# How to characterize a power law field?

• Moleculmol
In summary, central forces with the form -k/r^n are used to model actions directed along the line joining the object and the origin. To recognize a power law field, one can calculate the motion for different powers and compare with a given trajectory. The power n can be determined through theoretical models or heuristic reasoning. An example of application is in cosmology or interactions between particles.
Moleculmol
Hello, as part of the study of fields with central forces, I came across with fields called power law, defined by F = - K/r ^ n u
(u is radial vector passing through the origin O)
I would like to dismiss case n = 2, which refers to the Newtonian fields whose study was exhaustively conducted in class. My questions are:
-How to recognize a power law field from the trajectory of a moving material point in this field?
- Once the nature of the field identified (power law), the determination of the power n can be conjectured by reasoning on heuristic considerations, however are there any situations where one can resort to theoretical models to find this coefficient (I think for example to VanderWalls force but I would like another example).
- An example of application of these methods to a concrete physical situation, in cosmology (movement of the stars) or interactions between particles at infinitesimal scale.

Moleculmol said:
Summary: In classic mechanics, central forces with the form -k/r^n are often introduced to model actions that are directed along the line joining the object and the origin.

-How to recognize a power law field from the trajectory of a moving material point in this field?
Calculate how the motion will look like for different powers, then do the opposite if you have a given trajectory. n=2 and n=-1 are the only cases that lead to closed non-circular orbits.
Moleculmol said:
- An example of application of these methods to a concrete physical situation, in cosmology (movement of the stars) or interactions between particles at infinitesimal scale.
Well, on large scales these are all n=2, on small scales you probably have to consider quantum mechanics and things get more complicated.

## 1. What is a power law field?

A power law field is a type of mathematical relationship between two variables, where one variable is proportional to the other raised to a power. This type of relationship is often seen in natural phenomena, such as the distribution of wealth or the frequency of earthquakes.

## 2. How do you determine if a field follows a power law?

To determine if a field follows a power law, you can plot the data on a log-log scale and see if it forms a straight line. If the data points fall on a straight line, it is likely that the field follows a power law. Additionally, statistical tests such as the Kolmogorov-Smirnov test can be used to confirm the power law relationship.

## 3. What is the exponent of a power law field?

The exponent of a power law field is the number that represents the power to which the independent variable is raised. It is often denoted as "alpha" and can be calculated using regression analysis on the log-log plot of the data.

## 4. Can a power law field change over time?

Yes, a power law field can change over time. This can be due to various factors such as external influences, changes in the underlying system, or errors in data collection. It is important to regularly analyze and update the power law relationship to accurately characterize the field.

## 5. How is a power law field used in scientific research?

A power law field can be used in various scientific research fields, such as economics, physics, and biology. It can be used to understand and model complex systems, make predictions, and identify patterns and trends. Additionally, studying power law fields can provide insights into the underlying mechanisms and processes of natural phenomena.

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