Lagrangian to the Euler-Lagrange equation

In summary, the conversation discusses the formation of the Lagrangian, its use in the "Action" formula, and the extraction of the Euler-Lagrange equation in dynamics. The question posed is whether anyone has attempted to justify this final form, which is related to the principle of d'Alembert.
  • #1
Trying2Learn
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TL;DR Summary
Can one "reason" out thge Euler-Lagrange equation in dynamics
Hello all,

I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy.
(I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion.

Moving on...

One inserts the Lagrangian into the form of the "Action" and minimizes it.

One then extracts the Euler-Lagrange equation for a dynamical system: d(L/dq) = d( dL/dq-dot)/dt

So...

Has anyone ever attempted to "reason out" or "justify" THIS FINAL form to be used in dynamics?

I know one can begin with it, and demonstrate it leads to Newton's F=ma; but has anyone tried to justify this
Euler-Lagrange equation in and of itself? Is it "reasonable?"
 
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  • #2

1. What is the Lagrangian to the Euler-Lagrange equation?

The Lagrangian to the Euler-Lagrange equation is a mathematical function that describes the dynamics of a physical system. It takes into account the kinetic and potential energies of the system and is used to find the equations of motion for the system.

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

The Euler-Lagrange equation is significant because it provides a way to find the equations of motion for a system without having to use Newton's laws of motion. It is a more general and elegant approach to solving problems in classical mechanics.

3. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived from 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 integral. By varying the action integral with respect to the system's coordinates, the Euler-Lagrange equation can be obtained.

4. What types of systems can the Euler-Lagrange equation be applied to?

The Euler-Lagrange equation can be applied to a wide range of physical systems, including particles, rigid bodies, and fields. It is commonly used in classical mechanics, but it can also be applied in other areas such as quantum mechanics and relativity.

5. How does the Euler-Lagrange equation relate to the principle of least action?

The Euler-Lagrange equation is a direct result of the principle of least action. It is the mathematical expression of the principle and provides a way to find the path that minimizes the action integral for a given system. In this way, the Euler-Lagrange equation is intimately connected to the principle of least action.

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