A question on Lagrangian dynamics

In summary, the Lagrangian is a function that characterizes the state of a system at each point along a configuration path between two fixed configurations. This function depends on the coordinate values and their derivatives at each point, and the action for a given path is constructed by integrating the Lagrangian along the specified time interval. The motivation for this is to have a theory that agrees with experimental results, where we know that the state of a system is completely specified by the positions and velocities of its particles. This also explains why the Lagrangian should be, at most, a function of the positions and velocities of the particles, as higher-order derivatives are not needed to describe the system's evolution.
  • #1
"Don't panic!"
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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).
 
  • #3
Here is a classical take on this: https://archive.org/stream/Mechanics_541/LandauLifshitz-Mechanics#page/n9/mode/2up

Pay especial attention to the last paragraph on page 1.
 
  • #4
voko said:
Here is a classical take on this: https://archive.org/stream/Mechanics_541/LandauLifshitz-Mechanics#page/n9/mode/2up

Pay especial attention to the last paragraph on page 1.

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
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 let's 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
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
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
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).
 

1. What is Lagrangian dynamics?

Lagrangian dynamics is a mathematical framework used to describe the motion of a system of particles or rigid bodies. It is based on the principle of least action, which states that the motion of a system will follow a path that minimizes the action, a quantity representing the total energy of the system.

2. How is Lagrangian dynamics different from Newtonian mechanics?

While Newtonian mechanics is based on the concept of forces and accelerations, Lagrangian dynamics is based on the concept of energy and the path of least resistance. This makes it a more general and powerful approach, as it can be applied to a wider range of systems with complex constraints and interactions.

3. What are some real-world applications of Lagrangian dynamics?

Lagrangian dynamics has many applications in physics, engineering, and other fields. It is commonly used in celestial mechanics to study the motion of planets and satellites, in robotics to design efficient and stable movements, and in fluid dynamics to model the behavior of fluids.

4. What are the main components of a Lagrangian system?

A Lagrangian system consists of a set of particles or bodies, their positions and velocities, and the potential and kinetic energies of the system. These components are used to formulate the Lagrangian equations of motion, which describe the behavior of the system.

5. How is the Lagrangian calculated for a system?

The Lagrangian is calculated by summing the kinetic and potential energies of all particles or bodies in the system. It is a function of the generalized coordinates and velocities of the system, which represent the degrees of freedom of the system. The resulting Lagrangian equations of motion can then be solved to determine the behavior of the system over time.

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