In the physics community, classical mechanics is the part of mechanics that is not quantum mechanics. This means it includes special relativity. This is because special relativity is required to make complete sense of Maxwell's equations and round out our understanding of pre-quantum physics. This then gives a theory of E&M and mechanics that fit together and work in all non-quantum situations.
One does not need special relativity to make complete sense of Maxwell's equations at all. This is just modern drivel repeated by many 'educated' people. SR just takes as an axiom what is blatantly obvious within Maxwell's equation (what is _c_ measured with respect to). Furthermore, there are ether theories that literally make all the same predictions of SR and cannot be empirically differentiated between itself and SR; Einstein, in the early 1920s, changed his beliefs to that of the ether being real and that GR completely collapses without the ether. He held this belief until his death. This gets somewhat complicated, but the notion of spacetime is an incredibly stupid one; saying space warps is pure nonsense; the concept of space is a philosophical one, and the entire purpose is that it cannot warp, or wave, or do any such nonsense. What is really going on in GR is that the FIELD EQUATIONS are describing fields. Now get this; if I attach the coordinate system to the FIELD ITSELF, then 'space' becomes that field; or, rather, that field becomes space (more precisely, people naively now think that the field that is undulating is space that is undulating). So all you are doing via saying 'spacetime litearlly warps' is being an ignoramus and replacing the word 'field' for the word 'space' whilst ignoring all the abstractions that make space what it is. It is a pointless conflation of two different concepts into one; why do people do this? Because most scientists are actually midwits, and they socially latch onto certain theories and defend them to the death; this is why Max Planck said that science advances one funeral at a time. This would be like if I attached a coordinate system to a string, with distance travelled along the string as my generalized position coordinate; now, extrinsically, I can wave this string, and the string would be waving in a 'higher dimension;' naive people would then say that 'space itself is waving.' Intrinsically, one may be able to measure such 'waving,' and instead of concluding the string is waving, they would conclude space itself is waving merely because the position coordinate is attached to that string. Furthermore, any curved surface can be embedded into a higher dimensional 'flat' space. That is, a two-dimensional sphere can be embedded into three dimensions. But we can intrinsically measure the curvature of the sphere on the surface itself, thus naive people claim that the 'surface is not curving in higher dimensions; space is just curved.' 'Space' curving not into a higher dimension makes no sense. The field IS curving into a higher dimension. Lastly, we mathematically express all 'curved spaces' via a 'default flat' formulation. You cannot embed a flat space into a lower dimensional curved space. We cannot express a flat space via lower dimensional curved spaces, yet we can express any such curved 'spaces' in a higher dimensional flat space. This tells us that space is not 'curved,' it is not 'warped,' but it is merely you being limited to observing from a lower dimension, and, thus, have confused yourself. To reinforce, just because we can attach position coordinates to a FIELD does not mean that space itself is undulating; the FIELD is undulating.
For anyone interested in solid state physics and can spare $15-25, the text "Kittel, Charles. Introduction to Solid State Physics. 8th ed., John Wiley & Sons, 2004" is priceless although older editions are the affordable ones.
Truly Inspirational Video - thanks. During my background reading I noticed that the Cauchy stress tensor page on Wikipedia seems to have got the i & j indexes muddled up. I have suggested a correction to the Physics WikiProject. I tried to use your video as a reference but wikipedia does not seem to allow TH-cam links. That's ok because I have referenced two sets of university lecture notes that contain the formula that matches yours.
Awesome work as always. One point: U need Newtons 2. law as the first step in your derivation, which is in essence an axiom. Isnt then the whole derivation of Hamiltons principle an axiom itself then?
You've asked a loaded question which needs to be teased apart... 1. Newton's Law is not an axiom. Axioms are statements that are assumed to be true, but cannot be proven to be true or false - e.g. angles on a straight add up to 180°. This statement is really a definition and cannot be proven. However, having accepted it as a true statement, then many other rules/theorems can be inferred from the axiom. Newton's Law, on the other hand, is supported by empirical evidence - i.e. you can test/prove it in a lab. 2. Notwithstanding what I have written in (1.), the fact that the proof of a Law or a Principle might be based on an Axiom does not imply that the Law or Principle itself is an Axiom. For example, based on the Axiom that angles on a straight line add to 180°, I can prove the Theorem that in the case of parallel lines, the alternate interior angles are equal. This Theorem is provably true and so is NOT an Axiom; it would be true even if the initial Axiom had stated that angles on a straight line add up to 200° instead of 180°. Consequently, neither Newton's Law nor Hamilton's Principle are axioms. By definition, Laws and Principles are not axioms because they can be proven to be true, whereas Axioms cannot be proven to be true or false.
