I Playing with Lagrangian and I screw up

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The discussion centers on the correct application of the Lagrangian approach in physics, specifically regarding kinetic energy and the formulation of equations of motion. It emphasizes that the Lagrangian should be treated as an abstract function of independent variables rather than jumping ahead to specific solutions. The transition from abstract equations to specific particle trajectories occurs only when solving the Euler-Lagrange equations. Additionally, there is a noted inconsistency in how textbooks present this distinction. Understanding this process is crucial for accurately applying the Lagrangian method.
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Lagrangian, Kinetic Energy (where am I going wrong)
I am sorry for all these questions this morning.

Could someone read the attached and tell me where I am going wrong?
 

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Trying2Learn said:
Summary:: Lagrangian, Kinetic Energy (where am I going wrong)

I am sorry for all these questions this morning.

Could someone read the attached and tell me where I am going wrong?
The Lagrangian approach requires that you treat the Lagrangian as an abstract function of independent variables. You can't jump ahead, solve the equations of motion, and plug these back into the Lagrangian and start again.

It's only when you are solving the Euler-Lagrange equations that you transition from the equations in their abstract form, to the equations that represent a specific solution in terms of a particle trajectory etc.

Text books tend to vary to the extent that they make this distinction clear.
 
Ah ha!

Thank you!
 
Thread 'Gauss' law seems to imply instantaneous electric field'

Thread 'Gauss' law seems to imply instantaneous electric field'

Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
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Thread 'Gauss' law seems to imply instantaneous electric field (version 2)'

Thread 'Gauss' law seems to imply instantaneous electric field (version 2)'

This argument is another version of my previous post. Imagine that we have two long vertical wires connected either side of a charged sphere. We connect the two wires to the charged sphere simultaneously so that it is discharged by equal and opposite currents. Using the Lorenz gauge ##\nabla\cdot\mathbf{A}+(1/c^2)\partial \phi/\partial t=0##, Maxwell's equations have the following retarded wave solutions in the scalar and vector potentials. Starting from Gauss's law $$\nabla \cdot...
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