Christophe Delaere @ChristopheDelaere@qoto.org

Every solid object, such as the rods in the image below, has an inertia tensor. We can write the inertia tensor for the black rod, which keeps changing its orientation, as a sum of various multiples of the inertia tensors for the 6 fixed rods. The plot at the bottom shows how these 6 components change as the black rod points in different directions.

But why do we need 6 components? In 3D space, shouldn’t 3 be enough?

To see why we really need 6, let’s look at the inertia tensor. This is a matrix, I, that we use to multiply the angular velocity vector of an object, ω, to get the angular momentum, L.

The velocity of a point mass whose angular velocity is ω is given by:

v = ω × r

where r=(x,y,z) gives the coordinates of the point mass.

The linear momentum is:

p = m v = m ω × r

and the angular momentum is:

L = r × p = m r × (ω × r)

To get the same thing using matrices, note that we can write the cross product of r with any vector like ω as:

r × ω = C(r) ω

where C(r) is the matrix:

0 –z y
z 0 –x
–y x 0

This lets us write:

L = m r × (ω × r)
= -m r × (r × ω)
= -m C(r)^2 ω

Minus the square of the matrix C(r) is:

y^2+z^2 –xy –xz
–xy x^2+z^2 –yz
–xz –yz x^2+y^2

So the inertia tensor for a point mass is:

I = -m C(r)^2

This is the matrix above, multiplied by m.

In general, we add up (or integrate) over all the point masses in the body, to get the inertia tensor for the whole thing.

The inertia tensor we obtain this way will always be a symmetric matrix, so it can be specified with 6 numbers: 3 on the diagonal, and 3 above. So the inertia tensors belong to a 6-dimensional space.

Next time I have to advertise my "analytical mechanics" module to students I will just show them this comic and ask them whether they are interested in learning deep magic 🙃
https://xkcd.com/2904/

The Cylindrical Onion hits the big 10! 🎂
From CMS work experiences to engaging #outreach activities, our blogs have captured it all!
✍️Stay tuned as we revisit some of our favourite #blog posts from the past decade. Find some of them here:

- How did the Virtual Visit project start at CMS? 👉 https://cylindricalonion.web.cern.ch/blogs/how-did-virtual-visit-project-start-cms

- Friends, BBQs and the Universe 👉 https://cylindricalonion.web.cern.ch/blogs/friends-bbqs-and-universe

- Does CMS exist in the Disney Universe? 👉 https://cylindricalonion.web.cern.ch/blogs/does-cms-exist-disney-universe

#cern #ScienceMastodon #STEM #academia

Who did it better?

Supersymmetric Mario theory
#XKCD
https://xkcd.com/2873/

CMS is working at full intensity! ✨

Today the LHC reached another milestone: the beams passing through CMS now have 2400 bunches of protons! 👏

This is the maximum target for this year's #LHCRun3 and marks the start of the "cruising" mode in the production of data CMS scientists need to work on…

Why in the universe do people ask me to copy-paste a scan of my signature as a "proof" it was really me to write this or that e-document? It is no proof of anything (anybody can cut my signature from an older document and use it).
So, either trust me I am me, or ask for some true proof, but don't play pretend a scanned signature means anything.

What if proposed new particles are more massive than what the LHC can produce? 🤔

A quirk of quantum field theory may allow us to still detect their telltale signatures. CMS is on the hunt using a framework known as Effective Field Theory (EFT).
Find more: https://cms.cern/news/using-top-quarks-probe-natures-secrets

#moriond

Prof. Philip Kuznietsov hosted International Particle Physics Masterclasses at V.N. Karazin Kharkiv National University one year after Russian missiles prevented them. Students are shown here talking to undergraduates Fedir Boreiko and Mariia Tiulchenko and me in the @ATLASexperiment cavern.

#CMSPaper 1210: Making four #topquarks is crucial to test the #Higgsboson coupling to top quarks and the strong coupling (both also sensitive to undiscovered particles, predicted or otherwise).

This new paper gives 3.9 sigma (=chance of random fluctuation of 0.005 percent!) evidence for four-top production, in challenging collisions where most/all of the four top quarks decay to jets. There was a lot of (new!) #machinelearning methods required to be this sensitive, btw http://cms-results.web.cern.ch/cms-results/public-results/publications/TOP-21-005/index.html

Congrats NA62! Dark photon search in muon pairs on the arXiv this morning.
This is such a nice experiment! No new physics, but new parameter space probed at low mass and very low coupling.
https://arxiv.org/abs/2303.08666