# 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.

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!

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).

Thanks for the post.
I don't see why is not (∂L/∂v²)(v²+2v⋅ε)

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.

Thanks for your help, clear now! Should I study calculus of variations in order to continue with the book?

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.