Geometry Help: Solving Lagrangian Problem

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In summary, the conversation discusses a Lagrangian problem involving a cube on top of a cylinder and its relation to geometry. The formula for finding the height of the cube is derived and an unstated assumption is addressed. The formula is valid due to the cube's ability to roll without slipping.
  • #1
Xyius
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This is a Lagrangian problem, I am posting it here in introductory physics because what I need help with isn't in Lagrangian mechanics, but rather geometry.

http://img97.imageshack.us/img97/7504/what3.png [Broken]

I am confused as how they got those relations for x and y. I have tried to make sense out of it but cannot figure it out. I am sure it is something really simple!
 
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  • #2
Let's start with y. You can probably understand where the (r+b)cos(θ) comes from I hope. Now imagine that you roll the cube around the sphere by an angle θ. The length of the line CB is then what? That's right! It's rθ. So obviously this adds a height rθsin(θ). Now you have

[tex] y = (r+b)cos(θ) + rθsin(θ) [/tex]
 
  • #3
Hm. I agree with Xyius; I don't think that, as stated, the formula necessarily holds. Consider a much smaller cube in the same place, at the same angle.

The unstated assumption is that the cube was originally placed square and centered on top of the cylinder and rolled to its current location, but that hasn't been specified.
 
  • #4
Joffan said:
Hm. I agree with Xyius; I don't think that, as stated, the formula necessarily holds. Consider a much smaller cube in the same place, at the same angle.

A much smaller cube would not be able to get to as large of an angle, because the cube must roll without slipping. The formula is valid.
 
  • #5
Ah! I understand now! Thank you very much :]
 

1. What is a Lagrangian problem in geometry?

A Lagrangian problem in geometry involves finding the shortest path or optimal solution for a given geometric problem. This method was developed by mathematician Joseph-Louis Lagrange and is commonly used in various fields of mathematics, physics, and engineering.

2. How do you solve a Lagrangian problem in geometry?

To solve a Lagrangian problem in geometry, you first need to define the problem and determine the constraints. Then, you can use the Lagrangian function to find the critical points and solve for the optimal solution using the method of Lagrange multipliers.

3. What are the applications of Lagrangian problems in geometry?

Lagrangian problems have various applications in geometry, such as finding the shortest distance between two points, determining the optimal shape for a given area, and minimizing the surface area of a solid object. They are also used in physics to find the path of least resistance or least time for a moving object.

4. What is the difference between Lagrangian problems and other optimization methods?

The main difference between Lagrangian problems and other optimization methods is that they take into account the constraints of a problem while finding the optimal solution. This makes them useful for solving problems with multiple constraints, which cannot be solved using traditional optimization methods.

5. Are there any limitations to using Lagrangian problems in geometry?

One limitation of using Lagrangian problems in geometry is that they may not always provide the global optimal solution. In some cases, they may only give a local optimal solution, which may not be the best possible solution. Additionally, they may be more complex and time-consuming to solve compared to other optimization methods.

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