Mechanics by Landau: Solving a Lagrangian Problem

In summary, the conversation is about a question regarding the Lagrangian for a free particle and its expressions in different inertial frames of reference. The conversation delves into the use of calculus of variations and the derivation of the Euler-Lagrange equations. The person asking for help is advised that it is not necessary to study calculus of variations to continue with the book.
  • #1
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Hi!

Just reading the first book by Landau in the theoretical physics course, and I need some guidance about one point (cf. 4, Lagrangian for a free particle.) Notations: L and L' are Lagrangians referred to different inertial frames of reference. e is a element of velocity between L and L'.
It says " We have L' = L (v'^2) = L (v^2 + 2ve + e^2 ). Expanding this expression in powers of e and neglecting the terms above the first order, we obtain

L (v'^2) = L (v^2) + partial derivative of L respect to v^2 times 2ve​

So, if this is supossed to be a Taylor serie with n from zero to one, why appears the derivate of L with respect to v^2. Can someone, please, make all steps explicit?


Any help would be very appreciated!
 
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  • #2
L is a function of v², and when you change to the new intertial frame v² becomes v² + 2v⋅ε to first order in ε because (v + ε)² = v² + ε² + 2v⋅ε . The change in L to first order in ε is therefore (∂L/∂v²)(2v⋅ε).
 
  • #3
Thanks for the post.
I don't see why is not (∂L/∂v²)(v²+2v⋅ε)
 
  • #4
If you have a function f and you want the change in f from a to a + Δ, you have to multiply f'(a) by Δ, i.e. f(a + Δ) = f(a) + f'(a)Δ.

In our case, v² becomes v² + 2v⋅ε, so you multiply by 2v⋅ε.
 
  • #5
Thanks for your help, clear now! Should I study calculus of variations in order to continue with the book?
 
  • #6
No it's not really necessary. The only place where it is used in this book is in section 2, where the Euler-Lagrange equations are derived, and a quite clear description of it is given there.
 

1. What is Mechanics by Landau?

Mechanics by Landau is a book written by physicist Lev Landau and his student Evgeny Lifshitz. It is a comprehensive textbook on classical mechanics, covering topics such as Lagrangian and Hamiltonian mechanics, rigid body dynamics, and special relativity.

2. What is a Lagrangian problem?

A Lagrangian problem is a type of problem in classical mechanics that involves finding the equations of motion for a system using the Lagrangian formalism. This approach is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action integral.

3. What is the Lagrangian formalism?

The Lagrangian formalism is a mathematical framework used in classical mechanics to describe the motion of a system. It is based on the concept of a Lagrangian function, which is a mathematical function that encapsulates all of the information about a system's dynamics. The equations of motion can then be derived by minimizing the action integral of the Lagrangian.

4. How is Mechanics by Landau useful?

Mechanics by Landau is a valuable resource for students and researchers in physics and engineering. It provides a thorough and rigorous treatment of classical mechanics, covering a wide range of topics and providing detailed explanations and examples. It is also known for its concise and elegant mathematical approach, making it a useful reference for those studying advanced mechanics.

5. Is Mechanics by Landau suitable for beginners?

While Mechanics by Landau is a highly respected and widely used textbook, it is not typically recommended for beginners in classical mechanics. It assumes a solid foundation in mathematics and physics, and may be more challenging for those just starting to learn about the subject. However, it can be a valuable resource for those seeking a deeper understanding of mechanics once they have a solid grasp of the basics.

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