Graduate Lagrangian to the Euler-Lagrange equation

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SUMMARY

The discussion centers on the derivation and justification of the Euler-Lagrange equation from the Lagrangian formulation of mechanics, which is defined as the difference between kinetic energy and potential energy. The equation is expressed as d(L/dq) = d(dL/dq-dot)/dt, and participants explore whether this form can be justified independently, beyond its derivation from Newton's laws. The connection to d'Alembert's principle is also mentioned as a potential avenue for reasoning about the equation's validity.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with kinetic and potential energy concepts
  • Knowledge of the principle of least action
  • Basic grasp of differential calculus
NEXT STEPS
  • Research the principle of least action in classical mechanics
  • Study the derivation of the Euler-Lagrange equation in detail
  • Explore d'Alembert's principle and its implications in dynamics
  • Examine applications of the Euler-Lagrange equation in various physical systems
USEFUL FOR

Students and professionals in physics, particularly those studying classical mechanics, as well as researchers interested in the foundations of dynamical systems.

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TL;DR
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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