Hamilton's Equations, Canonical Transformations

Problem:
In a problem with one degree of freedom, a particle of mass m is subject to a force F(x,t)=F0t. The force is derivable from a potential.

(a)  Find the potential energy of the particle and the Lagrangian and Hamiltonian of the particle.
(b)  Solve Hamilton's equations of motion.

Solution:

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Problem:

(a)  Write the Hamiltonian H(p,q,t) in terms of the Lagrangian where are the ith generalized coordinate, velocity, and momentum, respectively. Derive the canonical equations of Hamilton from this definition of Lagrange's equations.

(b)  A particle moving in a central force field has a Lagrangian given by .   Find the Hamiltonian in terms of pr, pq , r, and q . Find the velocity dependent force represented by this Lagrangian.

Solution:

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Problem:

(a) If the Hamiltonian of a system is given by with b = constant, find the corresponding Lagrangian.

(b)  If a system has a Lagrangian show that the Hamiltonian can be written

Solution:

Problem:

Consider a mechanical system with one coordinate q(t) and Lagrangian L(q,dq/dt). Show that the Euler-Lagrange equation for the system implies Hamilton’s equations

,

where H(p,q) is the Hamiltonian of the system.

Student solution: