i started watching your videos when i was still deciding to study physics. one year later, when i was starting my undergrad first year i came back to your channel (and this video) and i thought "wow one day i'm gonna understand what you're doing". i remember reading the word "tensor" and not having idea what it meant. now, 3 years later, i'm taking my second course of classical mechanics and i finally came to this video hoping it'll help me undertstand the subject for my exam. i get the physics you're talking about now. it's such a good feeling!! keep it up guys, we're all gonna make it someday.
Little late but I’m an a level student doing physics next year, if you watch his tensor calculus series you should find a decent explanation in lesson 1 😁
As someone taking some graduate level courses in my last year of undergrad studies, I have really appreciated these types of videos. Keep them up Andrew, I'm loving them!
It's worth mentioning, that on 2:08 the first r_i doesn't necessarily have to be equal to the second r_i. That is because linear velocity desribed by cross product includes radius of rotation, while the first r_i is simply a radius vector, which in the most general case (like when there are several points of a rigid body lying on the rotation axis), again, may be different from the radius (the second r_i)
If we consider the equation where he defines L_i, then it is basically r_i x p_i. This is where he gets the expression you are referring to. But in this definition, it wouldn't make sense to define L_i = r_i x p_j right? So I'm confused as to how it is possible to have m_i r_i x ( w x r_j) in the next step?
@@rikthecuber yeah I’m almost done with the entire book in my class and I thought it’s pretty good in terms of explanation, especially compared to other mechanics books this is definitely the best choice
I don't understand anything he's talking about, I just find his voice and the sound of the marker relaxing. I'm not even a physics student, I just just find his videos interesting.
In the exterior calculus formalism, the angular momentum is actually not a vector, but a bivector, whose components are represented by a tensor of order 2. Similarly, the angular velocity W is a tensor of order 2, such that v = Wr, where v and r are the velocity and position vectors. Under this formalism, the inertia tensor is sometimes defined as a tensor of order 4, with four indices. For instance, if L and W are contraviant, then I is mixed. If W is contraviant and L covariant, then I is contravariant. In each case, it is suggested that W doubly contracts with I, which leaves a tensor of order 2, and this is L. The relationship L = IW is maintained as in your video, but each quantity has doubled in indeces. Can you make a video deriving this? I know it would be a pain, but I would love to see it.
Just a few pedantic notes; Einstein summation notation only applies when the index appears exactly twice. If the index appears three times you should explicitly write the sum. At 1:00 you left off the subscript for p_i making it look like the sum of all momenta At 6:40 you equated a scalar with a vector; you should have put (...)_i around it.
@Andrew Some people from Jlab are going to be presenting preliminary results from their 12 GeV upgrade at this APS regional conference I'll be attending in TN, so I'll def (maybe) attend these talks. (I'm more of a condensed matter guy so we'll see if I can make them...though there is an increasing overlap between CM and high energy physics because of all the quasiparticles we've been finding in CM, so that's pretty dope). Also, sorry this has nothing to do with this video...but twas a great video.
@@AndrewDotsonvideos Haha fair enough! I hope to get some experience in theory next summer if I can...perhaps an REU or another national lab internship. I saw your picture on the Jlab SULI internship program page! I have now worked almost 1.5 years through the SULI and HERE programs at Oak Ridge National Laboratory. I want to try my hand at theoretical work now so I guess SULI is a good way to try.
One question! You made it seem like the inertia tensor is defined for a point particle, and that for a rigid body you would only need to sum for every particle in the body. So is it correct to say that a point particle needs this tensor as well, when rotating about an arbitrary axis? Because I've never seen the tensor being used for a point mass. Maybe because it was always rotating on a plane normal to the axis it was rotating around and a bunch of terms drop to 0, but I'm just asking for completeness!
Ahh, this is neat, so you just write the angular momentum as cross producting the linear momentum with the position vector (analoguous to torque, so seems like a reasonable definition), then you expand the cross product with that vector identity I learned in multi calc, and hten you note that (sum)L_i is a linear function in w so you just take the matrix that the linear transformation corresponds to and call that the moment of inertia tensor, is that accurate? but then, why call it a tensor and not just a matrix?