@@Freeball99 Thanks for your time and response! I definetly have to look up the definition of axioms, laws and principles. Why I assumed that it is an axiom: - Newtons work/chapter is called "axiomata, sive leges motus" which contains the word axiom - I am german and on the german wikipedia article of Newtons laws the word "axiom" is used 9 times! Not so in the english article. Nevertheless, if a principle can be based on an axiom and thus does not become an axiom itself, that is explanation enough.
Are you sure of what you say about the libration of the moon and Lagrange studying optics? My understanding of "libration of the moon" is that, although the moon keeps one face towards the earth, it does swing slightly back and forth around that position, and that is called lunar libration. And what Lagrange was studying was something different, which was Lagrange points, which are basically attractor points in the earth-moon gravitational field, towards which an orbiting body would tend to return if subject to small displacements. Those are also called libration points. The "libr" in libration is the same as in equilibrium and refers to a tendency to return towards a mean position, so describes both the tendency of the moon to oscillate about its earth-facing orientation and the tendency of a body to oscillate about a Lagrange point. So I would say libration is not the twinkling of the moon and Lagrange was studying gravitation not optics.
From the Wikipedia article on LaGrange... "The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalised equations of motion..."
@@Freeball99 Ok, thank you. I stand corrected, so Lagrange was indeed studying the lunar libration at the time he came up with his idea (assuming Wikipedia is to be trusted) rather than the libration points that he must have studied at another time. Nevertheless, I hope you will agree that your words "the libration of the moon, that's the sort of twinkling and flickering effect on the moon, so he was actually using it to study optics" are not correct.
We do NOT assume small displacements or linear strain at any point in this derivation. Consequently, Hamilton's Principle is suitable for general strain.
@@Freeball99 When I refer to the general form I mean this: e_ij=0.5*(u_i,j+u_j,i+u_k,i*u_k,j) where the , denotes derivative. You seemed to use the linear part only.
Sorry, I missed your response until now...I think there is a slight disconnect here. I didn't enforce linear strain here. If we were going to linearize the strain, this this would be introduced when deriving the expression for the potential energy of the system. This would occur when applying Hamilton's Principle to an actual problem. Is there a time in the video or an equation where this is confusing?
@@Freeball99 No, I understood what you were doing, and it's one of the assumptions that I make when doing solid mechanics, but I wondered if there was a general derivation.
What if (starting from equation 9) we cannot assume that we can represent the external work as the negative of a potential? If generally the external work could be non-conservative? Does that change things considerably?
If the work is non-conservative, then it must remain as external work (ie it cannot be included in the internal potential for the system). I discuss it in this video: th-cam.com/video/LY-JcuYqSEk/w-d-xo.html
How is it sensible to use equation (1) (which you described as “F=ma” on the level of materials) in order to derive Hamilton’s principle and use Hamilton’s principle to ‘prove’ F=ma? It seems circular (I mean no disrespect, I like your videos: I’m a mathematician trying to grow more comfortable with physics)
You are exactly correct! Hamilton's Principle does not prove Newton's 2nd Law. Unfortunately I made the introduction sometime before completing the rest of the video and sometime after publishing it, I realized the folly of my words. Unfortunately TH-cam does not provide a means of editing the video after the fact without losing all the valuable comments that viewers such as yourself had added. That said, I let this go because the purpose of this video was not meant to prove Newton's Law, but rather to derive Hamilton's Principle - which it does correctly. If I had to do it over again (possibly when I remake it in the distant future) I would likely say the following instead... Lagrangian mechanics builds on the knowledge of Newtonian mechanics and Hamilton's principle builds on Lagrangian mechanics. As such, the equations derived from Hamilton's principle are equivalent to those derived from Newton's laws, but the advantage of Hamilton's principle is that it provides a more complete and elegant formulation of mechanics. Hamilton's principle is more general than Newton's laws and can be applied to a wider range of systems - like systems with constraints (statically indeterminate) and those containing non-conservative forces. Thanks for your comment.
I have a question, could one define the laws of mechanics as the hamilton principle and see newton's laws as a consequence of that? Are all the relations of this proof "equivalence" ?
