As I understand it, Hamilton kinda got his big break as a mathematician/physicist with the Hamiltonian, but as an alternate formalism, it wasn't well known at the time. Quaternions came afterward - which became the precursor for vector analysis, after Oliver Heaviside more or less invented the notion of a vector, which was essentially a quaternion, but instead of the relationship between i,j,k, he took the two parts of the product of quaternions - the scalar part and _the rest,_ i.e. the vector part - and just defined two operations that would return each part when applied to a vector.
Yay! Thank you! Let's hope I take you to the "upper bound" of knowledge. Hamiltonian systems are really cool, in themselves, and also a tool to understand different points of view of particle and rigid body systems.
@@ProfessorRoss Ill dedicate a significant amount of time studying dynamical systems and classical mechanics. Thank you for dragging me to the other side of knowledge😂😂🤣🤣
Why around 4:05 it was mentioned that Lagranigian is not always T-V? Is it hinting at non-mechanical systems? 2nd question: At around 38:28, it is said that the kinetic energy T can be a function of T(q_i, q_idot). Can you point to a system where T has explicit dependence on q_i. Moreover, at 46:13, we write {\partial L \over \partial q_i} = -{\partial H \over \partial q_i}. If the kinetic energy is a function of q_i in a general case, then {\partial H \over \partial q_i} = - {\partial L \over \partial q_i} + \sum_{i=1}^{n} \dot{q}_i {\partial p_i \over \partial q_i}, where {\partial p_i \over \partial q_i} may not turn out to be zero. So my question essentially is: how is kinetic energy a function of position vector explicitly? Thank you for your time
Lagrangian Mechanics are associated with the Principle of Least Action. Is there an equivalent Principle that is associated with Hamiltonian Mechanics? I feel like there should be and that if I knew what it was, I'd have a better grasp of the "why" of Hamiltonian Mechanics.
Yes, it's the same principle, but using the Hamiltonian (and canonical coordinates q(t) and p(t)) instead of just q(t). The principle of least action takes on special significance for Hamiltonian systems as it is the basis for developing new types of canonical transformations to simplify the dynamics. I have a lecture about this here: czcams.com/video/NFgJixB8tis/video.html
4:10 What are the exceptions? Is there a more general definition of the Lagrangian? I am aware that it defines the action S of the system, but when is that action S not defined by T - V, or at least not computed by T - V?
The example that comes to mind is point vortices, which just have an 'interaction' function that describes their motion. But more generally, for a given set of Lagrange equations, there is not a unique Lagrangian. There is a symmetry such that you can add any function of the form dF(q,t)/dt to the Lagrangian L, where F(q,t) is a scalar function of just the variables and time, and the Lagrangian equations are the same. Sometimes this symmetry is used to simplify, not the Lagrangian, but the Hamiltonian equations of motion.
This course is for advanced undergraduates and (post)graduate students. This course builds on prior knowledge of Lagrangian and rigid body mechanics, which have their own lecture series, czcams.com/play/PLUeHTafWecAUl2DuWWdRU1MckJv7M5LEH.html
Nice lecture. I have been trying to learn the geometry behind these ideas and have been reading the texts from Marsden, Abraham, and Ratiu but find them difficult to relate back to physics practices and notations that I am used to. Are there sources that tackle geometric mechanics but in a way more accessible for someone in physics or engineering?
Basically, I’m quite familiar with the view of mechanics and Lagrangian/Hamiltonian formalism presented in Goldstein’s Classical Mechanics. Now I’m trying to re-learn all that stuff using Abraham’s “Foundations of Mechanics” and Marsden and Ratiu’s “Mechanics and Symmetry”. But these two presentations are too far removed from one another for me to connect the dots well. Is there a good “middle ground” source?
J, Marsden was my PhD advisor and I also found the material difficult (!). I've heard good things about this Geometric Mechanics book series by Darryl D. Holm, which I might use if I teach a course on the subject: www.amazon.com/dp/184816775X
It means perturbation theory, but using special techniques to take advantage of the structure of Hamilton's CANONICAL equations (47:20). Some related techniques are in another lecture czcams.com/video/6OpyflhA2y4/video.html
Hello, can you recommend me some standard or otherwise good books for a good overview of 1) the relationship of calculus of variations and a) the lagrangian and b) the hamiltonians, 2) typical usecases of the lagrangian/hamiltonian formalism, 3) their relation to dynamic control and/or control theory ? For instance, what is the basics of say weather reports or fluid mechanics and there especially turbulent flow? If i am interested in such stuff, what are the basics i should get my hands on? In the video about the legendre transformation you recommended the book by hamell... thank you very much, cheers!
