# A question on Lagrangian dynamics

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1. Nov 17, 2014

### "Don't panic!"

Hi all,

I've recently been asked for an explanation as to why the Lagrangian is a function of the positions and velocities of the particles constituting a physical system. What follows is my attempt to answer this question. I would be grateful if you could offer your thoughts on whether this is a correct description, and if it is not, please could you enlighten me. Thanks for your time.

The Lagrangian characterizes the state of a system at each point along an arbitrary configuration path between two fixed configurations (arbitrary other than the fact that it must satisfy the boundary conditions). At each point along the configuration path we can give a local description of the path by specifying the coordinate values at that point as well as their first-order (and, in general, second-order and higher-order) derivatives at that point (i.e. a Taylor expansion about that point). The state of the system at a given point along the configuration path should depend on no more than the local description of the path around that point. This implies that the Lagrangian should be a function of the coordinate values, and their derivatives (along with higher order derivatives) at that point. The action for the given path is then constructed by integrating the value of the Lagrangian at each point along the configuration path within the specified time interval.

The classical equations of motion are second-order and therefore, as the principle of stationary action asserts that the actual physical configuration path taken by the system between the two fixed configurations (and hence the one in which the state of the system characterized by the Lagrangian at each point along the path has any physical meaning) is the one which gives a stationary value to the action, this implies that the Lagrangian should be, at most, first-order and hence a function of the positions and velocities of the particles within the system (and hence, its variation will be second order).

2. Nov 22, 2014

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 23, 2014

### voko

4. Nov 23, 2014

### "Don't panic!"

I think my problem lies in whether I'm understanding the motivation correctly. Is it that we wish the Lagrangian to be a characteristic property of the system that we are trying to describe. As such, at each given moment it should depend on no more than the state of the system at that instant. Empirically we know that the state of the system is completely specified by the positions and velocities of the particles constituting the system, and hence this implies the Lagrangian should be (at most) dependent on the positions and velocities of the particles constituting the system.
Would that be a correct motivation?

5. Nov 23, 2014

### voko

The correct motivation, as in the reference linked above, is that we want our theory to agree with the experiment. Experimentally, we know that certain second-order differential equations describe the time evolution of a given mechanical system very accurately, provided we supply accurate initial conditions. These initial conditions are positions and velocities, which follows from the nature of the equations

Mathematically, this is equivalent to a variational problem, where the integrand in the functional (the Lagrangian) must then be a function of positions and velocities.

Either way, we have a state - coordinates and velocities - and an apparatus - the action integral or the differential equations - which lets us predict a state given a past (or future) state. I do not think we know why this is true, we just know empirically that it is.

6. Nov 23, 2014

### "Don't panic!"

Ok, thanks for your help. I guess in my OP I was trying to motivate it in a general setting and why we use the formalism in field theory etc.
Would the description that I originally gave be acceptable?

7. Nov 23, 2014

### voko

I am unhappy with the OP because it mentions "higher-order derivatives" and does not articulate well as to why (empirical knowledge) they are in fact not needed in the Lagrangian formalism. That is very important, it is the physical part of the formalism because purely mathematically one could have any number of derivatives, just like the OP initially says.

8. Nov 23, 2014

### "Don't panic!"

Yeah, I mentioned the higher-orders as I was approaching it from a purely theoretical (mathematical) point of view initially as, a priori, we cannot say how much information is required to specify the state of the system. From empirical evidence we know that the classical equations of motion are second-order and therefore, as the principle of stationary action asserts that the actual physical configuration path taken by the system between the two fixed configurations (and hence the one in which the state of the system characterized by the Lagrangian at each point along the path has any physical meaning) is the one which gives a stationary value to the action, this implies that the Lagrangian should be, at most, first-order and hence a function of the positions and velocities of the particles within the system (and hence, its variation will be second order).