When the Lagrangians are equals?

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In summary, the conversation discusses a problem in Landau's book on classical mechanics and the solution proposed by the speaker. The problem involves finding the ratio of times for particles with different masses but the same potential energy. The speaker suggests using the Lagrangian for both paths but is unsure about its concept and meaning. They also request for the problem to be posted for better understanding.
  • #1
rmadsanmartin
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I’m not very good with english, it isn’t my native language..., but I’m going to explain my question...

I’m reading the first book of Landau's series ,it’s about clasical mechanics.
In the second chapter you can find a problem about the conservation's theorem

the problem says The first problem says:

Find the ratio of the times in the same path for particles having different masses but the same potential energy.

the solution is: t'/t=sqrt(m'/m)

My tentative solution is supposing that the lagrangian for both paths are the same...

then:

L'=L

1/2m'v'2-U=1/2mv2-U

Finally:

t'/t=sqrt(m'/m)

BUT, It’s that correct?

and why the lagrangians are the same? I’m not sure about the real concept (or meaning) of the lagrangian of a system...

thanks...
 
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  • #2
It would be helpful if you posted the problem since we don't all have a copy of Landau. Maybe this should be in the homework help section.
 

1. What is the significance of the Lagrangian equations?

The Lagrangian equations are a set of mathematical equations that are used to describe the motion of a system of particles. They are an alternative to Newton's laws of motion and are often used in physics and engineering to solve complex problems involving the motion of objects.

2. When are two Lagrangians considered to be equal?

Two Lagrangians are considered to be equal when they produce the same equations of motion for a given system. This means that both Lagrangians must result in the same values for position, velocity, and acceleration of the particles in the system.

3. How are Lagrangian equations used in mechanics?

Lagrangian equations are used in mechanics to solve problems involving the motion of particles and systems. They allow for a more efficient and elegant way of solving complex problems compared to traditional methods such as Newton's laws of motion. They are often used in fields such as astrophysics, robotics, and aerospace engineering.

4. Can Lagrangian equations be used for systems with constraints?

Yes, Lagrangian equations can be used for systems with constraints. In fact, they are particularly useful for solving problems involving systems with constraints, as they allow for the incorporation of these constraints into the equations of motion. This makes it easier to solve complex problems involving constrained systems.

5. What are the advantages of using Lagrangian equations over other methods?

There are several advantages to using Lagrangian equations over other methods. These include their ability to handle systems with constraints, their ability to take into account the entire system rather than just individual particles, and their elegant and efficient mathematical formulation. They also allow for the use of generalized coordinates, making it easier to solve problems involving complex geometries.

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