Question about L(v^2) Notation in Landau & Lifshitz's Mechanics

In summary, L(v^2) notation is a representation of the kinetic energy in Landau & Lifshitz's Mechanics, defined as the sum of the squares of velocities of all particles in a system. It is used because it simplifies equations and is closely related to Lagrangian mechanics. It can be applied to any system and is different from other kinetic energy notations.
  • #1
hagopbul
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Hello :

i am reading now landau & lifshitz book on mechanics and i have small question :

about L(v^2) notation it was not very clear in the book and i couldn't understand it correctly anyone can explain it or provide a link with explanation
page (4 - 5)

Best regards
Hagop
 
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  • #2
Do you have a problem to derive (3.1) ? I understand :

L is function of only ##\mathbf{v}##, not of ##r## and t. Further L is function of ##|\mathbf{v}|=\sqrt{v^2}##. Thus L is function of ##v^2##.
 
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1. What is L(v^2) notation in Landau & Lifshitz's Mechanics?

L(v^2) notation is a mathematical notation used in Landau & Lifshitz's Mechanics to represent the Lagrangian of a system in terms of the velocities squared (v^2) of the particles in the system. It is a compact and convenient way to express the equations of motion for a system.

2. How is L(v^2) notation different from traditional Lagrangian notation?

In traditional Lagrangian notation, the Lagrangian is expressed in terms of the velocities (v) of the particles in the system. L(v^2) notation, on the other hand, uses the squares of the velocities (v^2) in the Lagrangian. This notation is particularly useful for systems with non-linear forces, as it simplifies the equations of motion.

3. What are the advantages of using L(v^2) notation?

L(v^2) notation has several advantages. It simplifies the equations of motion for systems with non-linear forces, making them easier to solve. It also allows for a more compact and concise representation of the equations of motion, making it easier to work with complex systems. Additionally, L(v^2) notation is often used in theoretical physics and allows for a more elegant and general formulation of the equations of motion.

4. Can L(v^2) notation be used for all types of systems?

Yes, L(v^2) notation can be used for all types of systems, including mechanical, electrical, and quantum systems. It is particularly useful for systems with non-linear forces, but can also be applied to linear systems. However, it may not always be the most convenient notation for certain types of systems, and traditional Lagrangian notation may be more appropriate.

5. How is L(v^2) notation related to Hamilton's principle?

L(v^2) notation is closely related to Hamilton's principle, which states that the true path of a system is the one that minimizes the action (integral of the Lagrangian) between two points in time. L(v^2) notation allows for a more elegant and concise expression of Hamilton's principle, making it easier to apply in theoretical physics and mechanics.

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