Lagrangian, Hamilton and Variations with Constraints
Not what you're looking for?
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).
Purchase this Solution
Solution Summary
The file contains a detailed solution of the three problems posed regarding quantum mechanics and particle physics.
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Probability Quiz
Some questions on probability
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.