² = 2 closest to the origin.