Expanding a small oscillation potential in taylor series

In summary, the speaker is seeking help with understanding Goldstein's equation 6.3 from the 3rd edition, which deals with oscillations. They are struggling to understand how the Taylor series expansion is applied in this equation and are looking for assistance. They also mention that not many people have access to this book and suggest taking a screenshot and uploading it for further clarification. The speaker also questions the need for a Taylor series in the context of oscillating mass energy.
  • #1
shehry1
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I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.
 
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  • #2
see http://en.wikipedia.org/wiki/Taylor_series" [Broken]
 
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  • #3
shehry1 said:
I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.
Not many people have this book. Maybe you should take a screen shot and upload it? From what you're saying, I don't see why you would need a Taylor series for the energy of an oscillating mass. It is already quadratic with the displacement (for a spring-mass system at least).
 

1. What is a small oscillation potential?

A small oscillation potential is a function that describes the potential energy of a system, such as a spring or pendulum, when it is perturbed from its equilibrium position. It is often represented as a parabolic function, with the minimum point at the equilibrium position.

2. Why is it important to expand a small oscillation potential in a Taylor series?

Expanding a small oscillation potential in a Taylor series allows us to approximate the potential energy function and make predictions about the behavior of the system. It also helps us analyze the system's dynamics and determine the stability of the equilibrium position.

3. What is a Taylor series and how is it used to expand a potential?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function by evaluating its derivatives at a specific point. In the case of expanding a small oscillation potential, the Taylor series is used to approximate the potential energy function at the equilibrium point, with the first few terms representing the parabolic function.

4. What are the benefits of using a Taylor series to expand a potential?

Using a Taylor series to expand a potential allows us to simplify complex functions and make them more manageable for analysis. It also provides us with a way to approximate functions and make predictions about their behavior, which can be useful in many scientific fields, including physics and engineering.

5. Are there any limitations to using a Taylor series to expand a potential?

Yes, there are some limitations to using a Taylor series to expand a potential. For instance, it is only accurate for functions that are smooth and have continuous derivatives. In addition, the accuracy of the approximation depends on how many terms of the series are included in the expansion. As the number of terms increases, the approximation becomes more accurate, but the calculations become more complex.

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