My list of good references is as follows. I'll put your numbers next to the ones I think address that topic. Advanced Dynamics by Greenwood [2] Numerical Hamiltonian Problems by Sanz-Serna & Calvo Analytical Dynamics by Hand & Finch [1,2] A Student’s Guide to Lagrangians and Hamiltonians by Hamill [1,2 Classical Mechanics with Calculus of Variations & Optimal Control: An Intuitive Introduction by Levi [1,2,3] Classical Dynamics: A Contemporary Approach by José & Saletan [1,2] Classical Mechanics, 3rd Edition by Goldstein, Poole, & Safko [1,2] Additional texts that may be useful: Introduction to Mechanics & Symmetry by Marsden & Ratiu [1,2,3] Nonlinear Differential Equations and Dynamical Systems by Verhulst Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins Differential Equations, Dynamical Systems, & Linear Algebra by Hirsch & Smale
@@ProfessorRoss Dear Prof. Ross, thank you very much for that extensive list. Personally, I am an absolute, self-taught beginner. As of yet I have understood that much. The Lagrangian formalism includes the lagrange multiplier for constraints and the delta operator of the calculus of variations lets one find atationary points through some trickery with taylor rows and partial integration. I learned the contemporary dynamic optimization/optimal control works with the hamilton-jacobi-bellman eq. So it's based on the Hamiltonian rather than the Lagrangian. Second it still is based on Functionals in order to find optimal trajectories. One can transform the Lagrangian to the Hamiltonian theough the Legendre transformation. So amongst those books, is there any book that starts more or less on the most modern approach which is Optimal Control and the Hamilton-Jacobi-Bellman formalism and uses that as a reference point to explain modern mechanics. Maybe you know a better reference point. Bc I mean it seems like overhead to work yourself through books on mechanics first with hundreds of pages each such that you eventually grasp what is actually the bigger picture. Many Greets from Germany
@@friedrichwilhelmhufnagel3577 The most modern points of view in mechanics are based on the *differential geometry point-of-view*, which is well covered in Introduction to Mechanics & Symmetry by Marsden & Ratiu as well as Classical Dynamics: A Contemporary Approach by José & Saletan
As I understand it, Hamilton kinda got his big break as a mathematician/physicist with the Hamiltonian, but as an alternate formalism, it wasn't well known at the time. Quaternions came afterward - which became the precursor for vector analysis, after Oliver Heaviside more or less invented the notion of a vector, which was essentially a quaternion, but instead of the relationship between i,j,k, he took the two parts of the product of quaternions - the scalar part and _the rest,_ i.e. the vector part - and just defined two operations that would return each part when applied to a vector.
Your mechanics lectures are the best. Thank you, I prepare for the mechanics and quantum mechanics exam right now. Greetings from Berlin.
You are welcome!
Beautifully explained sir. Even a beginner can understand these concepts.
I have just been watching for 10 mins and my curiosity is on fire🔥🔥🔥🔥🔥🔥🔥🔥 You are such a great teacher😻😻 Thank you🙏
Yay! Thank you! Let's hope I take you to the "upper bound" of knowledge. Hamiltonian systems are really cool, in themselves, and also a tool to understand different points of view of particle and rigid body systems.
@@ProfessorRoss Ill dedicate a significant amount of time studying dynamical systems and classical mechanics. Thank you for dragging me to the other side of knowledge😂😂🤣🤣
Great lecture! Thank you for re-recording the video.
You are welcome. There is more to come
thank you very much for your great lecture on this fascinating subject
Quantum field theory is based in lagrangian formalism. The standard model is a lagrangian
Very good , keep going !
Why around 4:05 it was mentioned that Lagranigian is not always T-V? Is it hinting at non-mechanical systems?
2nd question: At around 38:28, it is said that the kinetic energy T can be a function of T(q_i, q_idot). Can you point to a system where T has explicit dependence on q_i. Moreover, at 46:13, we write {\partial L \over \partial q_i} = -{\partial H \over \partial q_i}. If the kinetic energy is a function of q_i in a general case, then {\partial H \over \partial q_i} = - {\partial L \over \partial q_i} + \sum_{i=1}^{n} \dot{q}_i {\partial p_i \over \partial q_i}, where {\partial p_i \over \partial q_i} may not turn out to be zero. So my question essentially is: how is kinetic energy a function of position vector explicitly?
Thank you for your time
For example, in polar coordinates, the r coordinate will appear in kinetic energy
Lagrangian Mechanics are associated with the Principle of Least Action. Is there an equivalent Principle that is associated with Hamiltonian Mechanics? I feel like there should be and that if I knew what it was, I'd have a better grasp of the "why" of Hamiltonian Mechanics.
Yes, it's the same principle, but using the Hamiltonian (and canonical coordinates q(t) and p(t)) instead of just q(t). The principle of least action takes on special significance for Hamiltonian systems as it is the basis for developing new types of canonical transformations to simplify the dynamics. I have a lecture about this here: czcams.com/video/NFgJixB8tis/video.html
@@ProfessorRoss Thank you!
4:10 What are the exceptions? Is there a more general definition of the Lagrangian? I am aware that it defines the action S of the system, but when is that action S not defined by T - V, or at least not computed by T - V?
