I'm tired of thinking about one set of technical thoughts, so it's time to take a break and think about a different set of technical thoughts. These are technical thoughts, but since trying to typeset mathematics in HTML is generally about as pleasant as self-performed dental surgery, I will try to keep myself to a minimum of notation. Perhaps I will also achieve a minimum level of incomprehensibility in the process?
There are three main frameworks for writing the equations of classical mechanics. Students who passed high school physics are familiar with Newton's formulation, the oldest of the three. In Newton's equations of motion for a particle, the central equation is
Less familiar are the Lagrangian and Hamiltonian formulations. In
Lagrange's formulation, also known as the principle of least action,
the equations of motion are related to a scalar function called the
action. The action S is the integral over [0,tfinal] of the
Lagrangian function L(q,v), where q is position and v is velocity.
The particle follows the path q(t) which minimizes the action . To find such a path, we use
the technique taught in elementary calculus courses: differentiate
the function with respect to the free variable, then set the
derivative to zero. The only complication -- which is not so much
of a complication, really -- is that we are differentiating with
respect to q(t), a variable which is itself a function. Taking
derivatives with respect to functions is a topic in the fancily
titled topic of
the calculus of variations. Anyhow, if we
write down the stationarity condition (derivative = 0), we find that
the minimizing function q satisfies the Euler-Lagrange
The Lagrangian is usually written as L(q,v) = T(v)-U(v), where T(v) = 1/2 mv2 is the kinetic energy and U(v) is the potential energy. Subsituting this form into the Euler-Lagrange equation, we have
To solve differential equations on a computer, it is necessary to approximate them by finite systems of difference equations. There are different ways to approach this approximation, but one of the most intuitive is simply to take the original equations of motions and replace derivatives by differences. For example, we might follow Euler and write
An alternative approach to discretizing the equations of motion is to use a discrete Lagrangian formulation. Instead of having a Lagrangian function that depends on q and v, we now have a function that depends on the value at consecutive time steps: L(qk,qk+1). And rather than having an action integral, we have an action sum. The minimization procedure is the same, however, and yields the discrete Euler-Lagrange equation:
So why would we choose to use a discrete Lagrangian formulation rather than discretizing Newton's law? It is because the discrete Lagrangian formulation can guide our choice of numerical methods. We define the momentum of the continuous system to be
p-k = -D2 L(qk,k+1).
momentumhave multiple reasonable analogues in the discrete world.
So we have two reasonable definitions for momentum in the discrete
world, and similarly we end up with two copies of other related
objects (e.g. there are two discrete canonical one-forms). But, by
an almost magical stroke, there is only one canonical
two-form associated with the discrete equations. Thus, the discrete
Euler-Lagrange equations for a conservative system mimic the
continuous Euler-Lagrange equations in that they
symplectic structure (i.e. they conserve the canonical
two-form). To this day, I have no good intuitive explanation of
what symplectic forms are and why they're important; nevertheless,
they are important, because with the exact
conservation of symplectic structure under discretization comes
better approximate conservation of better-known quantities
like energy and overall momentum. This behavior markedly differs
from the behavior of many direct discretizations of Newton's law in
which the energy of the discrete system tends to either grow or
decay over time.
So why am I thinking of this now? Some time ago, I was challenged
by one of my professors, W. Kahan, to show that a numerical scheme
for integrating a particular type of differential equation -- a
matrix Riccati equation -- would maintain a symmetry property
present in the original differential equation. He had in mind
something enormously complicated involving the group-theoretic
structure of the problem, which he had once proved but never
trusted, and which was lost in the morass of papers in his office.
I just attacked the problem with algebra guided by intuition, and to
our mutual surprise, found a proof. My proof is short (one page),
and relies on a set of coordinate changes to go from unknowns which
morally symmetric -- symmetric but for the error imposed
by discretization -- to unknowns which are symmetric even in the
discrete equations. A few months later (or earlier?), I gave a
brief summary of discrete Lagrangian mechanics in a computational
mechanics seminar that I was attending. While rummaging through old
files today, I found both my notes on discrete Lagrangian mechanics
and my proof of the symmetry in Kahan's integrator. It seems to me
that there is a philosophical connection betweek Kahan's integrator
and the integrators derived from the discrete Euler-Lagrange
equation: in both cases, the discretization is set up in such a way
that some interesting properties can't help but be conserved. I
wish I'd thought a little longer and made that connection more
precise; alas, that thinking will have to wait for another day.
1. Technically, the action need not be minimized, but need only be rendered stationary. This distinction is important in the study of dynamics of constrained systems, where the action is typically not minimized.