Can KE be reformulated using |v| instead of v^2?

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Discussion Overview

The discussion revolves around the potential reformulation of kinetic energy (KE) using the absolute value of velocity (|v|) instead of the square of velocity (v^2). Participants explore theoretical implications, comparisons between different norms, and the relevance of average velocity (v_avg) versus root mean square velocity (v_rms) in various contexts, including statistical mechanics and computer science.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether KE and classical laws can be reformulated using |v| instead of v^2, noting that while the units may not align, there could be alternative formulations worth exploring.
  • Another participant mentions the concept of momentum and its relation to velocity, suggesting that average momentum might be considered similarly to KE.
  • A participant raises the idea that using |v| could be beneficial in non-equilibrium situations where outliers may skew results derived from v^2.
  • There is a clarification regarding notation, with one participant expressing interest in using <|v|> instead of and questioning the utility of <|v|>^2.
  • Another participant points out that while total energy and momentum are conserved, total speed is not, which may limit the physical relevance of using |v|.
  • Concerns are raised about the mathematical complexities associated with the absolute value function, particularly its nonholomorphic nature.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility and implications of reformulating KE using |v|. There is no consensus on whether such reformulations are valid or useful, and the discussion remains unresolved regarding the practical applications of v_avg versus v_rms.

Contextual Notes

Participants acknowledge that changing the formulation of KE may lead to issues with units and physical relevance. The discussion also highlights the dependence on the frame of reference for energy and momentum, contrasting it with speed.

BallisticDisks
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While doing some calculations on v_rms using the Maxwell-Boltzmann distribution, I noticed that v_rms and v_avg are pretty similar (https://casper.berkeley.edu/astrobaki/index.php/File:MaxwellSpeedDist.png).
In fact, really it's just the choice of using the 1-norm (|v|_avg) vs. 2-norm sqrt(v^2 avg). When deriving v_rms or <KE> from the Maxwell-Boltzmann distribution , we get <KE> = 0.5m<v^2> = 3kT/2; however, using v_avg from the Maxwell-Boltzmann distribution we get v_avg = sqrt(8kT/(pi*m)) and then 0.5m<v>^2 = 4kT/pi.

In computer science/statistics, people often choose the distance metric <|x|> vs. sqrt<(x^2)> based on their needs, and generally <|x|> is thought to be better and more robust to outliers (yet has far worse properties, e.g., no convergence). I'm familiar with the work-energy theorem and the derivation of KE = 0.5mv^2 from Newton's 2nd law, but am trying to figure out if there are alternative formulations using |v|. I'm also familiar with the equipartition theorem with energies of form Ax^2 being assigned 0.5kT energy and am not sure how this would be reformulated using |v|.

So two questions:
1. Can KE/classical laws be reformulated using |v| instead of |v^2|, or has anything been published using the 1-norm (absolute value) or any other norm besides the 2-norm (square)? I realize the units of energy won't make sense with just changing how KE is computed. But, for example, speed could be defined as (sum |v|) instead of (sum v^2), though I'm not sure if people do this.

2. Are there cases where v_avg is useful to use instead of v_rms?

Thanks in advance for the help/dicussion! This has been gnawing at me since in CS/stats we usually have the choice of using |v| or v^2 to compare distributions, but I'm not sure if we're "forced" to use v^2 in physics because of fundamental laws (and if so which ones?).

Image is from: https://casper.berkeley.edu/astrobaki/index.php/Maxwellian_velocity_distribution
 
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Yep! I know that momentum = mv. I've only seen momentum used as a property of individual molecules as opposed to ensembles in stat mech (and also never seen a scalar formulation of momentum analogous to speed for velocity) . Is the average momentum of a system ever considered in a way similar to KE? Maybe (q1) is better posed as -- is there a macroscopic energy-like description of a system using |v| instead of v^2? For example, this could be useful in non-equillibrium settings where few particles have very high velocity and are outliers, and where v^2 could be misleading.

(q2) still remains -- is there a case where v_avg is useful to describe systems instead of v_rms? Or, is it really only used to describe the momentum of particles.
 
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BallisticDisks said:
Can KE/classical laws be reformulated using |v| instead of |v^2|,
The units don’t work out as written. However, since v is a vector over the reals then |v^2|=|v|^2
 
Poor notation on my part. I meant reformulated with <|v|> instead of <v^2>. But also curious about if <|v|>^2 is ever useful to use.
 
Both total energy and total momentum are conserved. But total speed is not conserved, so it is less physically relevant. Of course, there are some uses, but not nearly as many.

Energy, momentum, and speed will all change depending on your frame of reference. But energy and momentum form a 4-vector, so laws involving them are covariant. Speed isn't a component of a tensor, so any laws involving speed will have to change depending on your frame of reference.

Also, the absolute value function is hard to work with since it's nonholomorphic.
 

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