I would state it a little differently... Lagrangian mechanic builds on the knowledge of Newtonian mechanics and Hamilton's principle builds on Lagrangian mechanics. As such, the equations derived from Hamilton's principle are equivalent to those derived from Newton's laws, but the advantage of Hamilton's principle is that it provides a more complete and elegant formulation of mechanics. Hamilton's principle is more general than Newton's laws and can be applied to a wider range of systems - like systems with constraints (statically indeterminate) and those containing non-conservative forces.
@@Freeball99 You can literally derive Newtonian mechanics from either Lagrangian mechanics or Hamiltonian mechanics; it just depends on what axioms you start with. Lagrangian mechanics is not built on Newtonian mechanics at all, except in the historical sense.
f is not equal to ma but we are stuck with it.....it is only a best approximation based on 17th century deductions.....as a result ultimately einstein concluded that nothing can travel faster than light......what really needs to be derived is the relationship between force and energy....they are not the same, of course, and i suspect the difference between them involves a gravity-time quantity that has never been investigated.
This goes back to the original idea behind the calculus of variations...which I explained in a previous video (th-cam.com/video/VCHFCXgYdvY/w-d-xo.html). In that, I talked about path minimization problems and that the variation of the path had to be zero at the boundaries (since any solution to the optimal path problem must begin and end at the same points - boundary conditions). This is really just an extension of that...we are trying to find the optimal path that the system takes from it's state at t1 to its state at t2. Thus the variations of the state at the beginning and end of the interval must be zero.
I am sure there will be some who disagree with this and I might be splitting hairs...but if we want to get very technical about it, I would say the general definition of a 'force' in the scientific sense is an interaction between two objects that can change an object's energy by influencing its motion, structure, or thermal state. By contrast, I would argue that F=ma is not the definition of a force, but rather a mathematical description of the effects that a NET force has on a particle.
I would define a force as a push or a pull on an object. In my view, Newton's 2nd Law is a statement of how force and mass are related to acceleration (or motion). It's a Law of Motion.
@@Freeball99 I think that good definition of basic physical quantities (charge, mass, force etc) can be made only through measurement method. For instance, we can define mass as quantity which proportional to volume if the bodies in experiment made from same material and Speed
Chúc em luôn thành công và được mọi người yêu quý nhiều hơn. Lộc may mắn có gia đình nhà vợ rất tốt , hiền hậu ... chúc em Lộc và đại gia đình nhiều sức khỏe và may mắn ❤️❤️❤️❤️
Could you give us a source where that "index notation" came from? A textbook would be nice. I am not a structural engineer, so I have difficulties of finding one.
Why do you want a source? It is just a notation that someone made up. Unless you meant "Can you provide a resource that explains this notation in detail?" Then that makes sense, but just asking for a source for "index notation" is quite odd.
@@pyropulseIXXI Different disciplines like to use different sets of notations. I'm curious about the context of this notation and how useful it is in it. Also, if I want to dive further into the topic of variational calculus, it would be better if there is a book serving as a reference that uses the same symbols, so that it may have tighter connection to the videos.
You keep it clean, crisp and above all brief .....this is basic......eager to watch what is up next......
You made things very clear. By reading the J N Reddy book, I did not get that much clarity what I got after seeing your video. Thanks a lot sir.
It is really beautiful. The best explanation I've ever seen. Two thumbs up for this video...
Yours is a wonderful channel. Just discovered it, and love your approach to everything. Bravo.
I've just found your channel and it's awesome. Thank you for sharing your knowledge with us.
Have a good day
Thanks for watching!
Cauchy momentum equation (1)
Cauchy stress tensor (2) converts from stress tensor to traction vector
Wonderful! Your channel save my course of optical control, which confused me a lot.😂 Hope u can make more great videos.❤
In the physics community, classical mechanics is the part of mechanics that is not quantum mechanics. This means it includes special relativity. This is because special relativity is required to make complete sense of Maxwell's equations and round out our understanding of pre-quantum physics. This then gives a theory of E&M and mechanics that fit together and work in all non-quantum situations.
One does not need special relativity to make complete sense of Maxwell's equations at all. This is just modern drivel repeated by many 'educated' people. SR just takes as an axiom what is blatantly obvious within Maxwell's equation (what is _c_ measured with respect to).
Furthermore, there are ether theories that literally make all the same predictions of SR and cannot be empirically differentiated between itself and SR; Einstein, in the early 1920s, changed his beliefs to that of the ether being real and that GR completely collapses without the ether. He held this belief until his death.
