Problem about Lagrangian mechanics

In summary, the conversation discusses the potential energy of a wedge in Lagrangian and the approach of systematically plugging in all values of potential energy, even if they are constant and will disappear. There is also a mention of the Lagrangian being defined up to an additive function and the importance of correctly identifying forces of constraint in a problem.
  • #1
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Homework Statement
Find the Lagrangian of system.
In the question say the wedge can move without friction on a smooth surface.
Why does the potential energy of the wedge appear in Lagrangian?
Relevant Equations
##\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q}##
CM 2. 20.png

In Solution https://www.slader.com/textbook/9780201657029-classical-mechanics-3rd-edition/67/derivations-and-exercises/20/

In the question say the wedge can move without friction on a smooth surface.

Why does the potential energy of the wedge appear in Lagrangian?

(You can see the Larangian of this system at below.)
CM 2. 20 .1.png

CM 2. 20 .2.png
 
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  • #2
The potential energy of the wedge is constant anyway, so isn't going to affect the equation of motion.

But in any case the expression ##U = Mgy_M## isn't correct, because the centre of mass of the wedge is not the coordinate ##y_M## (that is the coordinate of the left corner). In any case they will still stumble upon the correct answer, because the mistake happens to be constant and drops out.
 
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  • #3
It looks like strange thing to do. I wouldn't have the terms involving ##y_M## and ##\dot y_M## as the wedge is constrained to move in the x-direction only.
 
  • #4
They are just following a systematic approach and plugging-in all values of potential energy, even the constant one which is going to disappear. If you do this for all problems, it helps you to establish a standard approach and makes it less likely you'll forget some term(s) in more complex problems. Not essential but probably a useful practice for some students.

As etotheipi points out, they've done it incorrectly, which is a good example of irony.
 
  • #5
Moreover, the Lagrangian is defined up to an additive function
$$\dot f(t,q)=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial q^i}\dot q^i$$
that is the Lagrangians ##L## and ##L'=L+\dot f(t,q)## generate the same equations
 
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  • #6
Steve4Physics said:
They are just following a systematic approach and plugging-in all values of potential energy, even the constant one which is going to disappear. If you do this for all problems, it helps you to establish a standard approach and makes it less likely you'll forget some term(s) in more complex problems. Not essential but probably a useful practice for some students.
I don't think this was the motivation here. The problem asks for the forces of constraint as well, so you don't want to impose the constraint right from the start. It's not entirely clear to me from the problem statement, however, if it was asking for just the forces of constraint on just the particle or for all of the forces of constraint within the system.
 

Related to Problem about Lagrangian mechanics

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of particles and systems. It is based on the principle of least action, which states that a system will follow the path that minimizes the action, a quantity related to the energy of the system.

2. How is Lagrangian mechanics different from Newtonian mechanics?

Lagrangian mechanics is a more general and elegant approach to describing the motion of particles and systems compared to Newtonian mechanics. It takes into account all forces acting on a system, not just those caused by interactions with other objects. It also allows for the use of generalized coordinates, making it easier to solve complex problems.

3. What is the role of the Lagrangian in this framework?

The Lagrangian is a mathematical function that describes the kinetic and potential energy of a system. It is used to derive the equations of motion for the system and is a key component of the principle of least action in Lagrangian mechanics.

4. Can Lagrangian mechanics be applied to all types of systems?

Yes, Lagrangian mechanics can be applied to a wide range of systems, including classical mechanics, quantum mechanics, and even relativistic systems. It is a versatile framework that can be adapted to various types of problems and scenarios.

5. What are some real-world applications of Lagrangian mechanics?

Lagrangian mechanics has many practical applications, including predicting the motion of celestial bodies, analyzing the behavior of mechanical systems such as pendulums and springs, and understanding the dynamics of particles in fields such as electromagnetism and quantum mechanics. It is also used in engineering, robotics, and other fields to model and optimize the behavior of complex systems.

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