hey Chuck:
first a little background: some time ago Michael Perelman and i discussed offlist lagrangian and maximisation methods as used in economics, so when i replied to his post, i thought that's what he was talking about ...
> In my fumbling through math hobby stuff, I always thought the
> prinicple of least action was considered part of the `conservation'
> laws, so that a point (i.e mass) took it's curve unless influenced by
> some external force...
>
think about this, for much of physics, there is complete equivalence of Lagrangian and Newtonian mechanics, right? and in Newtonian mechanics you can have energy conservation or not, depending on what you chose for the system you are analyzing. in some cases an external force can change the energy of your system, and Newton's methods could care less, oui?
so too then in Lagrangian mechanics, you can have Lagrangians and associated dynamics without energy conservation.
on the other hand, if there is a conserved energy in the system, the Lagrangian/ Hamiltonian scheme is the easiest way to make this explicit, and also to find approximate conserved quantities for complicated dynamical systems.
the thing about least action: its not necessarily least: that is, if you take a couple of your curves in extended configuration space (that is position and velocity space) nearby to the actual curve, it turns out that the action along these other curves just don't change much from the "true" curve. but there is nothing in mechanics that the true curve must have the *minimum* action. it could be some are slightly above and some slightly below. in terms of your calculus, the derivative (except this is a functional derivative) is zero, but the second derivative can be concave up, down, or not at all, which is the saddle point. in modern parlance, lagrangian mechanics can be derived from a *stationary* principle -- action hardly changes for other curves near the true curve -- not a *least action* principle.
so the whole teleological viewpoint that nature takes the path of
*least* action just ain't so. first (functional) derivative equals zero
is enough for equivalence of Newton and Lagrangian mechanics. i just
read somewhere that in general relativity, someone showed under
reasonable conditions the laws are saddle-point like, not minima in the
action.
> Is that a right or wrong way to think about the lagrangian or
> hamiltonian formulation?
minus details and nit picking: in energy conserving systems, the Hamiltonian function will be independent of time. so energy and time form a pair (just like in the uncertainty principle). and if the Hamiltonian is independent of a spatial coordinate, momentum will be conserved in that direction (position/momentum pair, just like in the .... ).
two side notes:
Lagrangian mechanics is also very good for mechanical systems with certain kinds of (non-holonomic) constraints, the famous example is the bicycle. it is a conservative mechanical system with a stable fixed point ... errr, the bike stays up. and its not due to gyroscopic effects.
also, there is a principle of least time in optics. you'll find something on google.
Les