# Why did angular momentum get the letter L

Note - this question was inspired by this questions on physics.SE.

Many (most) physical quantities are denoted with a single letter - latin or greek. For many, the letter chosen makes sense: $t$ for time, $m$ for mass, etc. Often the letter chosen either relates to the word for the quantity (in the language of the person first describing it, or Latin or Greek) or indeed the name of the discoverer. Occasionally, the letter is borrowed from another field with a similar use. For example, the letter $n$ is often used for counting natural numbers, and thus became associated with quantum numbers.

All of which leaves me to wonder: why did angular momentum get the letter $L$? I saw a few "answers" online that I found very unsatisfactory - that it related to "Left", or the right angle made between the velocity and position vectors... I suspect there is a deeper explanation but it has eluded me. I tried looking at some of the early descriptions by Newton, Hooke and Kepler but didn't find an answer.

• If it stands for anything, I'd bet a week's salary the $L$ is for "lever-arm" or some such. But you should be warned that if you continue asking questions like this for each for physical quantity that has it's own associated letter, you should expect nothing but unsatisfactory answers. For example, the magnetic field $B$ and its associated vector-potential $A$ and the $D$ and $H$ fields were assigned those letters for no other reason than Maxwell and the gang started at the beginning of the alphabet and worked their way forwards through the letters that hadn't been used yet. – David H Dec 26 '14 at 2:37
• @DavidH I realize it may be a fool's errand but angular momentum is quite old and oddly in the middle of the alphabet. – Floris Dec 26 '14 at 2:42
• I suspect that you are probably right. Mainly I just wanted to put the disclaimer out there that uninteresting answers are not uncommon. Also, it occurs to me that Newton didn't use $L$, so that probably sets a lower bound in its origins. I'd be curious what letter Euler used since he's the guy who pretty much hammered out the details of rotational dynamics. – David H Dec 26 '14 at 3:09
• I did read somewhere that it might be L for Leonard (Euler)... – Floris Dec 26 '14 at 3:10

## 1 Answer

I just want to comment that the agreement on letters, by which we write $\frac d{dt}\mathbf L=\mathbf M$ for the law of angular momentum, must have come very late -- after 1964. As evidence, note that it is still written

1. $\frac d{dt}\mathfrak N=\mathfrak M$ by Sommerfeld in Mechanik (1943, p.63);

2. $\frac d{dt}\mathbf M=\mathbf L$ by Sommerfeld in Mechanics (1952, p.72);

3. $\frac d{dt}\mathbf P=\mathbf M$ by Joos in Theoretical Physics (3rd edition, 1958, p.110);

4. $\frac d{dt}\mathbf M=\mathbf K$ by Landau-Lifshitz in Mechanics (1960, p.108);

5. $\frac d{dt}\mathbf H=\mathbf L$ by Truesdell (quoting Joos) in Whence the Law of Moment of Momentum? (1964);

6. $\frac d{dt}\mathbf M=\mathbf L$ by Truesdell in Die Entwicklung des Drallsatzes (1964), which is a translation of 5.

Thus, it's likely going to be impossible to attribute the choice of $\mathbf L$ to anyone in particular.

EDIT: Much more promising seems to be the hypothesis that the choice was first made in quantum mechanics, to denote orbital angular momentum. As evidence, we have the following nice unanimity:

1. $L_x = \frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Weyl, Gruppentheorie und Quantenmechanik (1928, p.167);

2. $L_x = -\frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Eckart, The Application of Group theory... (1930, p.350);

3. $\pmb{\mathsf L}_z = \frac1i\bigl(y\frac\partial{\partial x}-x\frac\partial{\partial y}\bigr)$ in Wigner, Gruppentheorie und ihre Anwendung... (1931, p.219);

4. $L_x = \frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in van der Waerden, Die gruppentheoretische Methode... (1932, p.19);

5. $\pmb l_3 = \frac1i\bigl(x_1\frac\partial{\partial x_2}-x_2\frac\partial{\partial x_1}\bigr)$ in Pauli, Die allgemeinen Prinzipien der Wellenmechanik (1933, p.185);

6. $L_x = \frac h{2\pi i}\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Bauer, Introduction à la théorie des groupes... (1933, p.38).

The point was, of course, that the eigenvalue of $L_x^2+L_y^2+L_z^2$ would relate to the azimuthal quantum number already denoted $l$ — a letter choice which seems attributed by Sommerfeld (1926) to Russell and Saunders, by Russell et al. (1929) to Hund (1927, p.27), and by Hund (1925, p.347) to Heisenberg (1925, p.850)...

• That is quite surprising - thanks for this! – Floris Jan 19 '15 at 7:05
• Who would think the notation adopted in quantum mechanics could influence its classical homologue.. – hjhjhj57 Jan 20 '15 at 10:00
• I am convinced by the edit. Great set of references. – Floris Jan 22 '15 at 22:57
• Hey, this is wonderful: I wasn't aware of van der Waerden's work here. – WetSavannaAnimal May 29 '15 at 8:11