What is the Lagrangian with constraint forces?

In summary: Now you can solve the Euler-Lagrange equations for the one coordinate ##x## and you have a single equation of motion with no second derivative."In summary, For mechanics problems involving an inclined plane, a ball rolling off of a half hemisphere, or a pendulum, after deriving the Lagrangian, a constraint must be chosen. This constraint should restrict the generalized coordinates to specific values, such as in the case of a block sliding down a frictionless inclined plane. The chosen constraint can then be incorporated into the Lagrangian to solve for the equations of motion."
  • #1
Eh6794
1
0
TL;DR Summary
I am confused about the constraint I choose for my mechanics problems.

Let's say I have an inclined plane, a ball rolling off of a half hemisphere or a pendulum and I need to find a tension, friction or normal force.

After I derive the lagrangian, I need to find a constraint, but how do I choose a constraint?

I have worked through a couple of problems with solutions, but it seems like I find the coordinate/variable in the direction of the force I want? Ex. pendulum tension would be L?
ii
 
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  • #2
Eh6794 said:
Summary:: I am confused about the constraint I choose for my mechanics problems.

Let's say I have an inclined plane, a ball rolling off of a half hemisphere or a pendulum and I need to find a tension, friction or normal force.

After I derive the lagrangian, I need to find a constraint, but how do I choose a constraint?

I have worked through a couple of problems with solutions, but it seems like I find the coordinate/variable in the direction of the force I want? Ex. pendulum tension would be L?

ii
"After I derive the lagrangian, I need to find a constraint, but how do I choose a constraint?" Maybe |R| = cte?
That is, ##r = a## => ##r-a = 0##, it is a good holonomic constraint. And i don't know another constraint in this cases you cited. You find a constraint studying the restrictions of the system, let's say like this.

"I have worked through a couple of problems with solutions, but it seems like I find the coordinate/variable in the direction of the force I want? Ex. pendulum tension would be L?"
I do not understand. What do you mean?
 
  • #3
You choose the constraint that, ahem, constrains your generalized coordinates not to be just anything. For example, in the case of a block sliding down a frictionless inclined plane you can choose generalized coordinates ##x## and ##y## along the vertical and the horizontal and write $$\mathcal L=\frac{1}{2}m\dot x^2+\frac{1}{2}m\dot y^2+mgy.$$The constraint is that the block must stay on the plane so if you pick a value for one coordinate, the other is strictly specified. Here the relevant constraint is ##y=x~\tan\theta## where ##\theta## is the angle of the incline relative to the horizontal. Then the Lagrangian becomes $$\mathcal L=\frac{1}{2}m\dot x^2+\frac{1}{2}m\dot x^2\tan^2\theta+mgx\tan\theta.$$
 

Related to What is the Lagrangian with constraint forces?

1. What are Lagrangian Constraint Forces?

Lagrangian Constraint Forces are forces that arise due to constraints on the motion of a system. These constraints can be physical, such as a fixed point of rotation, or mathematical, such as a constraint equation. These forces are incorporated into the Lagrangian equations of motion to accurately describe the motion of a system.

2. How are Lagrangian Constraint Forces different from other forces?

Lagrangian Constraint Forces are different from other forces because they do not appear in the traditional Newtonian equations of motion. Instead, they are incorporated into the Lagrangian equations, which take into account the constraints on the motion of a system. This allows for a more comprehensive and accurate description of the system's motion.

3. Can Lagrangian Constraint Forces be ignored?

No, Lagrangian Constraint Forces cannot be ignored as they play a crucial role in determining the motion of a system. Ignoring these forces can lead to incorrect predictions and inaccuracies in the description of the system's motion.

4. How are Lagrangian Constraint Forces calculated?

Lagrangian Constraint Forces are typically calculated using the method of Lagrange multipliers. This involves taking the partial derivatives of the constraint equations with respect to the generalized coordinates and incorporating them into the Lagrangian equations of motion.

5. What are some examples of Lagrangian Constraint Forces?

Some examples of Lagrangian Constraint Forces include tension in a string, normal force, and frictional force. These forces are often present in systems with constraints, such as a pendulum swinging from a fixed point or a block sliding on a surface with friction.

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