This gets somewhat complicated, but the notion of spacetime is an incredibly stupid one; saying space warps is pure nonsense; the concept of space is a philosophical one, and the entire purpose is that it cannot warp, or wave, or do any such nonsense. What is really going on in GR is that the FIELD EQUATIONS are describing fields. Now get this; if I attach the coordinate system to the FIELD ITSELF, then 'space' becomes that field; or, rather, that field becomes space (more precisely, people naively now think that the field that is undulating is space that is undulating). So all you are doing via saying 'spacetime litearlly warps' is being an ignoramus and replacing the word 'field' for the word 'space' whilst ignoring all the abstractions that make space what it is. It is a pointless conflation of two different concepts into one; why do people do this? Because most scientists are actually midwits, and they socially latch onto certain theories and defend them to the death; this is why Max Planck said that science advances one funeral at a time.
This would be like if I attached a coordinate system to a string, with distance travelled along the string as my generalized position coordinate; now, extrinsically, I can wave this string, and the string would be waving in a 'higher dimension;' naive people would then say that 'space itself is waving.' Intrinsically, one may be able to measure such 'waving,' and instead of concluding the string is waving, they would conclude space itself is waving merely because the position coordinate is attached to that string.
Furthermore, any curved surface can be embedded into a higher dimensional 'flat' space. That is, a two-dimensional sphere can be embedded into three dimensions. But we can intrinsically measure the curvature of the sphere on the surface itself, thus naive people claim that the 'surface is not curving in higher dimensions; space is just curved.' 'Space' curving not into a higher dimension makes no sense. The field IS curving into a higher dimension.
Lastly, we mathematically express all 'curved spaces' via a 'default flat' formulation. You cannot embed a flat space into a lower dimensional curved space. We cannot express a flat space via lower dimensional curved spaces, yet we can express any such curved 'spaces' in a higher dimensional flat space.
This tells us that space is not 'curved,' it is not 'warped,' but it is merely you being limited to observing from a lower dimension, and, thus, have confused yourself. To reinforce, just because we can attach position coordinates to a FIELD does not mean that space itself is undulating; the FIELD is undulating.
Learning a lot with these videos!! Very nice...
For anyone interested in solid state physics and can spare $15-25, the text "Kittel, Charles. Introduction to Solid State Physics. 8th ed., John Wiley & Sons, 2004" is priceless although older editions are the affordable ones.
Very enjoyable, took me back to Continuum mechanics
Thank you so much for your valuable information and interesting explanations in your videos.😍
I didn't unerstand it, and I never will. I wanted to see the amazing transformation from a complex set of equations to something simple and beautiful.
Go read a textbook and solve problems. I don't think you understand the erroneous nature of your expectations.
Like button has been judiciously smashed.
Truly Inspirational Video - thanks. During my background reading I noticed that the Cauchy stress tensor page on Wikipedia seems to have got the i & j indexes muddled up. I have suggested a correction to the Physics WikiProject. I tried to use your video as a reference but wikipedia does not seem to allow TH-cam links. That's ok because I have referenced two sets of university lecture notes that contain the formula that matches yours.
Great presentation. Thanks for this.
Incredibly useful thanks a lot
Lovely. Thanks sir
I’m tutoring in Maths and Physics and wanted to ask what is this nice annotation interface or application used here?
I'm using an app called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.
Just outstanding ... Thanks
Most welcome!
Awesome work as always. One point: U need Newtons 2. law as the first step in your derivation, which is in essence an axiom. Isnt then the whole derivation of Hamiltons principle an axiom itself then?
You've asked a loaded question which needs to be teased apart...
1. Newton's Law is not an axiom. Axioms are statements that are assumed to be true, but cannot be proven to be true or false - e.g. angles on a straight add up to 180°. This statement is really a definition and cannot be proven. However, having accepted it as a true statement, then many other rules/theorems can be inferred from the axiom. Newton's Law, on the other hand, is supported by empirical evidence - i.e. you can test/prove it in a lab.
2. Notwithstanding what I have written in (1.), the fact that the proof of a Law or a Principle might be based on an Axiom does not imply that the Law or Principle itself is an Axiom. For example, based on the Axiom that angles on a straight line add to 180°, I can prove the Theorem that in the case of parallel lines, the alternate interior angles are equal. This Theorem is provably true and so is NOT an Axiom; it would be true even if the initial Axiom had stated that angles on a straight line add up to 200° instead of 180°.
