Principle of Least Action & Euler-Lagrange Equations

In summary: This is the condition that the second derivative of the Lagrangian along the path is positive definite. In summary, the Euler-Lagrange equations are a necessary condition for the path to have least action, but they are not sufficient. The second-variational inequality must also be satisfied for the action to have a minimum value along the path.
  • #1
ObsessiveMathsFreak
406
8
I'll just throw down some definitions and then ask my question on this one.

In a conservative system, the Lagrangian, in generalised coordinates, is defined as the kinetic energy minus the potential energy.

[tex]L=L(q_i,\dot{q}_i,t) = K(q_i,\dot{q}_i,t) - P(q_i,t).[/tex]
All [tex]q_i[/tex] here being functions of t.

It satisfies the Euler-Lagrange equations in all its generalised coordinates.

[tex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \ \ \ forall i=1,2,\ldots,n[/tex]

The "Action" of a path betwwen two points is the integral of the Lagrangian along that path.
[tex]A = \int_{t_1}^{t_2} L(q_i,\dot{q}_i,t) dt[/tex]

The Principle of least action states that the path actually taken is the path with with least Action.(The path that minimises the integral). For the path to have least action, the Euler-Lagrange equations are a necessary condition .

Now here is my question. Are the Euler-Lagrange equations a sufficient condition for the path to have least action? It seems so to me, but can anyone confirm this?
 
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  • #2
hi,

I would say yes, because:
the Euler-Lagrange equations can be derived from the "action" integral.
The solution of Lagrange equations minimises the integral
 
  • #3
I think the principle actually says that the physical system will take a path of stationary action (ie. it could be a maximum, or just a local minimum, etc), and that this is equivalent to being a solution of the Euler-Lagrange equations. Hence, no, I don't think the equations are sufficient if you want the least action (especially not the global minimum).
 
  • #4
Expanding on what cesiumfrog said, a solution to the Euler-lagrange equations has maximal action, minimal action, or something in between corresponding to an "inflection point".

We can use the Euler-Lagrange equations to find the shortest path between two points A and B on the surface of a sphere. This path is the segment of the "great circle" that passes through A and B. This segment has minimal distance, but the complement of this segment on the great circle has maximal distance, but it also satisfies the euler-lagrange equations.
 
  • #5
OK, but if we ammend to the Euler-Lagrange equations the condition that is I believe [tex]\frac{\partial^2 L}{\partial \dot{q}_i^2}>0[/tex] as a necessary condition for a path of least action.

Is this condition, along with Euler-Lagrange, sufficient for the path to be of least action?
 
  • #6
ObsessiveMathsFreak said:
I'll just throw down some definitions and then ask my question on this one.

In a conservative system, the Lagrangian, in generalised coordinates, is defined as the kinetic energy minus the potential energy.

[tex]L=L(q_i,\dot{q}_i,t) = K(q_i,\dot{q}_i,t) - P(q_i,t).[/tex]
All [tex]q_i[/tex] here being functions of t.

It satisfies the Euler-Lagrange equations in all its generalised coordinates.

[tex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \ \ \ forall i=1,2,\ldots,n[/tex]

The "Action" of a path betwwen two points is the integral of the Lagrangian along that path.
[tex]A = \int_{t_1}^{t_2} L(q_i,\dot{q}_i,t) dt[/tex]

The Principle of least action states that the path actually taken is the path with with least Action.(The path that minimises the integral). For the path to have least action, the Euler-Lagrange equations are a necessary condition .

Now here is my question. Are the Euler-Lagrange equations a sufficient condition for the path to have least action? It seems so to me, but can anyone confirm this?

The principle of least action is actually a misnomer, at least in so far as it is related to physics. What you're actually looking for are those paths in the configuration space for which the value of the action is stationary, i.e., those paths along which the action takes either its minimum, maximum, or shoulder-point value.

The Euler-Lagrange equations are certainly a necessary condition for the action to have a least value, but they are not a sufficient condition. For the action to have a minimum value along a path, one needs the path to satisfy not only the Euler-Lagrange equations, but also the second-variational inequality.
 

1. What is the principle of least action?

The principle of least action, also known as the principle of least action and stationary action, is a fundamental concept in classical mechanics that states that a physical system will always follow the path that minimizes the total action.

2. What is the mathematical equation used to represent the principle of least action?

The mathematical equation used to represent the principle of least action is the Euler-Lagrange equation. This equation is derived from the Lagrangian function, which is a mathematical expression that represents the total energy of a system.

3. How is the principle of least action applied in physics?

The principle of least action is applied in physics to describe the motion of particles and systems. It allows us to predict the path that a system will take by minimizing the total action, which is the integral of the Lagrangian function over time.

4. What is the significance of the Euler-Lagrange equation?

The Euler-Lagrange equation is significant because it provides a way to determine the path that a system will take in the most efficient way possible. This equation is also used in other fields of physics, such as quantum mechanics and relativity, to describe the behavior of particles and systems.

5. Are there any limitations to the principle of least action and Euler-Lagrange equations?

Yes, there are some limitations to the principle of least action and Euler-Lagrange equations. They are only applicable to systems that can be described by a Lagrangian function, and they do not account for dissipative forces or non-conservative systems. Additionally, they do not apply to systems at the quantum level.

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