Angular momentum - kleppner and kolenkow - derivation or definition?

[3102]

I am reading "An Introduction to Mechanics" by Kleppner and Kolenkow (2014). On page 241 is the definition of the angular momentum:

"Here is the formal definition of the angular momentum $$\vec{L}$$ of a particle that has momentum $$\vec{p}$$ and is at position $$\vec{r}$$ with respect to a given coordinate system: $$\vec{L}=\vec{r} \times \vec{p}$$"

In the book there is no explanation why this formula should be true. From this equation the formulas for torque and moment of inertia are derived.

My question is: Why is the formula above correct? Why isn't the formula for angular momentum something completely different, like $$\vec{L}=\sqrt{(\vec{r} \cdot \vec{p})2\pi M}$$? Is the formula for angular momentum just an arbitrary definition? If not, how to derive it? How did people come across that particular formula?

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If anyone read the book mentioned above: Is there any derivation that I haven't found?

 

[Fredrik]

That formula is usually taken as a definition. You don't prove definitions. They just associate an English word with something mathematical.

 

[Dale]

In the book there is no explanation why this formula should be true.

Definitions are always true, by definition. You cannot derive a definition, and it doesn't make sense to ask why it is true, it is tautologically true because it is defined to be true.

The better question about a definition is whether or not it is useful.

 

[3102]

BUT: An gyroscope behaves in a way that is not influenced by our definitions. If I took another definition for angular momentum (e.g.: the one mentioned above) then I would compute a totally wrong "forecast" about the gyroscopes motion. So, at least for my part, it does not seem like an arbitrary definition.

 

[Nugatory]

Is the formula for angular momentum just an arbitrary definition? If not, how to derive it? How did people come across that particular formula?

The quantity $\vec{r}\times\vec{p}$ has been observed to be conserved, which makes useful enough to be worth studying. The arbitrary choice happened when we decided to call it "angular momentum" instead of something else.

(As an aside, there are deeper reasons why this thing we call angular momentum is conserved, but they were not discovered until centuries after angular momentum had been discovered and named).

 

[guitarphysics]

Just because the quantity is useful doesn't mean the definition's not arbitrary. By arbitrary we mean that it could have been called anything, not necessarily angular momentum (and obviously not necessarily represented by $\vec{L}$). This quantity, the cross product between the radius vector and the momentum of a particle, turns out to be pretty useful, so we decide to make it an official physical quantity by giving it a name and all that. That part was arbitrary.

 

[Fredrik]

BUT: An gyroscope behaves in a way that is not influenced by our definitions. If I took another definition for angular momentum (e.g.: the one mentioned above) then I would compute a totally wrong "forecast" about the gyroscopes motion. So, at least for my part, it does not seem like an arbitrary definition.

Regardless of what term you associate with $\vec r\times\vec p$ and what piece of mathematics you associate with the term "angular momentum", it would be $\vec r\times\vec p$ that you use in the calculation. For example, the formula $\frac{d}{dt}\vec r\times\vec p=\vec r\times\vec F$ holds regardless of what terms you associate with mathematical things.

 

[3102]

Just because the quantity is useful doesn't mean the definition's not arbitrary. By arbitrary we mean that it could have been called anything, not necessarily angular momentum (and obviously not necessarily represented by $\vec{L}$). This quantity, the cross product between the radius vector and the momentum of a particle, turns out to be pretty useful, so we decide to make it an official physical quantity by giving it a name and all that. That part was arbitrary.

But how do you know, that this particular quantity r x p is so useful? Is there any passage in the book that shoes you that this quantity is so much more useful than any other quantity?

 

[3102]

Definitions are always true, by definition. You cannot derive a definition, and it doesn't make sense to ask why it is true, it is tautologically true because it is defined to be true.

The better question about a definition is whether or not it is useful.

okay, why is the particular quantity "r x p" so useful? And for me way more important: How to find out which quantity is important and which not?

 

[Nugatory]

okay, why is the particular quantity "r x p" so useful? And for me way more important: How to find out which quantity is important and which not?

It is so useful because it is conserved. For example, the solution to the problem of planetary orbits depends on conservation of angular momentum.

 

[Fredrik]

But how do you know, that this particular quantity r x p is so useful? Is there any passage in the book that shoes you that this quantity is so much more useful than any other quantity?

I haven't looked inside K & K in many years, but you're the one who talked about using it to calculate the motion of a gyroscope, so you must be aware of some uses for it already. If not, the sections or chapter(s) about angular momentum in the book should make it clear in what ways it's useful.