Consequently, neither Newton's Law nor Hamilton's Principle are axioms. By definition, Laws and Principles are not axioms because they can be proven to be true, whereas Axioms cannot be proven to be true or false.
@@Freeball99 Thanks for your time and response!
I definetly have to look up the definition of axioms, laws and principles.
Why I assumed that it is an axiom:
- Newtons work/chapter is called "axiomata, sive leges motus" which contains the word axiom
- I am german and on the german wikipedia article of Newtons laws the word "axiom" is used 9 times! Not so in the english article.
Nevertheless, if a principle can be based on an axiom and thus does not become an axiom itself, that is explanation enough.
Excellent Excellent Excellent........
Are you sure of what you say about the libration of the moon and Lagrange studying optics? My understanding of "libration of the moon" is that, although the moon keeps one face towards the earth, it does swing slightly back and forth around that position, and that is called lunar libration. And what Lagrange was studying was something different, which was Lagrange points, which are basically attractor points in the earth-moon gravitational field, towards which an orbiting body would tend to return if subject to small displacements. Those are also called libration points. The "libr" in libration is the same as in equilibrium and refers to a tendency to return towards a mean position, so describes both the tendency of the moon to oscillate about its earth-facing orientation and the tendency of a body to oscillate about a Lagrange point. So I would say libration is not the twinkling of the moon and Lagrange was studying gravitation not optics.
From the Wikipedia article on LaGrange... "The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalised equations of motion..."
@@Freeball99 Ok, thank you. I stand corrected, so Lagrange was indeed studying the lunar libration at the time he came up with his idea (assuming Wikipedia is to be trusted) rather than the libration points that he must have studied at another time. Nevertheless, I hope you will agree that your words "the libration of the moon, that's the sort of twinkling and flickering effect on the moon, so he was actually using it to study optics" are not correct.
Excellent!
If you only consider this for a point mass, you can simplify it to 1.5 lines.
Is it applicable to non conservative systems in the same form?
Yes, non-conservative forces can be included in the external work term.
Perfect!!
Very nicely done. Subbed,
Great job
Do you have a video of lagrange equation from Hamilton principle
th-cam.com/video/LY-JcuYqSEk/w-d-xo.html
Will your calculation work for te general strain rather than the linear strain?
We do NOT assume small displacements or linear strain at any point in this derivation. Consequently, Hamilton's Principle is suitable for general strain.
@@Freeball99 When I refer to the general form I mean this: e_ij=0.5*(u_i,j+u_j,i+u_k,i*u_k,j) where the , denotes derivative. You seemed to use the linear part only.
Sorry, I missed your response until now...I think there is a slight disconnect here. I didn't enforce linear strain here. If we were going to linearize the strain, this this would be introduced when deriving the expression for the potential energy of the system. This would occur when applying Hamilton's Principle to an actual problem. Is there a time in the video or an equation where this is confusing?
@@Freeball99 No, I understood what you were doing, and it's one of the assumptions that I make when doing solid mechanics, but I wondered if there was a general derivation.
Anyone who don't understand Hamilton's Principle don't understand Physics.
What if (starting from equation 9) we cannot assume that we can represent the external work as the negative of a potential? If generally the external work could be non-conservative? Does that change things considerably?
If the work is non-conservative, then it must remain as external work (ie it cannot be included in the internal potential for the system). I discuss it in this video: th-cam.com/video/LY-JcuYqSEk/w-d-xo.html
@@Freeball99 thank you very much!
How is it sensible to use equation (1) (which you described as “F=ma” on the level of materials) in order to derive Hamilton’s principle and use Hamilton’s principle to ‘prove’ F=ma? It seems circular (I mean no disrespect, I like your videos: I’m a mathematician trying to grow more comfortable with physics)
You are exactly correct! Hamilton's Principle does not prove Newton's 2nd Law. Unfortunately I made the introduction sometime before completing the rest of the video and sometime after publishing it, I realized the folly of my words. Unfortunately TH-cam does not provide a means of editing the video after the fact without losing all the valuable comments that viewers such as yourself had added. That said, I let this go because the purpose of this video was not meant to prove Newton's Law, but rather to derive Hamilton's Principle - which it does correctly.
