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Lagrangian, Hamilton and Variations with Constraints

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1. The ground-state energy of a quantum particle of mass m in a pillbox (right-circular cylinder) is given by the following equation (see attachment). Find the ratio of R to H that will minimize the energy for a fixed volume.

2. A particle, mass m, is on a frictionless horizontal surface. It is constrained to move so that theta = wt (rotating radial arm, no friction).
(a) Find the radial positions as a function of time
(b) Find the force exerted on the particle by the constraint

3. A point mass m is moving over a flat, horizontal, frictional plane. The mass is constrained by a string to move radially inward at a constant rate. Using plane polar coordinates:
(a) Set up the Lagrangian
(b) Obtain the constrained Lagrange equations
(c) Solve the phi-dependent Lagrange equation to obtain w(t), the angular velocity. What is the physical significance of the constant of integration that you get from your `free`integrationÉ
(d) Using the w(t) from part (b), solve the p-dependent (constrained) Lagrange equation to obtain theta(t). In order words, explain what is happening to the force of constraint as p-->0.

(See attached file for detailed questions).

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The file contains a detailed solution of the three problems posed regarding quantum mechanics and particle physics.

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