The example that comes to mind is point vortices, which just have an 'interaction' function that describes their motion. But more generally, for a given set of Lagrange equations, there is not a unique Lagrangian. There is a symmetry such that you can add any function of the form dF(q,t)/dt to the Lagrangian L, where F(q,t) is a scalar function of just the variables and time, and the Lagrangian equations are the same. Sometimes this symmetry is used to simplify, not the Lagrangian, but the Hamiltonian equations of motion.
That is symmetry theorem is Noether’s theorem, correct?
Thanks for the response!
Doesn't QFT borrow from Lagrangian formalism as well?
this course is for which semester students?
This course is for advanced undergraduates and (post)graduate students. This course builds on prior knowledge of Lagrangian and rigid body mechanics, which have their own lecture series, czcams.com/play/PLUeHTafWecAUl2DuWWdRU1MckJv7M5LEH.html
Nice lecture. I have been trying to learn the geometry behind these ideas and have been reading the texts from Marsden, Abraham, and Ratiu but find them difficult to relate back to physics practices and notations that I am used to. Are there sources that tackle geometric mechanics but in a way more accessible for someone in physics or engineering?
Basically, I’m quite familiar with the view of mechanics and Lagrangian/Hamiltonian formalism presented in Goldstein’s Classical Mechanics. Now I’m trying to re-learn all that stuff using Abraham’s “Foundations of Mechanics” and Marsden and Ratiu’s “Mechanics and Symmetry”. But these two presentations are too far removed from one another for me to connect the dots well. Is there a good “middle ground” source?
J, Marsden was my PhD advisor and I also found the material difficult (!). I've heard good things about this Geometric Mechanics book series by Darryl D. Holm, which I might use if I teach a course on the subject: www.amazon.com/dp/184816775X
What does "canonical" mean in the context of canonical perturbation theory. Thank you Ross
It means perturbation theory, but using special techniques to take advantage of the structure of Hamilton's CANONICAL equations (47:20). Some related techniques are in another lecture czcams.com/video/6OpyflhA2y4/video.html
@@ProfessorRoss thank you Ross!
Hello, can you recommend me some standard or otherwise good books for a good overview of 1) the relationship of calculus of variations and a) the lagrangian and b) the hamiltonians, 2) typical usecases of the lagrangian/hamiltonian formalism, 3) their relation to dynamic control and/or control theory ? For instance, what is the basics of say weather reports or fluid mechanics and there especially turbulent flow? If i am interested in such stuff, what are the basics i should get my hands on? In the video about the legendre transformation you recommended the book by hamell... thank you very much, cheers!
My list of good references is as follows. I'll put your numbers next to the ones I think address that topic.
Advanced Dynamics by Greenwood [2]
Numerical Hamiltonian Problems by Sanz-Serna & Calvo
Analytical Dynamics by Hand & Finch [1,2]
A Student’s Guide to Lagrangians and Hamiltonians by Hamill [1,2
Classical Mechanics with Calculus of Variations & Optimal Control: An Intuitive Introduction by Levi [1,2,3]
Classical Dynamics: A Contemporary Approach by José & Saletan [1,2]
Classical Mechanics, 3rd Edition by Goldstein, Poole, & Safko [1,2]
Additional texts that may be useful:
Introduction to Mechanics & Symmetry by Marsden & Ratiu [1,2,3]
Nonlinear Differential Equations and Dynamical Systems by Verhulst
Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins
Differential Equations, Dynamical Systems, & Linear Algebra by Hirsch & Smale
@@ProfessorRoss Dear Prof. Ross,
thank you very much for that extensive list. Personally, I am an absolute, self-taught beginner. As of yet I have understood that much. The Lagrangian formalism includes the lagrange multiplier for constraints and the delta operator of the calculus of variations lets one find atationary points through some trickery with taylor rows and partial integration. I learned the contemporary dynamic optimization/optimal control works with the hamilton-jacobi-bellman eq. So it's based on the Hamiltonian rather than the Lagrangian. Second it still is based on Functionals in order to find optimal trajectories. One can transform the Lagrangian to the Hamiltonian theough the Legendre transformation. So amongst those books, is there any book that starts more or less on the most modern approach which is Optimal Control and the Hamilton-Jacobi-Bellman formalism and uses that as a reference point to explain modern mechanics. Maybe you know a better reference point. Bc I mean it seems like overhead to work yourself through books on mechanics first with hundreds of pages each such that you eventually grasp what is actually the bigger picture.
Many Greets from Germany
@@friedrichwilhelmhufnagel3577 The most modern points of view in mechanics are based on the *differential geometry point-of-view*, which is well covered in
Introduction to Mechanics & Symmetry by Marsden & Ratiu
as well as
Classical Dynamics: A Contemporary Approach by José & Saletan
@@ProfessorRoss neat! Thanks a lot!!
His first names are William and Rowan.