More useful than any other quantity? Useful for what? You'd have to answer the latter question to get an answer to the former. Since I just proved that $\frac{d}{dt}\vec r\times\vec p=\vec r\times F$ (I didn't post the proof), I can tell you that it's extremely useful for that.

 

[3102]

It is so useful because it is conserved. For example, the solution to the problem of planetary orbits depends on conservation of angular momentum.

Thanks for your answer. Now I have got 2 more questions:
1. How did physicists find out that the quantity "r x p" is the useful quantity? I assume that they did not try all possible combinations of quantities.
2. How did they know that they should use the cross product? I don't think that they were just randomly combining mathematical tools.

A short answer to my two questions would be very nice.

 

[Nugatory]

Thanks for your answer. Now I have got 2 more questions:
1. How did physicists find out that the quantity "r x p" is the useful quantity? I assume that they did not try all possible combinations of quantities.
2. How did they know that they should use the cross product? I don't think that they were just randomly combining mathematical tools.

Newton described the concept of angular momentum using a combination of geometrical reasoning and his three laws. At the time he was looking for an explanation of Kepler's Law of Areas, which was an empirical observation about planetary orbits.

The modern formulation in terms of $\vec{r}\times\vec{p}$ came later, with the development of modern vector calculus.

If you google for "angular momentum history" you'll find a few more detailed descriptions. I don't know of any comprehensive explanation that's available online though.

 

[Dale]

1. How did physicists find out that the quantity "r x p" is the useful quantity? I assume that they did not try all possible combinations of quantities.

I don't know historically how it happened for angular momentum. However, typically the way this works is that they don't have the concept, but the quantity keeps on showing up in calculations. Once it has shown up 3 or 4 times they get tired of writing it so they assign it to a variable to make their formulas easier.

2. How did they know that they should use the cross product? I don't think that they were just randomly combining mathematical tools.

This is just a matter of recognition. If you expand out the cross product in terms of vector components you get a very characteristic form of expression. Whenever you see that form you recognize it as the cross product.

 

[D H]

A deep answer:

• The laws of physics haven't changed since yesterday, last week, or billions of years ago.
  That the laws of physics are invariant with respect to time means that energy is a conserved quantity.
• The laws of physics are the same in London and Moscow, and on Mars, and in the Andromeda Galaxy.
  That the laws of physics are invariant with respect to displacement means that linear momentum is a conserved quantity.
• The laws of physics are the same whether you look south or north, east or west, up or down.
  That the laws of physics are invariant with respect to orientation means that angular momentum is a conserved quantity.

These deep answers are a consequence of Noether's[/PLAIN] theorem. Emmy Noether developed her theorems four hundreds of years after Kepler first proposed his laws of planetary motion. A number of physicists worked on these problems between Kepler and Noether. There's a whole lot of science that was developed between Kepler and Noether.

Here's a dirty little secret of science education: The nice picture you are presented with in physics classes in high school and the undergraduate college level represents hundreds of years of stumbling around by physicists. You aren't taught that "stumbling around" because teaching that has been deemed to be counterproductive. Once you hit graduate school, you'll see exactly how much stumbling around scientists do before arriving at a simple and consistent explanation. This is the case not just in physics but throughout the sciences.

With regard to the cross product, Kepler, Newton, d'Alembert, Euler, Lagrange, Laplace, along with many others who worked on rotational motion did not use the cross product. They didn't even use vectors! They instead "stumbled around." The concept of a vector is a rather recent invention, dating to the final part of the $19^\text{th}$ century -- just a few decades before Emmy Noether developed her theorems. You'll see an amazing amount of "stumbling around" if you read texts or journal articles written prior to the $20^\text{th}$ century.

 

[jtbell]

The modern formulation in terms of $\vec{r}\times\vec{p}$ came later, with the development of modern vector calculus.

This was during the 1890-1910 period, more or less, as I recall reading. The physics of angular momentum (at least at the first-year introductory level) had surely been settled long before that (probably in the 1700s, or at the latest, early 1800s). I don't know what kind of notation people were using before then, but I suspect some of them were using quaternions.

 

[D H]

I don't know what kind of notation people were using before then, but I suspect some of them were using quaternions.

Hamilton developed the quaternions in 1843. What those people who preceded Hamilton used falls into the category of what I not so nicely called "stumbling around".

