Tips on writing the Lagrangian

In summary, the conversation discusses the difficulty in writing the Lagrangian for a system and asks for tips on how to approach it. An example problem is given and the conversation ends with the suggestion to practice and try different methods to gain confidence and improve intuition.
  • #1
Hey guys. I'm trying to gather some tips that people have acquired that helps them write the Lagrangian for a system. Obviously, the classic examples are drilled into our heads over and over, but just when you think you can tackle any problem the professor throws at you, there is that tricky one that gets you.

So I guess I'm asking, is there somewhat a systematic way to figure what the Lagrangian of a system is..?

One example that stumped me was a system where there is a circle radius, R, and in it there is a rod length, l, with uniform mass that can move frictionlessly. Find the Lagrangian of the system and the frequency of small oscillations.

Now conceptually I get that you reduce it down to a center of mass problem and that this center of mass oscillates like a simple harmonic. But I get stuck on how to actually go about writing the Lagrangian. (When l ≥ 2R, there should be no oscillations. But how to implement an inequality into the Lagrangian..? Most constraints that I've dealt with were "simple" equations, (something = somethingsomething).

This really frustrates me, because the other problems I tackled them easily. But a complete left field problem like this is making me doubt if I really understand how to write the Lagrangian of any arbitrary system.

So please if you guys can list some general tips that helps in writing the Lagrangian, it would greatly help me understand this skill/ability better.

That example problem isn't the main point of this thread. Just an example of a problem that is not a variant of classics like the single or double pendulum, simple harmonic oscillator, etc. So you don't have to use that as an example to teach me.
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  • #2
Do you mean "l ≥ 2R gives no oscillations"?
This is an unphysical case anyway (the rod cannot be inside), you can simply ignore it.

If both sides of the rod stay in contact with the circle all the time, your system has a single degree of freedom, and there is a quite natural way to express kinetic energy in terms of this value. Hint: rotation.
  • #3
^ Yeah sorry, that is what I meant.

That hint helped me thanks. It is like watching a magic trick. Once you know the trick, it becomes simple. Haha.

So do you have any tips in general on how to better this "intuition"...? Or is it just about gaining more confidence...?
  • #4
Practice. And test several different methods, one of them might give nice equations.
  • #5
Thanks. I'll just try to do as much problems with different scenarios as I can. And I'll keep your tip in mind.

1. What is the Lagrangian?

The Lagrangian is a mathematical function that describes the dynamics of a physical system in terms of its positions and velocities. It is commonly used in classical mechanics and field theory.

2. Why is it important to write the Lagrangian?

Writing the Lagrangian allows us to describe and analyze the behavior of a physical system in a more elegant and concise way. It also allows us to use powerful techniques such as the Euler-Lagrange equations to solve for the equations of motion.

3. How do I write the Lagrangian for a specific system?

The Lagrangian is typically written in terms of the system's generalized coordinates and their corresponding velocities. To write the Lagrangian, you must first identify all the relevant parameters and variables in the system, and then use the appropriate mathematical equations to express their relationships.

4. Are there any tips for writing the Lagrangian?

Yes, here are some tips: 1) Keep your notation and terminology consistent throughout your work. 2) Double-check your calculations to avoid mistakes. 3) Make sure your Lagrangian satisfies the principle of least action. 4) Use symmetries and conservation laws to simplify your Lagrangian.

5. Can the Lagrangian be used in other areas of science?

Yes, the Lagrangian has applications in various fields such as quantum mechanics, electromagnetism, and even economics. It is a powerful tool for describing and analyzing dynamical systems in general, not just in classical mechanics.

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