Problems involving the Hamiltonian

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:

Problem:

A block of mass m moves on a frictionless surface with a displacement x under the influence of a spring force F = -kx,  where k is the spring constant.
(a)  Calculate the potential energy U(x), and the Lagrangian.
(b)  Find Lagrange's equations of motion.
(c)  Find the Hamiltonian.
(d)  Find Hamilton's equations of motion.

Solution:

Problem:

A point particle of mass m is constrained to move in one dimension on the x-axis.
The force acting on the particle can be derived from the velocity-dependent potential
U = F(dx/dt)t.
Write down
(a)  the Lagrangian,
(b)  the Hamiltonian,
(c)  Hamilton’s equations.

Solution:

Problem:

A particle of mass m moves in one dimension under the influence of a force

F(x,t) = -k x exp(-t/τ),

where k and τ are positive constants.  (Note carefully the dependence on x to the first power).
(a)  Compute the Lagrangian function.
(b)  Use Lagrange's equation to determine the equation of motion explicitly.
(c)  Compute the Hamiltonian function in terms of the generalized coordinate and generalized momentum.  (Show clearly how you get this.)
(d)  Determine Hamilton's equations of motion explicitly for this particular problem (not just general formulae).
(e)  Does the Hamiltonian equal the total energy?
(f)  Is the total energy of the mass conserved?
(g)  What is it about the force F which supports your answer to part f?

Solution:

Problem:

The Lagrangian for a simple spring is given by  L = ½m(dx/dt)2 - ½kx2.
Find the Hamiltonian and the equations of motion using the Hamiltonian formulation.  Identify any conserved quantities.

Solution:

Problem:

Consider the Lagrangian L = m(dx/dt)(dy/dt) - mω02xy.
(a)  Write down Lagrange's equations associated with this Lagrangian and solve them.  What physical system does this Lagrangian describe?
(b)  Determine the Hamiltonian of the system.
(c)  Define new generalized coordinates x' and y' such that
x = 2 (x' + iy'), y = 2 (x' – iy'). 
Write down the Lagrangian and Lagrange's equations in terms of the new generalized coordinates and velocities and solve them.
(d)  Express the total energy of the system in terms of the new generalized coordinates and velocities, assuming these coordinates are real.

Solution:

Problem:

Consider a simple plane pendulum consisting of a mass m attached to a string of length l.  After the pendulum is set into motion, the length of the string is shortened at a constant rate

dl/dt = -Θ = constant.

The suspension point remains fixed.  Consider only times for which l0 - Θt > 0, i.e. for which the mass has not yet contacted the suspension point.
(a)  Compute the Lagrangian, write down Lagrange's equation of motion, and solve it for small displacements from equilibrium.
(b)  Compute the Hamiltonian function.
(c)  Compare the Hamiltonian and the total energy and discuss the conservation of energy for this system.

Solution:

Problem:

Consider the 2D problem of a free particle of mass m moving in the xy plane.
(a)  Use the Lagrangian formalism to find the equations of motion of the particle using Cartesian coordinates (x, y) in an inertial reference frame.
(b)  Now switch to explicitly time dependent coordinates.
X = xcos(ωt) + ysin(ωt),  Y = -xsin(ωt) + ycos(ωt). 
These would be the Cartesian coordinates of the particle in a reference frame rotating with constant angular speed ω counterclockwise about the z-axis of the inertial reference frame, i.e. ω = ωk.  Write down the Lagrangian in terms of the coordinate X and Y and the corresponding velocities.   Note:  x = Xcos(ωt) - Ysin(ωt),  y = Xsin(ωt) + Ycos(ωt).
(c)  Find the equations of motion in terms of the new coordinates.  Identify the fictitious forces appearing in the rotating frame.
(d)  What are the conditions for a circular orbit of radius R = (X2 + Y2)½ about the origin in the rotating frame?
(d)  Find the Hamiltonian H, and use Hamilton's equations to find the equations of motion.

Solution:

Problem:

Consider a system described by the Lagrangian L = ½mv2 - ½kx2, where v = dx/dt.
(a)  Express L in terms of the generalized coordinate q = (km)¼x and the associated generalized velocity.
(b)  In terms of q and its canonical momentum p derive the form of the Hamiltonian H.  Is H a constant of motion?  Is it the energy E of the system?
(c)  Derive Hamilton's equations of motion and solve for q(t) and p(t).
(d)  Elaborate on the trajectories of the system in phase space.

Solution: