Delta amplitude and nabla amplitude

[Jhenrique]

Delta amplitude and "nabla amplitude"

Why all jacobi theory and all ellipitc integrals is based in $\Delta(\theta) = \sqrt{1-m \sin(\theta)^2}$ ?

You already think that this definition is just midle of history, cause' you can define other elementar function: $$\nabla(\theta) = \sqrt{1-m \cos(\theta)^2}$$ So, a "nabla amplitude", will imply in more interesting definitions:

if: $$\int_{0}^{\phi}\frac{d\theta}{\sqrt{1-m \sin(\theta)^2}}=u$$ thus: $$\int_{0}^{\phi}\frac{d\theta}{\sqrt{1-m \cos(\theta)^2}}=v$$ and so: $$\\ u=\int \frac{d\phi}{\Delta(\phi)} \;\;\;\Rightarrow \;\;\;\frac{du}{d\phi}=\frac{1}{\Delta(\phi)} \;\;\;\Rightarrow \;\;\;\frac{d\phi}{du} = \Delta(\phi) = dn(u) \;\;\;\Rightarrow \;\;\; \phi = \int dn(u) du \\ \\ \\ \\ v=\int \frac{d\phi}{\nabla(\phi)} \;\;\;\Rightarrow \;\;\;\frac{dv}{d\phi}=\frac{1}{\nabla(\phi)} \;\;\;\Rightarrow \;\;\;\frac{d\phi}{dv} = \nabla(\phi) = qn(v) \;\;\;\Rightarrow \;\;\; \phi = \int qn(v)dv$$
And the implications continues... So, why the elliptic integrals are based only in the root square of a sine, why not exist definition based in the root square of a cosine too?

 

[jvicens]

Thank you for your question regarding the use of the delta amplitude in Jacobi theory and elliptic integrals. The reason why the delta amplitude is commonly used in these theories is because it is a fundamental function that is necessary for calculating the elliptic integrals. This function is closely related to the elliptic integrals and their inverses, which are used to solve various problems in mathematics and physics.

While the nabla amplitude you have proposed may also have some interesting properties, it is not as commonly used or studied as the delta amplitude. This is likely due to historical reasons, as you have mentioned. However, it is important to note that the delta amplitude and the nabla amplitude are not interchangeable and cannot be used interchangeably in all situations.

Furthermore, the use of the delta amplitude is not limited to just the root square of a sine. It can also be used with other trigonometric functions such as cosine, tangent, and cotangent. This is because the delta amplitude is a generalization of the cosine amplitude, which is defined as the square root of 1 minus the square of a trigonometric function. Therefore, the use of the delta amplitude allows for a more versatile and general approach to solving problems involving elliptic integrals.

In conclusion, while the nabla amplitude may have some interesting implications, the delta amplitude remains an important and fundamental function in Jacobi theory and elliptic integrals. Its use is not limited to just the root square of a sine and it is a necessary tool for solving various mathematical and physical problems.