Re: [Phys-l] solving an energy equation

[John Denker <jsd@av8n.com>]

Carl Mungan wrote:

> But I'm still bothered by the question of whether something is lost 
> in the energy approach.

Executive summary:  You can get either answer depending on what you mean
by "THE" energy approach.  In particular, does knowing the energy also
mean knowing the derivatives of the energy?  Or not????


The extremes are illustrated by the following examples.

A) Suppose we have a potential energy surface specified as V(x), and
  a particle moving on this surface.  We are also given *two* initial
  conditions, x(0) and xdot(0).  Then "typically" we can timestep the
  equations of motion, solving for x(t) and xdot(t) for all t.

  The unspoken assumption here is that V(x) completely specifies the
  problem.  That is, we assume that from problem to problem, V(x) might
  change but the particle's intrinsic properties stay the same.  That
  might be OK for a certain range of problems, but it is not OK in
  general.  For example, in one problem we might have a puck that simply
  glides across the surface, while in the next problem we might have
  a billiard ball the rolls without slipping.  These two objects will
  generally follow different trajectories across any given surface
  V(x).

B1) In this problem we assume a point particle, to avoid the complexities
  pointed out in example (A).

  Again we plan on timestepping the equation of motion.  We are given
  the primordial initial conditions x(0) and xdot(0), and the force
  field F(x) for all x  Each timestep can be considered a little
  "initial value" problem unto itself:
     x(0) xdot(0) and F(x(0)) are used to calculate x(1) and xdot(1).
     x(1) xdot(1) and F(x(1)) are used to calculate x(2) and xdot(2).
     x(2) xdot(2) and F(x(2)) are used to calculate x(3) and xdot(3).
     et cetera.

  That just works.

B2a) In contrast, consider the following version of "the" energy approach:
     x(0) xdot(0) and V(x(0)) are used to calculate x(1) and xdot(1).
     x(1) xdot(1) and V(x(1)) are used to calculate x(2) and xdot(2).
     x(2) xdot(2) and V(x(2)) are used to calculate x(3) and xdot(3).
     et cetera.

  That doesn't quite work, if we consider V(x) as an oracle t

B2b) It doe  however work if we are allowed to ask multiple questions of the
  oracle, like this:
     x(0) xdot(0) V(x(0)) V(delta1 + x(0)) V(delta2 + x(0))
                                 are used to calculate x(1) and xdot(1).
     x(1) xdot(1) V(x(1)) V(delta1 + x(1)) V(delta2 + x(1))
                                 are used to calculate x(2) and xdot(2).
     x(2) xdot(2) V(x(2)) V(delta1 + x(2)) V(delta2 + x(2))
                                 are used to calculate x(3) and xdot(3).
     et cetera.

   where delta1 and delta2 are tiny vectors that allow us to explore
   the neighborhood.

By contrasting example B1b with B2a, we see that "the" energy approach either
works or doesn't, depending on whether we know the energy in our neighborhood,
or just at our exact pointlike location.

C) If you try to timestep the equations of motion, but at each step you keep
  only the energy of the particle (or position and energy) rather than position
  and momentum, you have to work harder.  This is a replay of item (B) above.

  The behavior of the particle is first-order Markovian if I am allowed to
  remember x and xdot.  That is, x(t) and xdot(t) are a function of only
  one earlier point, x(t-1) and xdot(t-1).

  In contrast, the behavior is second-order Markovian if I am allowed to
  remember the energy instead.  I need x(t-1), E(t-1), x(t-2), and E(t-2)
  in order to predict x(t) and E(t).

Once again we see that we need to know derivatives of the energy.


This can be understood in terms of Sturm-Liouville theory.  In physics we
encounter tons of *second*-order differential equations.  Because they are
second order, they require *two* initial conditions.

In the equation F = m x dot dot, the x variable is two derivatives removed
from the F variable.  In contrast, in the equation F = - V prime, V is only
one derivative removed from F.  That's just not enough.