Introduction to Liouville's Theorem
Paths in Simple Phase Spaces: the SHO and Falling Bodies
Let's first think further about paths in phase space. For example, the simple harmonic oscillator, with Hamiltonian , describes circles in phase space parameterized with the variables . (A more usual notation is to write the potential term as .)
Question: are these circles the only possible paths for the oscillator to follow?
Answer: yes: any other path would intersect a circle, and at that point, with both position and velocity defined, there is only one path forward (and back) in time possible, so the intersection can't happen.
Here's an example from Taylor of paths in phase space: four identical falling bodies are released simultaneously, see figure, measures distance vertically down. Tw ... Read more »
It's clear that Lagrange's equations are correct for any reasonable choice of parameters labeling the system configuration. Let's call our first choice Now transform to a new set, maybe even time dependent, The derivation of Lagrange's equations by minimizing the action still works, so Hamilton's equations must still also be OK too. This is called a point transformation: we've just moved to a different coordinate system, we're relabeling the points in configuration space (but possibly in a time-dependent way).
General and Canonical Transformations
In the Hamiltonian approach, we're in phase space with a coordinate system having positions and momenta on an equal footing. It is therefore possible to think of more general transformations than the point transformation (which was restricted to the position coordinates).
We can have transformations that mix up position and momentum variables:
where ... Read more »
Maupertuis' Principle: Minimum Action Path at Fixed Energy
Incredibly, Maupertuis came up with a kind of principle of least action in 1747, long before the work of Lagrange and Hamilton. Maupertuis thought a body moved along a path such that the sum of products of mass, speed and displacement taken over time was minimized, and he saw that as the hand of God at work. This didn't go over well with his skeptical fellow countrymen, such as Voltaire, and in fact his formulation wasn't quite right, but history has given him partial credit, his name on a least action principle.
Suppose we are considering the motion of a particle moving around in a plane from some initial point and time to some final . Suppose its potential energy is a function of position, For example, imagine aiming for the hole on a rather bumpy putting green, but also requiring that the ball take a definite time, say two seconds, from being hit to falling in the hole. The action principle we've talked about so far will give the path, parameterized by time, ... Read more »
A New Way to Write the Action Integral
Following Landau, we'll first find how the action integral responds to incremental changes in the endpoint coordinates and times, then use the result to write the action integral itself in a new, more intuitive way. This new formulation shows very directly the link to quantum mechanics, and variation of the action in this form gives Hamilton's equations immediately.
Function of Endpoint Position
We'll now think of varying the action in a slightly different way. (Note: We're using Landau's notation.) Previously, we considered the integral of the Lagrangian over all possible different paths from the initial place and time to the final place and time and found the path of minimum action. Now, though, we'll start with that path, the actual physical path, and investigate the corresponding action as a function of the final endpoint variables, given a fixed beginning place and time.
Taking one degree of freedom (the generalization is straightfor ... Read more »
Time Evolution in Phase Space: Poisson Brackets and Constants of the Motion
The Poisson Bracket
A functionof the phase space coordinates of the system and time has total time derivative
This is often written as
is called the Poisson bracket. (Warning! This is Landau's definition: many others use the opposite sign.)
If, for a phase space function (that is, no explicit time dependence) then is a constant of the motion, also called an integral of the motion.
In fact, the Poisson bracket can be defined for any two functio ... Read more »