Trouble understanding coordinates for the Lagrangian

In summary, the conversation discusses a problem in Landau's book on mechanics and how to arrive at the infinitesimal displacement for the particles m1. The solution involves using Pythagoras' theorem on orthogonal infinitesimal displacements and can also be approached using a formal approach in terms of the holonomic basis. The infinitesimal motion of m1 can be disassembled into two parts - horizontal rotation and rotation about point A in the plane of the book. This explanation is done in spherical coordinates.
  • #1
p1ndol
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Hello, I'm having some trouble understanding this solution provided in Landau's book on mechanics. I'd like to understand how they arrived at the infinitesimal displacement for the particles m1. I appreciate any kind of help regarding this problem, thank you!
 

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  • #2
It's nothing more than Pythagoras applied to orthogonal infinitesimal displacements ##ad\theta## and ##a\sin{\theta} d\phi##, however if you want a (very slightly) more formal approach in terms of the holonomic basis...

if ##\phi## is held constant then ##\mathbf{E}_{\theta} = \dfrac{\partial \mathbf{r}}{\partial \theta} = a \hat{\boldsymbol{e}}_{\theta}## whilst if ##\theta## is held constant then ##\mathbf{E}_{\phi} = \dfrac{\partial \mathbf{r}}{\partial \phi} = a\sin{\theta} \hat{\boldsymbol{e}}_{\phi}##. Since ##\mathbf{E}_{\theta}## and ##\mathbf{E}_{\phi}## are orthogonal you have $$dl^2 = \displaystyle{\sum_i \sum_j }dx^i \mathbf{E}_i \cdot dx^j \mathbf{E}_j= {E_{\theta}}^2 d\theta^2 + {E_{\phi}}^2 d\phi^2 = a^2 d\theta^2 + a^2 \sin^2{\theta} d\phi^2$$
 
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  • #3
Thank you very much!
 
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  • #4
I think he did it in spherical coordinates. The infinitesimal motion of m1 can be disassembled into two parts; this disassembly is correct since the displacements in the two directions are small (meaning they are kinda linear) and orthogonal:

Displacement^2 caused by horizontal rotation ##\Omega##:
$$dl^2_{horizontal}=R^2(\Omega\mathrm{dt})^2=a^2\sin^2 \theta(\Omega\mathrm{dt})^2$$

Displacement^2 caused by the rotation of m1 about A in the plane of book:
$$v=r\omega\implies dl^2_{vertical}=(a\mathrm d{\theta})^2$$

Hope this helps.
 
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  • #5
Thanks, you couldn't have been clearer!
 

What is the Lagrangian?

The Lagrangian is a mathematical function that describes the dynamics of a system in classical mechanics. It takes into account the position, velocity, and potential energy of the system's particles.

Why is it important to understand coordinates for the Lagrangian?

Coordinates for the Lagrangian are important because they allow us to mathematically model and analyze the behavior of a system. They provide a way to describe the position and motion of particles in a system and how they interact with each other.

What are the different types of coordinates used for the Lagrangian?

The most commonly used coordinates for the Lagrangian are Cartesian coordinates (x, y, z), cylindrical coordinates (r, θ, z), and spherical coordinates (r, θ, φ). However, other coordinate systems can also be used depending on the specific system being studied.

How are coordinates for the Lagrangian related to Newton's laws of motion?

Coordinates for the Lagrangian are related to Newton's laws of motion through the principle of least action. This principle states that the path a particle takes between two points is the one that minimizes the action, which is a mathematical expression involving the Lagrangian and the particle's coordinates.

What are some common challenges when understanding coordinates for the Lagrangian?

Some common challenges when understanding coordinates for the Lagrangian include visualizing and interpreting the mathematical equations, understanding the relationship between different coordinate systems, and applying the concept to real-world systems. It may also require a solid understanding of calculus and physics principles.

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