Constrained Lagrangian equetion (barbell)

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In summary, my results were wrong because my Langrangian was not correct. I derived L + λf by x_1, x_2, y_1, y_2 and λ from equation (1) and (2), substituting it into the constraint equation, and got an equation that looked like this: (m \ddot x_1 + \frac{\partial U}{\partial x_1} ) * <something> = 0. However, when I tried to apply this equation to my problem, it looked like it was wrong. I think my mistake was in not correctly calculating my Langrangian.
  • #1
Jengalex
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Hi!

I tried to compute an ideal barbell-shaped object's dynamics, but my results were wrong.
My Langrangian is:

## L = \frac{m}{2} ( \dot{x_1}^2 + \dot{x_2}^2 + \dot{y_1}^2 + \dot{y_2}^2 ) - U( x_1 , y_1 ) - U ( x_2 , y_2 ) ##

And the constraint is:

## f = ( x_1 - x_2 )^2 + ( y_1 - y_2 )^2 - L^2 = 0 ##

I dervated L + λf by ## x_1, x_2, y_1, y_2 ## and λ:

## m \ddot x = - \frac{\partial U}{\partial x_1 } + \lambda ( x_1 - x_2 ) ## (1)
Four equtions similar to this and the constraint.

Then I expressed ## \ddot x_2 , \ddot y_1 , \ddot y_2 ## with ## \ddot x_1 , x_1 , x_2 , y_1 , y_2 ## and U's partial derivates' local values:

## m \ddot x_2 + \frac{\partial U}{\partial x_2} = - m \ddot x_1 - \frac{\partial U}{\partial x_1} ##
## m \ddot y_2 + \frac{\partial U}{\partial y_2} = - m \ddot y_1 - \frac{\partial U}{\partial y_1} ##
## ( m \ddot y_1 + \frac{\partial U}{\partial y_1} )(x_1 - x_2) = (- m \ddot x_1 - \frac{\partial U}{\partial x_1})(y_1 - y_2) ## (2)

After expressing ## (x_1 - x_2) , (y_1 - y_2) ## and ## \lambda ## from equation (1) and (2), substituted it into the constraint equation I got an equation like this:

## (m \ddot x_1 + \frac{\partial U}{\partial x_1} ) * <something> = 0 ##

I think it's wrong.
Can you confirm or point on my mistake?
Thanks :)
 
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  • #2
Jengalex said:
Hi!

I tried to compute an ideal barbell-shaped object's dynamics, but my results were wrong.
My Langrangian is:

## L = \frac{m}{2} ( \dot{x_1}^2 + \dot{x_2}^2 + \dot{y_1}^2 + \dot{y_2}^2 ) - U( x_1 , y_1 ) - U ( x_2 , y_2 ) ##

And the constraint is:

## f = ( x_1 - x_2 )^2 + ( y_1 - y_2 )^2 - L^2 = 0 ##I dervated L + λf by ## x_1, x_2, y_1, y_2 ## and λ:

## m \ddot x = - \frac{\partial U}{\partial x_1 } + \lambda ( x_1 - x_2 ) ## (1)
Four equtions similar to this and the constraint.

Then I expressed ## \ddot x_2 , \ddot y_1 , \ddot y_2 ## with ## \ddot x_1 , x_1 , x_2 , y_1 , y_2 ## and U's partial derivates' local values:

## m \ddot x_2 + \frac{\partial U}{\partial x_2} = - m \ddot x_1 - \frac{\partial U}{\partial x_1} ##
## m \ddot y_2 + \frac{\partial U}{\partial y_2} = - m \ddot y_1 - \frac{\partial U}{\partial y_1} ##
## ( m \ddot y_1 + \frac{\partial U}{\partial y_1} )(x_1 - x_2) = (- m \ddot x_1 - \frac{\partial U}{\partial x_1})(y_1 - y_2) ## (2)

After expressing ## (x_1 - x_2) , (y_1 - y_2) ## and ## \lambda ## from equation (1) and (2), substituted it into the constraint equation I got an equation like this:

## (m \ddot x_1 + \frac{\partial U}{\partial x_1} ) * <something> = 0 ##

I think it's wrong.
Can you confirm or point on my mistake?
Thanks :)
I don't know what a "barbell-shaped dynamics" is, but if in your problem in the plane with two points there is a constraint, the degrees of freedom are 3, not 4, so I would have written the Lagrangian as function of 3 independent generalized coordinates, for example x1, y1 and the angle between the line connecting the two points (P1 = (x1,y1); P2 = (x2,y2)) and the x axis.
 
  • #3
lightarrow said:
I don't know what a "barbell-shaped dynamics" is, but if in your problem in the plane with two points there is a constraint, the degrees of freedom are 3, not 4, so I would have written the Lagrangian as function of 3 independent generalized coordinates, for example x1, y1 and the angle between the line connecting the two points (P1 = (x1,y1); P2 = (x2,y2)) and the x axis.

Yes I've tried it now, it looks fine. Thanks!
 

1. What is a constrained Lagrangian equation in the context of a barbell?

A constrained Lagrangian equation in the context of a barbell is a mathematical equation that describes the motion of a barbell subject to external forces and constraints, such as the weight of the barbell and the length of the bar.

2. How is a constrained Lagrangian equation different from a regular Lagrangian equation?

A constrained Lagrangian equation takes into account any external forces or constraints acting on the system, while a regular Lagrangian equation only considers the kinetic and potential energy of the system. This makes the constrained Lagrangian equation more complex and difficult to solve.

3. What are some common constraints encountered in a barbell system?

Some common constraints encountered in a barbell system include the length of the bar, the weight of the bar and the weights attached to the bar, and any fixed points or surfaces that the barbell may come into contact with during its motion.

4. How is a constrained Lagrangian equation used to solve for the motion of a barbell?

The constrained Lagrangian equation is used to derive the equations of motion for the barbell, which can then be solved to determine the position, velocity, and acceleration of the barbell at any given time. This allows for a more comprehensive understanding of the motion of the barbell and can be used to make predictions about its behavior.

5. Can a constrained Lagrangian equation be applied to other systems besides a barbell?

Yes, a constrained Lagrangian equation can be applied to any system that is subject to external forces and constraints. This includes systems in mechanics, physics, and engineering, making it a versatile and useful tool for understanding and analyzing complex systems.

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