How to derive this (Lagrangian of mattress) from Zee's book

In summary, the potential expression includes the first term in a power series and will eventually include all higher powers. The phi^3 term is not included due to the assumption of symmetry, but the next term will include a phi^4 term. This is due to taking the continuum limit, which results in terms like (partial_x phi)^2 phi^2.
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Clara Chung
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I am not sure where does the dy term, phi^2 and phi^4 terms come from.
I guess there are dx and dy because we have to account for the nearest neighbour pairs in the x and y axis?
I guess there is a phi^2 term because 2q_a*q_b=(q_a-q_b)^2-q_a^2-q_b^2, the term q_a^2-q_b^2?
How about the phi^4 term?
 
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Clara Chung said:
I guess there is a phi^2 term because 2q_a*q_b=(q_a-q_b)^2-q_a^2-q_b^2, the term q_a^2-q_b^2?
That is correct.
Clara Chung said:
How about the phi^4 term?

In his expression for ##V##, he is only writing the first term in a power series, and the power series will really include all higher powers too. Although he is presumably assuming that the potential is symmetric under taking all the ##q_a## to ##-q_a##, otherwise there would be a ##\phi^3## term too. (Actually, looking at my copy of Zee, it seems that the very next equation in the book does include ##\phi^3## so this might just be a typo.) So the next term will look like

$$
V = \sum_{ab} \frac{1}{2} k_{ab} q_a q_b + \sum_{abcd} k_{abcd} q_a q_b q_c q_d + \cdots
$$
Now when you take the continuum limit of this, you'll get a bunch of terms including a ##\phi^4## term (and also terms like ##(\partial_x \phi)^2 \phi^2## which are presumably included in the ellipses along with higher order terms). If I had included a cubic term it would also allow terms like ##\phi^3##.
 
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Related to How to derive this (Lagrangian of mattress) from Zee's book

1. How do I derive the Lagrangian of a mattress from Zee's book?

To derive the Lagrangian of a mattress from Zee's book, you will need to follow the steps outlined in the book. First, you will need to understand the basic principles of Lagrangian mechanics and how it applies to the system of a mattress. Then, you can use Zee's equations and examples to guide you through the derivation process.

2. Is it difficult to derive the Lagrangian of a mattress from Zee's book?

The difficulty of deriving the Lagrangian of a mattress from Zee's book will depend on your level of understanding of Lagrangian mechanics and your familiarity with the equations and concepts in the book. If you have a strong background in physics and mathematics, it may not be too difficult. However, if you are new to the subject, it may take more time and effort to fully grasp the derivation process.

3. What are the key equations and principles used in deriving the Lagrangian of a mattress?

The key equations and principles used in deriving the Lagrangian of a mattress include the Lagrangian equation, the principle of least action, and the equations of motion for a system. These concepts are explained in detail in Zee's book and are essential for understanding the derivation process.

4. Can I use Zee's book to derive the Lagrangian of any type of mattress?

Yes, Zee's book provides a general framework for deriving the Lagrangian of any type of system, including a mattress. However, the specific equations and examples used in the book may vary depending on the type of system being studied. It is important to understand the underlying principles and adapt them to your specific system.

5. What are the practical applications of deriving the Lagrangian of a mattress?

The Lagrangian of a mattress can be used to analyze the dynamics and behavior of the system, such as the motion and energy of the mattress. It can also be used to study the effects of external forces and constraints on the mattress. This information can be useful in designing and optimizing mattresses for various purposes, such as improving comfort and support for sleepers.

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