First Lagrange and then Hamilton offered alternative formulations of Newton's F=ma dynamical law. Both formulations have advantages over the Newton's formulation for systems that involve complicated constraints on the position and velocities of the constituents of the system (rigid rods for example have ends that have to stay the same distance apart). In simple mechanical systems the Lagrangian is just the difference T--V of the kinetic and potential energy thought of as a function of the positions and velocities of the particles that comprise the system. Often it is fairly simple to write down an expression for the Lagrangian and then the equations of motion for the system follow from the principal of "least action" . The actual path, time=t-->position at time t=x(t), is such as to "minimize" the action. The action is the integral of L(x(t),x'(t)) from an initial time t=a to a final time t=b and the motion is constrained by the requirement that there is a given starting position x(a) and a given final position x(b) (x'(t) is the velocity at time t). In actual systems the action is often not "minimized" -- instead the action is critical(and like a flat tangent (critical point) on a curve this could occur at a local minimum or a local maximum or an inflection point).
Perhaps the most interesting aspect of the Lagrangian formulation of mechanics is a connection with quantum mechanics discovered by R. Feynman. There is a famous 2 slit experiment where light gets from x on one side of the screen to y on the other side but has to pass through one or the other of two slits. The characteristic interference pattern is "explained" by adding complex amplitudes associated with the two paths. Feynman imagined that one could think of quantum amplitudes propagating from x to y *in the absence of a screen* in the following way. Put lots of screens in between x and y and then make lots of slits. To get from x to y one would have to travel from one screen to the next each time finding a slit to go through. Each such "broken line segment" (path) going from x to y is associated with a quantum amplitude. The appropriate quantum amplitude to go from x to y is just the sum of all the amplitudes associated with all the paths. Now make the screens "go away" by passing to a limit of infinitely many infinitely close screens with infinitely many infinitely close slits. The quantum amplitude that Feynman associated with each path going from x to y is the exponential of "i" times the action of the path divided by h (Plank's constant). Here's the interesting part. As h-->0 (the quantum world becomes classical) the principal contribution to Feynman's path integral should come from those trajectories along which the action is stationary (in the Lagrangian formulation of mechanics these are just the trajectories of classical mechanics).
Sadly, mathematicians have never quite been able to establish a setting in which Feynman's heuristic insight is mathematically intelligible, but his path integral formulation of quantum mechanics is used to formulate all the modern gauge theories of Physics (Feynman famously remarked that the absence of Mathematics would have set back Physics about 2 weeks). The principal reason is that the Lagrangian is much easier to guess at than Schrodinger's equation (the Hamiltonian formulation) and the Lagrangian is an effective repository for all the symmetries one cares to name (relativistic invariance, space inversion, time reversal, or more mysteriously gauge invariance). John Palmer
Charles A. Grimes wrote:
> ``...I remember thinking how stupid was this notion that `nature took
> the path of least action'...as physicists and applied mathematicians
> we had little interest in the quasi-relgious aspects of the
> theory...'' Les Schaffer
>
> -------
>
> This was over on pen-l from a thread on the irrelevance of workers in
> economic theory....
>
> 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...
>
> Is that a right or wrong way to think about the lagrangian or
> hamiltonian formulation?
>
> Please elaborate a little on the concept of least action...
>
> CG
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