If I had to do it over again (possibly when I remake it in the distant future) I would likely say the following instead... Lagrangian mechanics builds on the knowledge of Newtonian mechanics and Hamilton's principle builds on Lagrangian mechanics. As such, the equations derived from Hamilton's principle are equivalent to those derived from Newton's laws, but the advantage of Hamilton's principle is that it provides a more complete and elegant formulation of mechanics. Hamilton's principle is more general than Newton's laws and can be applied to a wider range of systems - like systems with constraints (statically indeterminate) and those containing non-conservative forces.
Thanks for your comment.
@@Freeball99 And thank you for your reply and your transparency!
I have a question, could one define the laws of mechanics as the hamilton principle and see newton's laws as a consequence of that? Are all the relations of this proof "equivalence" ?
I would state it a little differently...
Lagrangian mechanic builds on the knowledge of Newtonian mechanics and Hamilton's principle builds on Lagrangian mechanics. As such, the equations derived from Hamilton's principle are equivalent to those derived from Newton's laws, but the advantage of Hamilton's principle is that it provides a more complete and elegant formulation of mechanics. Hamilton's principle is more general than Newton's laws and can be applied to a wider range of systems - like systems with constraints (statically indeterminate) and those containing non-conservative forces.
@@Freeball99 You can literally derive Newtonian mechanics from either Lagrangian mechanics or Hamiltonian mechanics; it just depends on what axioms you start with. Lagrangian mechanics is not built on Newtonian mechanics at all, except in the historical sense.
Shouldn't u on the right side of the first equation have an index of i?
Yes, it should.
f is not equal to ma but we are stuck with it.....it is only a best approximation based on 17th century deductions.....as a result ultimately einstein concluded that nothing can travel faster than light......what really needs to be derived is the relationship between force and energy....they are not the same, of course, and i suspect the difference between them involves a gravity-time quantity that has never been investigated.
Uol nice video man !!!
i wonder why can the delta u be supposed as 0 for t1 and t2 ?
This goes back to the original idea behind the calculus of variations...which I explained in a previous video (th-cam.com/video/VCHFCXgYdvY/w-d-xo.html). In that, I talked about path minimization problems and that the variation of the path had to be zero at the boundaries (since any solution to the optimal path problem must begin and end at the same points - boundary conditions). This is really just an extension of that...we are trying to find the optimal path that the system takes from it's state at t1 to its state at t2. Thus the variations of the state at the beginning and end of the interval must be zero.
F=ma ...it's the definition of Force, no?
I am sure there will be some who disagree with this and I might be splitting hairs...but if we want to get very technical about it, I would say the general definition of a 'force' in the scientific sense is an interaction between two objects that can change an object's energy by influencing its motion, structure, or thermal state.
By contrast, I would argue that F=ma is not the definition of a force, but rather a mathematical description of the effects that a NET force has on a particle.
@@Freeball99 it's the first mathematical expression of that literal definition.
F=ma
Isn't that simply the definition of F?
I would define a force as a push or a pull on an object. In my view, Newton's 2nd Law is a statement of how force and mass are related to acceleration (or motion). It's a Law of Motion.
@@Freeball99 I think that good definition of basic physical quantities (charge, mass, force etc) can be made only through measurement method. For instance, we can define mass as quantity which proportional to volume if the bodies in experiment made from same material and Speed
Cool. I can't say more than that
Chúc em luôn thành công và được mọi người yêu quý nhiều hơn. Lộc may mắn có gia đình nhà vợ rất tốt , hiền hậu ... chúc em Lộc và đại gia đình nhiều sức khỏe và may mắn ❤️❤️❤️❤️
AAA+++
Wow. This guy is telling me that you can ‘prove’ a physical law without empirical work? Total crap.
Well, he used that dV = - F dr and F = m a
these are indeed good vibrations!! Great video
Could you give us a source where that "index notation" came from? A textbook would be nice. I am not a structural engineer, so I have difficulties of finding one.
Here is a good explanation: www.ees.nmt.edu/outside/courses/GEOP523/Docs/index-notation.pdf
Why do you want a source? It is just a notation that someone made up. Unless you meant "Can you provide a resource that explains this notation in detail?" Then that makes sense, but just asking for a source for "index notation" is quite odd.
@@pyropulseIXXI Different disciplines like to use different sets of notations. I'm curious about the context of this notation and how useful it is in it. Also, if I want to dive further into the topic of variational calculus, it would be better if there is a book serving as a reference that uses the same symbols, so that it may have tighter connection to the videos.