Lagrangian Multipliers with messy Solution

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SUMMARY

The discussion centers on the application of Lagrangian multipliers to derive equations for a mechanical system involving kinetic and potential energy equations alongside a holonomic constraint. The user, Mishal Mohanlal, seeks assistance in progressing from the derived equations, which include squared terms leading to multiple equations. The conversation highlights the need for clarity in solving these equations, particularly through numerical methods like the Runge-Kutta technique. The context suggests a connection to solid state physics, although the user is primarily focused on mechanical simulation.

PREREQUISITES
  • Lagrangian mechanics fundamentals
  • Understanding of holonomic constraints
  • Knowledge of kinetic and potential energy equations
  • Familiarity with numerical methods, specifically Runge-Kutta
NEXT STEPS
  • Study the application of Lagrangian multipliers in mechanical systems
  • Explore methods for solving differential equations using the Runge-Kutta technique
  • Research holonomic constraints and their implications in physics
  • Learn about matrix methods for solving systems of equations in solid state physics
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Engineers, physicists, and students interested in mechanical systems and numerical simulation techniques, particularly those working with Lagrangian mechanics and differential equations.

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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