Odd order motion equations from a Lagrangian

[lasm2000]

Does anyone knows why it is impossible to obtain an odd order equation of motion from a Lagrangian? I recently heard that, but I can't find a nice discussion anywhere.

It is also interesting to note that this might not even valid for field theory, take as an example the Schrodinger equation which can indeed be retrieved from a Lagrangian and is of odd order (because of the first order time derivative), anyone has an idea of why this happens? I mean why we can't get an odd order equation of motion for discrete system with n degrees of freedom but we can get it for some field (which obviously has infinite degrees of freedom).

Thanks in advance.

 

[LightHero]

The reason why it is usually impossible to obtain an odd order equation of motion from a Lagrangian is because it would violate the fundamental laws of physics. In classical mechanics, Newton's Second Law states that the rate of change of momentum in a system is proportional to the net force acting on it. This law can be expressed in the form of a second-order differential equation, which states that the acceleration of a particle is proportional to the net force acting upon it. The Lagrangian formalism in classical mechanics is based on this law and thus produces second-order equations of motion.In the case of field theory, the situation is different. Instead of a single particle, we have an infinite number of particles interacting with each other. This requires a different approach, and it is possible to derive a first-order Schrodinger equation from a Lagrangian. This is because the Schrodinger equation describes how the wave function of a quantum system evolves in time, and thus the Lagrangian must take into account not only the interactions between individual particles but also the interactions between different waves.

 

[astrokreng]

I can provide some insights into this question. The Lagrangian formalism is a powerful mathematical tool used to derive equations of motion for physical systems. It is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action (a quantity related to the energy) of the system.

In general, the Lagrangian formalism can be applied to both continuous systems, such as fields, and discrete systems with a finite number of degrees of freedom. However, there are some fundamental differences between these two types of systems that can explain why it is not possible to obtain odd order equations of motion from a Lagrangian for discrete systems.

One key difference is that continuous systems, such as fields, have an infinite number of degrees of freedom, while discrete systems have a finite number. This means that for a continuous system, the Lagrangian can be written as a functional of the fields, while for a discrete system, it is a function of the generalized coordinates and their time derivatives.

In the Lagrangian formalism, the equations of motion are obtained by varying the action with respect to the generalized coordinates. For continuous systems, this variation can lead to odd order equations of motion because the Lagrangian is a functional. However, for discrete systems, the variation only produces even order equations of motion because the Lagrangian is a function of the generalized coordinates.

Additionally, the Schrodinger equation, which is a fundamental equation in quantum mechanics, is an exception to this rule. It can be derived from a Lagrangian, even though it is an odd order equation. This is because the Schrodinger equation is a wave equation, and waves are inherently continuous and have an infinite number of degrees of freedom.

In summary, the reason why it is not possible to obtain odd order equations of motion from a Lagrangian for discrete systems is due to the fundamental differences between continuous and discrete systems, specifically in terms of their degrees of freedom and the form of the Lagrangian. The exception of the Schrodinger equation can be explained by its nature as a wave equation. I hope this explanation helps to clarify the issue.