That all sounds great! There are additional assumptions you can make when you call something a vector/tensor. It means the components transform a specific way when you change your basis (coordinate system). The inertia tensor is an example of something called a cartesian tensor, which means it's components transform a certain way if you rotate your coordinate system. Think of it like starting with a vector pointing in the "x-direction". If you rotate your coordinate system such that the vector no longer points in your new x direction, the components should change in such a way that leaves the length of the vector unchanged. A vector is just a rank-1 tensor, and a square matrix whos components obey that transformation rule is a 2nd rank tensor (two indices). Not just any combination of quantities organized in a square matrix will transform this way, and so not all square matrices are tensors.
My algebra teacher told us a tensor is a map of (V x V x V ...) onto R ( V being a vector of a vector field E). Nevertheless, the inertia tensor prensented here just looks like a matrix that maps a vector of E to another vector of E. Someone please explain, Im confused.
gah im confused. how can we say v = w x r if v can have components parallel to r? (edit - maybe it doesn't matter anyway because you cross v with r after)
Angel - His question is valid; the cross product of two vectors is orthogonal to both vectors, hence the component parallel to r should be zero. Rasp - The answer is breifly stated at 0:45, namely, we are assuming that the object is purely rotating around the origin. We could have taken an arbitrary origin and express the velocity as the velocity of the center of mass plus a rotation around the center of mass, in which case we get an additional term which is the orbital angular momentum of the system as a whole (that is, treating it like a point particle at the CoM).
Ahmed Al-shabi I think Simon Clark is making a science one so I’m just going to wait til after he does. Odds are he’s putting more effort into his video, so I’d hate to post a bunch of memes that maybe he was going to use
It's a weird tensor thing. Basically, if you have a triple cross product, you can simplify it by doing some stuff that turns the triple cross product into a term minus another term.
That's actually one of the way of defining the cross product. Maybe what you ask is a proof of an equivalence between that definition and some other, and I would like to know what other definition you are refering to by the way
I'm late. But for anyone still interested, it's because the product of 2 levi cevita can be written as a 3x3 matrix of cronecker deltas then getting the determinant. If you notice, the first 2 terms are the same for this special case in physics, so you can do a lot of simplifying and end up with the identity (called the contracted epsilon identity). I found it on wiki too if u wanna look.
It would be cool if you used this linear algebra/tensor approach to solve a simple University physics problem. Something like a ball rolling on a plane
It would be a big deal..if I was actually being sexist.That hashtag is filled with what people stereotype "girlythings" to be. I'm not sarcastically suggesting girls don't do physics. I'm directly making fun of the hashtag by generalizing it to be anything. Just like how you see those memes about it where it'll just be a person breathing and say #justgirlythings.
i started watching your videos when i was still deciding to study physics.
one year later, when i was starting my undergrad first year i came back to your channel (and this video) and i thought "wow one day i'm gonna understand what you're doing". i remember reading the word "tensor" and not having idea what it meant.
now, 3 years later, i'm taking my second course of classical mechanics and i finally came to this video hoping it'll help me undertstand the subject for my exam. i get the physics you're talking about now. it's such a good feeling!!
keep it up guys, we're all gonna make it someday.
Right on!
Staring at Goldstein for hours and Andrew comes along and clarifies everything in 10 minutes! Bless up
First year undergrad here, I was just thinking "wow, I'll be able to understand this, isn't this nice" and then some epsilons and deltas came in
Oh darling.... I am so sorry you'll still have to go through year 2 and 3 physics and may all the bloody Force be with you!
Don’t worry. It’s not that bad by the time you get there.
Dude I totally feel you
Little late but I’m an a level student doing physics next year, if you watch his tensor calculus series you should find a decent explanation in lesson 1 😁
SO TRUE
As someone taking some graduate level courses in my last year of undergrad studies, I have really appreciated these types of videos. Keep them up Andrew, I'm loving them!
It's worth mentioning, that on 2:08 the first r_i doesn't necessarily have to be equal to the second r_i. That is because linear velocity desribed by cross product includes radius of rotation, while the first r_i is simply a radius vector, which in the most general case (like when there are several points of a rigid body lying on the rotation axis), again, may be different from the radius (the second r_i)
If we consider the equation where he defines L_i, then it is basically r_i x p_i. This is where he gets the expression you are referring to. But in this definition, it wouldn't make sense to define L_i = r_i x p_j right? So I'm confused as to how it is possible to have m_i r_i x ( w x r_j) in the next step?
