Graduate Lagrangian Multipliers with messy Solution

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The discussion revolves around applying Lagrangian multipliers to a mechanical system involving kinetic and potential energy equations alongside a holonomic constraint. The user has derived four equations corresponding to each coordinate but is uncertain about the next steps in solving them. They mention the challenge of handling squared terms that lead to two equations when isolating a variable. The user seeks assistance in developing the system of equations further, with the intention of using the Runge-Kutta method for simulation. The conversation highlights the complexities of Lagrangian mechanics, particularly for those with limited formal training in the subject.
Mishal0488
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Hi Guys

Please refer to the attached file.
I have not included any of the derivatives or partial derivatives as it does get messy, I just just included the kinetic and potential energy equations and the holonomic constraint.

The holonomic constraint can be considered using Lagrange multipliers. The result is 4 equations, one for each coordinate and the holonomic constraint.

I am not sure what to do once I am at this point, can someone please assist?
With regards to the holonomic constraint, I can make one of the variables the subject of the formula, however due to the squared terms there are two equations which will arise.

Kind regards
Mishal Mohanlal
 

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look like a known problem in solid state physics but i am not able to remember it now did you try matrix method for differential equations
 
Why solid state physics? The image is a mechanical system which I am trying to simulate.

Note that I am an engineer and my understanding of Lagrangian mechanics is limited since it is not taught as part of engineering. I have learned through self study.

I was hoping to develop the system of equations and thereafter solve it using Runge Kutta
 

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