Hamilton's Equations, Canonical Transformations

Problem:
Given is the Lagrangian   .

(a)  Find the Hamiltonian.
(b)  Find the generator of the canonical transformation that converts this Hamiltonian into the form P2/2m+(1/2)kQ2.  Use Poisson brackets to verify explicitly that the transformation is canonical.

Solution:

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Problem:
A point mass m is constrained to move along a straight massless rod, which rotates with given angular velocity W around an axis fixed in space and perpendicular to the rod. Let q be a linear coordinate along the rod with q=0 at the axis of rotation. The point mass is bound to q=0 by a harmonic force Fq=-mw02q, where w0 is the vibrational frequency when the rod does not rotate.

(a)  Write the Lagrangian and Hamiltonian function for this system and show that, if  , the solutions to the canonical equations of motion are , with , where f0 and A are constants of integration and .

(b)  Show that the transformation from generalized coordinates q and p to f and A (or its inverse), which is implied by the result of part (a), is a canonical transformation, and find the generating function F1=F1(q,f ,t).

(c)  Now consider the case that   Find a suitable new Hamiltonian in the coordinates f and A and develop the equations of motion for f and A.

Solution:

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Problem:
A mass moves without friction on a plane, subject to the force Fq=keq . At t=0, q=0, r=r0¹ 0, .

(a)  Write down the Lagrangian and Hamiltonian functions, L an H.
(b)  State the Poisson bracket condition on a constant of motion and use the condition to determine whether or not pr is a constant of motion.
(c)  Find  .
(d)  It is proposed that the problem may be simplified by transforming qq , pq to coordinates Qq=eq , Pq=pq . Show whether or not this is a canonical transformation.

Solution:

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Problem:
The set of equations

defines a canonical transformation for the dynamical system with the Hamiltonian function

.

(a)  Find the transformed Hamilton function K.
(b)  Solve Hamilton's equations of motion for the transformed Hamiltonian K, and hence for the original system.
(c)  Explain the physical meaning of l1, l2, P1, P2, Q1, Q2.

Solution:

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Problem:
A set of generalized coordinates, (q1,p1), (q2,p2) is transformed to a new set (Q1,P1), (Q2,P2) through

Q1=q12, P1=P1(q1,p1,q2,p2)

Q2=q1+q2, P2=P2(q1,p1,q2,p2).

(a)  Find the most general expression for P1 and P2 if the transformation is canonical.
(b)  Show that a particular choice will reduce to .
(c)  Solve Hamilton's equations in terms of the new variables.

Solution:

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Problem:

Consider the transformation from the canonical position and momentum variables q and p to a new set of variables Q and P given by Q=qaebp, P=qae-bp, where a and b are constants. For which values of a and b is this a canonical transformation? Obtain a generating function for the canonical transformation.

Solution: