Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: That jerk again!



John Gastineau wrote:
Ok, Ok, I was sloppy. What I was trying to say was that the value of a
parameter, say, velocity, at one moment in time, is independent of the value
of its rate of change. This is true in general, and can be seen in the
necessity of specifying both an object's location and velocity to completely
specify its condition. I emphasize that this is at one moment in time.

The point made in the first sentence above is valid, but the example given
as justification for it in the second sentence is not a valid reason. The
reason that the value of a differentiable function at a point is not relevant
for the value of its derivative at that point is that only *differences*
between the function's values at that point and points neighboring to it are
taken when the derivative is evaluated. Taking the difference subtracts off
any background functional value. The reason that the classical dynamical
state of a particle is given by both its position and its velocity/momentum
is because the dynamical equation of motion for that particle (Newton's
2nd law, or equivalently the Euler-Lagrange equation) is *second* order in
time. The solution for that equation only becomes determined once both an
initial value for the function (i.e. the position) and an initial value for
its first derivative (i.e. the velocity) are specified. If instead of a
second order formulation the first order Hamiltonian formulation is used,
then the equations of motion become a *pair* of first order DEs for both the
position and the momentum, both of which need to have their initial values
specified to determine the subsequent motion. In this formulation no initial
time derivatives are needed, but the number of dynamical variables is doubled
instead, so the total needed amount of initial information is the same in
either case.

David Bowman
dbowman@gtc.georgetown.ky.us