Functionals and calculus of variations

[Fredrik]

Probably yes. I can't think of any other reason.

 

["Don't panic!"]

Ok, thanks for all your input. Really appreciate you spending time on the matter! :)

 

[Stephen Tashi]

I'm just trying to justify to myself a bit more what the lagrangian is, why it's a function of both coordinates and velocities (I assume that to be able to fully specify the dynamics of the system at each instant in the time interval considered, one needs to know the positions of all the particles and the rate of change of those positions at that point?!)

There are two different questions.
1) What is the physical interpretation of the integrand $L( q(t) , q'(t), t)$?
2) Why does the actual path of a physical system follow the particular q(t) that minimizes the action integral?

I wouldn't say that the integrand itself (the Lagrangian) determines or describes the dynamics of a physical system since physical systems follow the special path that minimzes an integral of the integrand. So the value of the integrand at point in time is not a complete physical description of a system.

One can take the attitude of expositions like http://www.physicsinsights.org/lagrange_1.html that the Lagrangian is is an artificial mathematical construction. That outlook is: I want to solve a differential equation with given initial and final conditions. I will invent an integrand such that finding the path that minimizes an integral of it gives the solution to the differential equation. (The case of motion under a constant frictional force discussed in that link emphasizes the "artificial" nature of the Lagrangian.)

 

["Don't panic!"]

I had thought of the Lagrangian as a function that characterises the state of the system at each point along along an allowed path (i.e. one which satisfies the boundary conditions), and is as such a function of the coordinate values and the velocity values at that point (as the state of the system can be completely specified at a given instant by the coordinates and the rate of change of those coordinates, i.e. the velocities, at that instant). In this sense, for each allowed path we can specify the coordinate and velocity values each point along the path and hence inserting these values into the Lagrangian would (in principle) allow one to characterise the state of the system at each point along the path. Integrating over a given interval will give a characteristic value to each given path (the action), and allowing one to distinguish between them and determine the actual physical path taken by the system.
We then assert that the actual path taken is the one which gives a stationary value to the action, and hence (again, in principle) evaluating the Lagrangian (upon inserting the coordinate and velocity values at each point along this path) at each point along this path will characterise the actual state of the system at each point along the path.

Sorry this is a bit wordy, just trying to express in words how I've been trying to think about it in my mind (it has really been bugging me, as you can probably tell!).