Lagrangian Multipliers with messy Solution

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In summary, The conversation is a discussion about a mechanical system and using Lagrangian mechanics to solve it. The holonomic constraint is introduced and can be solved using Lagrange multipliers, resulting in 4 equations. The person is unsure of what to do after this point and asks for assistance. They mention using the matrix method for differential equations, but the other person clarifies that it is a solid state physics problem. The person explains that they are an engineer and have self-studied Lagrangian mechanics, and they plan to use the Runge Kutta method to solve the system of equations.
  • #1
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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  • #2
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
 
  • #3
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
 

FAQ: Lagrangian Multipliers with messy Solution

What are Lagrangian Multipliers?

Lagrangian Multipliers are a mathematical technique used to optimize a function subject to a set of constraints. They allow us to find the maximum or minimum value of a function while taking into account the constraints placed on the variables.

How do Lagrangian Multipliers work?

Lagrangian Multipliers work by introducing a new variable, known as the Lagrange multiplier, into the original function. This variable is then used to create a new function, known as the Lagrangian, which is optimized to find the maximum or minimum value of the original function subject to the given constraints.

What is a "messy" solution in the context of Lagrangian Multipliers?

A "messy" solution in the context of Lagrangian Multipliers refers to a solution that involves a large number of variables and complex equations. This can happen when the original function and constraints are complicated, making it difficult to find a simple and elegant solution.

How do you solve for Lagrangian Multipliers with a messy solution?

Solving for Lagrangian Multipliers with a messy solution requires a combination of algebraic manipulation, calculus, and sometimes numerical methods. It is important to carefully identify and simplify the equations involved to make the solution process more manageable.

What are some real-world applications of Lagrangian Multipliers?

Lagrangian Multipliers have various applications in fields such as economics, engineering, and physics. They can be used to optimize production processes, allocate resources efficiently, and solve optimization problems in physics, such as finding the path of least resistance in a fluid flow. They are also commonly used in machine learning and data analysis to find the best fit for a given set of data points.

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