PHY5210 W15

Chapter 13: Hamiltonian Mechanics

Reading:

In Taylor, read sections 13.5 to 13.6 for today, and 14.1 to 14.2 for Monday.

Recall

H = T + U

for natural generalized coordinates. Hamilton's equations in multiple dimension are

pidot = -∂H /∂ qi        qidot = ∂H /∂pi

Lagrange's Equations versus Hamilton's Equations

It is worth exploring some of the reasons that Hamilton's equations appear in more advanced work. You might expect that it involves a connection to quantum mechanics, but this is not the reason.

Hamilton's equations can be thought of as 2n first order differential equations of the form

zdot = h(z)

where z is a 1×2n dimensionsal matrix (vector) of all the generalized coordinates and conjugate momenta, z = (q, p). To completely specify the initial conditions of a system, one must specify all the coordinates and all the momenta (or velocities, but it is convenient to use momenta here). This corresponds to specifying an initial value of z = z0. Then Hamilton's equations specify how z evolves in time, that is the trajectory of the motion in phase--space.

From this, one can erive the following corollary: trajectories in phase space do not cross. For if they did, then at the point where they cross, the trajectories would have to be the same going forward in time. And since Hamilton's equations can be reversed in time, the two trajectors would have to be the same going backward in time. Therefore, the two trajectories would be the same, not two different trajectories that cross.

Phase--Space Orbits

A One--Dimensional Harmonic Oscillator

Set up Hamilton's equations for a one--dimensional simple--harmonic oscillator with mass m and force constant k, and describe the possible orbits in the phase space defined by the coordinates (x, p).

The Hamiltonian is

H = p²/2m + ½ m ω² x²

Hamilton's equations are

xdot = ∂H/∂p = p/m   and   pdot = -∂H/∂x = -m ω² x

These are as expected. The phase--space orbits are ellipses. This is most easily seen by noting that the Hamiltonian is the total energy E, so we can rewrite it as

p²/2mE + x²/(2E/mω²) = 1

This is the equation for an ellipse since the constants are all positive. The ellipses for different orbits are the same shape, scaled up or down depending on the energy of the orbit. It is clear that the orbits do not cross.

Falling Bodies

Consider a body folling vertically downward subject to the force mg xhat, where the x direction is increasing downward, and describe the orbits in the phase space defined by the coordinates (x, p).

Consider the phase--space orbits for bodies started with initial conditions (x0, p0) = (0, 0), (X, 0), (0, P), and (X, P). Follow their trajectories for time t and compare the phase--space region defined by these four bodies at time 0 and time t.

The orbits are parabola. Again they do not cross, and following the initial points for to a later time we find that the region defined by the four points changes shape from a rectangle to a parallelogram. But the region of the rectangle and parallelogram are the same. This is an example of the result of Liouville's theorem.

Liouville's Theorem in a Nutshell

The falling body example is a simple demonstration of Liouville's theorem. This theorem basically says that for a collection of particle trajectories, the area occupied by the trajectories in phase--space at a particular instant of time is conserved.

This is an example of a more advanced, abstract topic that can be addressed with the Hamiltonian technique. Although abstract, it is directly applicable to bunches of charged particles in an accelerator structure. The bunch of particles occupies some region in phase space, and that region must be conserved. If one wants to focus the bunch into a small region of space, the spread of momenta must increase so that the size in phase space remains constant. There are ways to avoid this limitation, but it involves additional particles that can absorb the growth in phase space and be thrown away.

Virial Theorem in General

For more details see, for example, Fowles and Cassidy. It is possible to demonstrate the Virial theorem with Lagrangian or even Newtonian techniques. But a nice general derivation can be made with Hamiltonian techniques.

The Virial theorem states that for bound systems, averaged over time, the kinetic energy is related to the potential energy. For a potential proportional to rn, the time averaged kinetic energy is (see Prob. 8.17 in Taylor)

< T > = n < U >/2

Poisson Brackets

For more details on this material, see for instance, Landau and Lifshitz. Consider the time derivative of a function f = f(q, p, t)

df/dt = ∑i [ (∂f/∂qi) qidot + (∂f/∂pi)pidot ] + ∂f/∂t = ∑i [ (∂f/∂qi)(∂H/∂pi) - (∂f/∂pi)(∂H/∂qi) ] + ∂f/∂t

The quantity in square brackets is defined as the Poisson bracket of f with H

[ H, f ] = ∑i [ (∂H/∂pi)(∂f/∂qi) - (∂H/∂qi)(∂f/∂pi) ]

With the Poisson bracket notation the total time derivative is written

df/dt = [ H, f ] + ∂f/∂t

We can also generalize the Poisson bracket to any two functions of the coordinates and momenta, f and g

[ f, g ] = ∑i [ (∂f/∂pi)(∂g/∂qi) - (∂f/∂qi)(∂g/∂pi) ]

This is not exactly the Poisson bracket of quantum mechanics. Despite this fact, they do share some mathematical properties, for example

[ f, g ] = -[ g, f ]
[ f, c ] = 0
[ f1 + f2, g ] = [ f1, g ] + [ f2, g ]
[ f1 f2, g ] = f2 [ f1, g ] + f1 [ f2, g ]
(∂/∂t)[ f, g ] = [ ∂f/∂t, g ] + [ f, ∂g/∂t ]

© 2015 Robert Harr