Not that many people used Hamilton's quaternions at first.They remained a rather esoteric concept for a good twenty or so years after Hamilton first described them. Maxwell took interest and noted that his description of electrodynamics might be a lot easier if expressed in terms of quaternions. Followers of Hamilton, most notably Peter Tait, tried to eschew the value of quaternions. Others, most notably Josiah Gibbs and Oliver Heaviside, absolutely hated Hamilton's quaternions and instead developed the vector calculus that is now used almost universally in the teaching of introductory level physics.

 

[Nugatory]

Here's a dirty little secret of science education: The nice picture you are presented with in physics classes in high school and the undergraduate college level represents hundreds of years of stumbling around by physicists. You aren't taught that "stumbling around" because teaching that has been deemed to be counterproductive.

Interestingly, a lot more of the stumbling around is shown when talking about elementary QM at that level: Bohr atoms, wave-particle duality, uncertainty caused by measurements disturbing the system, photons like little bullets, observers collapsing the wave function, cats neither dead nor alive... And it seems pretty clear to me that it is counterproductive - an amazing number of threads in the QM forum are about getting people to unlearn these concepts.

 

[3102]

Wow, thanks for all your answers.

But, now I've got 2 more questions:
A) Was the cross product because of the long formulas in physics or was it independently developed in mathematics?
B) If the cross product was developed in physics, how did they get to these long formulas for rotational motion?

 

[D H]

The three dimensional dot product and cross product were originally minor side shows in Hamilton's quaternions. Gibbs and Heaviside worked on developing a quaternion-free notation that centralized these minor side shows. The result is the vector analysis that students are now taught, worldwide.

With regard to those "long formulas for rotational motion", what long formulae? Euler's equations are long and there are three of them. The modern version is rather compact in comparison: $\boldsymbol {\tau}_{\text{ext}} = \mathbf I \dot{\boldsymbol \omega} + \boldsymbol \omega \times (\mathbf I \boldsymbol \omega)$.

This happens a lot. Developments in mathematics help make physics more compact, and developments in physics motivates mathematical developments. For example, if you proceed in studying physics you'll inevitably run across "Maxwell's equations," four short equations that succinctly represent all of classical electromagnetism. Maxwell did not write those four equations. He instead wrote 21 scalar equations, some of them rather long. If you proceed in studying physics even further, you'll find that you can express those four short equations in an even shorter form, as one single equation.

 

[3102]

With regard to those "long formulas for rotational motion", what long formulae? Euler's equations are long and there are three of them. The modern version is rather compact in comparison: $\boldsymbol {\tau}_{\text{ext}} = \mathbf I \dot{\boldsymbol \omega} + \boldsymbol \omega \times (\mathbf I \boldsymbol \omega)$.

By this I mean the 3 formulae, one for each coordinate direction x,y,z; the expansion of the cross product.
$\tau_{z}=xF_y-yF_x$

Where do these equations come from?

 

[D H]

They come from the definition of angular momentum and from the strong form of Newton's third law.

Suppose Physicist A writes about some particular obscure concept. Suppose that concept is used very little afterwards, and whenever it is used, it is used in its original form. It's going to stay in its original form.

Suppose on the other hand that Physicist B writes about some concept, Physicist C writes about some apparently different concept, Physicist D writes about yet another apparently different concept. Physicist E comes along and says "Hey! You guys were writing about the exact same thing! My paper presents a generic framework for analyzing this class of problems."

Next, Physicists F, G, H, I, and J, each of whom is unaware of Physicist E's unifying work, write about some other apparently different concepts. Physicist K comes along and says "Hey! All of you guys were writing about the exact same thing! Did you not read Physicist E's paper? My paper presents an even more generic framework for analyzing this class of problems."

Things that pop up over and over and over are given special names, special analyses. Physicists look for common features and create a common nomenclature when this happens. Other physicists look for ways to reduce the mathematics to a canonical form. This is what happened with rotational motion. The initial descriptions were rather verbose and perhaps a bit unclear. Now it is extremely compact.

 

[D H]

But how do you know, that this particular quantity r x p is so useful? Is there any passage in the book that shoes you that this quantity is so much more useful than any other quantity?

It's important because angular momentum stands side-by-side with energy and linear momentum (and also electrical charge, and a very small number of other things) that are conserved quantities. One of the key differences between Newton's mechanics and the Lagrangian / Hamiltonian formulations of Newtonian mechanics was a focus on those conserved quantities. This became even more important in the development of quantum mechanics.