Your integral is very revealing of how to derive the inertia tensor directly by identifying from r²ω-r(r•ω) the r²δ and the dyadic/cross product
My classical mechanics textbook skipped the part where you evaluated the double cross products so this definitely helped. Thank you
John R Taylor ?
@@thematrix1101 I am thinking of using that, currently doing chapter 1 haha. Is that as good as I have heard? Just wanted to ask.
@@rikthecuber yeah I’m almost done with the entire book in my class and I thought it’s pretty good in terms of explanation, especially compared to other mechanics books this is definitely the best choice
I don't understand anything he's talking about, I just find his voice and the sound of the marker relaxing. I'm not even a physics student, I just just find his videos interesting.
really good teaching style, thanks!
The best way of learning is by explaining the material. Awesome video, helped a lot!
I am a physics grad student and I appreciate your videos very much! Keep up the good work!
That was super informative and just helped me with what I am working on. Thanks!
I dare to say, you just saved my theoretical mechanics test tomorrow. gj
did he?
In the exterior calculus formalism, the angular momentum is actually not a vector, but a bivector, whose components are represented by a tensor of order 2. Similarly, the angular velocity W is a tensor of order 2, such that v = Wr, where v and r are the velocity and position vectors. Under this formalism, the inertia tensor is sometimes defined as a tensor of order 4, with four indices. For instance, if L and W are contraviant, then I is mixed. If W is contraviant and L covariant, then I is contravariant. In each case, it is suggested that W doubly contracts with I, which leaves a tensor of order 2, and this is L. The relationship L = IW is maintained as in your video, but each quantity has doubled in indeces.
Can you make a video deriving this? I know it would be a pain, but I would love to see it.
Great video.
this was very helpful, thank you!!
i wish i had a teacher like you
Excellent. Thanks.
Very good video. Thanks.
man...as a bsc physics student i have to say that you saved me now
Best video on the topic!
I remember when my dynamics teacher did this, we were so confused
Just a few pedantic notes;
Einstein summation notation only applies when the index appears exactly twice. If the index appears three times you should explicitly write the sum.
At 1:00 you left off the subscript for p_i making it look like the sum of all momenta
At 6:40 you equated a scalar with a vector; you should have put (...)_i around it.
Yeah... I got a D in calculus, this is a no from me chief. still watched the video though
what a nice video to watch
Video is enjoyed
@Andrew Some people from Jlab are going to be presenting preliminary results from their 12 GeV upgrade at this APS regional conference I'll be attending in TN, so I'll def (maybe) attend these talks. (I'm more of a condensed matter guy so we'll see if I can make them...though there is an increasing overlap between CM and high energy physics because of all the quasiparticles we've been finding in CM, so that's pretty dope). Also, sorry this has nothing to do with this video...but twas a great video.
That sounds pretty cool! Personally I can't stomach hearing experimental results talked at me. Too much jargon.
@@AndrewDotsonvideos Haha fair enough! I hope to get some experience in theory next summer if I can...perhaps an REU or another national lab internship. I saw your picture on the Jlab SULI internship program page! I have now worked almost 1.5 years through the SULI and HERE programs at Oak Ridge National Laboratory. I want to try my hand at theoretical work now so I guess SULI is a good way to try.
Thanks for you
Papa Andrew has uploaded for us peasants
omlll I love the title!
One question! You made it seem like the inertia tensor is defined for a point particle, and that for a rigid body you would only need to sum for every particle in the body. So is it correct to say that a point particle needs this tensor as well, when rotating about an arbitrary axis? Because I've never seen the tensor being used for a point mass. Maybe because it was always rotating on a plane normal to the axis it was rotating around and a bunch of terms drop to 0, but I'm just asking for completeness!
Ahh, this is neat, so you just write the angular momentum as cross producting the linear momentum with the position vector (analoguous to torque, so seems like a reasonable definition), then you expand the cross product with that vector identity I learned in multi calc, and hten you note that (sum)L_i is a linear function in w
so you just take the matrix that the linear transformation corresponds to and call that the moment of inertia tensor, is that accurate?
but then, why call it a tensor and not just a matrix?
That all sounds great! There are additional assumptions you can make when you call something a vector/tensor. It means the components transform a specific way when you change your basis (coordinate system). The inertia tensor is an example of something called a cartesian tensor, which means it's components transform a certain way if you rotate your coordinate system.
