Yes, I have just started. I got home from work a few minutes ago, and poured a stiff martinis to consider Les and John's great answers. Thanks to both of you. It is difficult to get good answers from very vague and confused questions, and I appreciate the effort.
While this is probably gobbly-gok to most LBO, I'll explain my interest in this stuff. First what is this shit about? It is about how to look at physical problems. You can look at them as problems of points and their changing positions, or you can look at them as energy and time.
In a really great little book, two Russians, Lev Landau and Evgeny Lifshitz, open their lectures by recasting classical Newtonian mechanics of a point and its motions by introducing:
``The most general formulation of the law governing the motion of mechanical systems [that] is the _principle of least action_ or _Hamilton's principle_, according to which every mechanical system is characterized by a definite function ... or briefly L(q, dot-q, t)....
Let the system occupy, at the instants t_1 and t_2, positions defined by two sets of values of the coordinates q^(1) and q^(2). The condition in the system moves between these positions in such a way that the integral
S = t_2 INT t_1 L(q, dot-q, t) dt, (2.1)
takes the least possble value. The function is called the Lagrangian of the system concerned, and the integral (2.1) is called the action.... '' (4p, Landau LD, Lifshitz EM, Mechanics and Electrodynamics, Pergamon Press: 1972.)
Here the coordinates are q_i, and the velocities are dot-q_i.
I thought Les's comment on the religious features of the Lagrangian was hilarious.
I will meditate on these posts. Thanks again to you both.
CG