Is the Bead's Motion on a Parabola Correctly Modeled by Lagrangian Mechanics?

[devon cook]

Homework Statement

Honestly, this site is more complicated than the problem I'm trying to solve. Like ... everybody... I've changed the equation of the curve. Think simple! A stupid bead slides down a parabola (y=(1/2)x^2 from x=1 to x=0 to the left, ok? I've done Lagrangian mechanics on it and got the DE
x''(1+x^2)+x(x')^2 +gx=0 ! (The dash " ' " is with repect to time) . Let's say my maths is ok on that, right?
POINT 1. If you analyze it using Newtonian mechanics (according to me) the x(x') term is missing. That's not good. That's not even close to good.
POINT 2. I've solved this DE (x''(1+x^2)+x(x')^2 +gx=0 ) and got a function for the horizontal velocity in terms of distance which looks sensible.
BASICALLY! I just want some feedback here, coz it's interesting and I don't know if I'm right. Hey, you guys in the US or whatever, contact me.
Dev

Homework Equations

The Attempt at a Solution

 

[jbsteele]

It is difficult to provide feedback on this problem without knowing the exact equations and variables that have been used. Additionally, we would need to know what type of Newtonian mechanics was being used in order to assess whether or not the x(x') term should be included. It may be beneficial to provide more information such as what the variables x and g represent in the equation and any other equations that have been used.

 

[thomate1]

I understand your frustration with the complexity of this problem. However, it is important to approach it with a clear and logical mindset. The equation you have provided for the bead sliding down a parabola appears to be a valid one, and it is good that you have solved it and obtained a sensible function for the horizontal velocity.

As for your first point, the missing x(x') term in the Newtonian mechanics analysis could be due to the simplification of the problem, as the bead is only moving in one dimension. However, it would be worth investigating further to see if this term has any significant impact on the overall solution.

In terms of feedback, it would be helpful to see your solution and any assumptions made in order to provide a more thorough analysis. Additionally, reaching out to other scientists or experts in the field for their input and validation of your findings would be beneficial. Keep up the good work and continue to explore and question your results.