Think of it like starting with a vector pointing in the "x-direction". If you rotate your coordinate system such that the vector no longer points in your new x direction, the components should change in such a way that leaves the length of the vector unchanged. A vector is just a rank-1 tensor, and a square matrix whos components obey that transformation rule is a 2nd rank tensor (two indices). Not just any combination of quantities organized in a square matrix will transform this way, and so not all square matrices are tensors.
Andrew, does this count as one of the Tensor Calc. series video? If not, how's that going?
My algebra teacher told us a tensor is a map of (V x V x V ...) onto R ( V being a vector of a vector field E). Nevertheless, the inertia tensor prensented here just looks like a matrix that maps a vector of E to another vector of E. Someone please explain, Im confused.
gah im confused. how can we say v = w x r if v can have components parallel to r? (edit - maybe it doesn't matter anyway because you cross v with r after)
Rasp Berry That v can have components parallel to r does not imply v is parallel to r. For example, v = v_parallel + v_perpendicular.
Angel - His question is valid; the cross product of two vectors is orthogonal to both vectors, hence the component parallel to r should be zero.
Rasp - The answer is breifly stated at 0:45, namely, we are assuming that the object is purely rotating around the origin. We could have taken an arbitrary origin and express the velocity as the velocity of the center of mass plus a rotation around the center of mass, in which case we get an additional term which is the orbital angular momentum of the system as a whole (that is, treating it like a point particle at the CoM).
time for another physics meme review
before my meme sinks to the bottom
Ahmed Al-shabi I think Simon Clark is making a science one so I’m just going to wait til after he does. Odds are he’s putting more effort into his video, so I’d hate to post a bunch of memes that maybe he was going to use
@@AndrewDotsonvideos that's really thoughtful of you.
mine was my Halloween costume
Should be tittled moment of inertia Tensor.
Don't you already have a video on the inertia tensor?
Zoltán Kürti I have a video on 2 index tenders, and I derived it in I think maybe a more confusing way in that video
@@AndrewDotsonvideos you are right! Now I remember.
A random question, is this collage or school level in the US?
Definitely college level
I'm confused, I didn't think there was supposed to be a negative sign in the product of inertia terms
It's a weird tensor thing. Basically, if you have a triple cross product, you can simplify it by doing some stuff that turns the triple cross product into a term minus another term.
You often erase stuff a little too much for those of us learning it for the first time
can you prove the levi-cevita symbol expansion identity you are using,
That's actually one of the way of defining the cross product. Maybe what you ask is a proof of an equivalence between that definition and some other, and I would like to know what other definition you are refering to by the way
I'm late. But for anyone still interested, it's because the product of 2 levi cevita can be written as a 3x3 matrix of cronecker deltas then getting the determinant. If you notice, the first 2 terms are the same for this special case in physics, so you can do a lot of simplifying and end up with the identity (called the contracted epsilon identity). I found it on wiki too if u wanna look.
@@TasX send me a link
Just showed up in my freshman mechanics class and I'm so confused
the tensor notation for the cross product gave me cancer. Your video 'cross products using levi civita' was useful though.
LiveFreeOrDie Tensor notation is superior
un grande este weon jjjajaj
toy entrando
It would be cool if you used this linear algebra/tensor approach to solve a simple University physics problem. Something like a ball rolling on a plane
Can you redo this video for beginner who just got a grasp of what angular momentum is😂
🤯
First one🎈🎈
13:53 How did you know?? Ha ha
Why does it have that hashtag..... Honestly I am kinda dissapointed :(
Indian fans hit like.
Are you sure deriving Inertia tensor is just girly things LMAO>
Quahntasy - Animating Universe He’s clearly being sarcastic, I mean, as if girls ever do any physics. 🙃
Science with Katie wow I didn’t expect him to be that sexist
Rose L I just thought it was a random funny hashtag...
It would be a big deal..if I was actually being sexist.That hashtag is filled with what people stereotype "girlythings" to be. I'm not sarcastically suggesting girls don't do physics. I'm directly making fun of the hashtag by generalizing it to be anything. Just like how you see those memes about it where it'll just be a person breathing and say #justgirlythings.
Andrew Dotson oh ok, glad that got cleared up! Good video!
Last
what the fuck
Ew you're using index notation. Excuse whilst I throw up in my mouth