Hamiltonian Systems and Liouville Integrability

In summary, a mathematics junior is struggling with a research project on Hamiltonian systems and Liouville integrability using a graduate level book by Vilasi. The book is poorly written and assumes prior knowledge of Hamiltonian dynamics. The student is looking for a simpler book on Liouville integrability, accessible to a junior math student with limited physics knowledge. They are currently using Goldstein as a reference.
  • #1
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Hi

I am a mathematics junior and I am doing a research project on hamiltonian systems and liouville integrability (don't ask why...). I am using the book by Vilasi, a graduate level book, but I am finding it quite difficult and badly written; for instance he uses functional analysis and differential geometry concepts without defining things, explaining or proving certain things. He also expects you o already know Hamiltonian dynamics. So I was wondering if there is a relatively simple book that would cover Liouville integrability (NOT Liouville's theorem as given "[URL [Broken] for phase space)? By relatively simple, I mean ofcourse accessible to me, a junior math student (all the physics I once knew I have mostly forgotten, but I am using Goldstein to help me along the way).
 
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  • #2
I guess this means no one can help me?
 
  • #3



Hello there,

I completely understand your frustration with the book you are using for your research project. It can be challenging to find a good resource that explains complex mathematical concepts in a clear and accessible way. However, I believe there are a few options you can explore to help you better understand Hamiltonian systems and Liouville integrability.

First, I would recommend checking out some online resources such as lecture notes or video lectures on these topics. There are many universities that offer open access to their course materials, and you may be able to find some helpful resources there. Additionally, there are also online forums and discussion groups where you can ask questions and get help from other students or experts in the field.

Secondly, you can also try looking for other books on the subject that may be more suitable for your level of understanding. A few recommendations that I have are "Hamiltonian Dynamics" by John Hubbard and Beverly West, "Classical Mechanics: Point Particles and Relativity" by Walter Greiner, and "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem" by Kenneth R. Meyer and Glen R. Hall. These books are written at a more introductory level and may provide a better foundation for understanding Hamiltonian systems and Liouville integrability.

Lastly, don't be afraid to reach out to your professor or a math tutor for additional help and guidance. They may be able to offer some additional resources or explanations that can help clarify any confusion you may have.

I hope these suggestions are helpful and wish you the best of luck with your research project!
 

1. What is a Hamiltonian system?

A Hamiltonian system is a mathematical model used to describe the motion of a physical system, such as a particle or a collection of particles. It is based on the concept of energy, known as the Hamiltonian, and the equations of motion are derived from this energy function.

2. What is Liouville integrability?

Liouville integrability is a property of Hamiltonian systems in which the equations of motion can be solved exactly, meaning that the position and momentum of the system can be determined at any point in time. This is possible because the system has a sufficient number of conserved quantities, known as integrals of motion, which allow for the system's behavior to be predicted.

3. What is the significance of Liouville integrability in Hamiltonian systems?

Liouville integrability is significant because it allows for the exact solution of Hamiltonian systems, making them more predictable and easier to analyze. It also provides insight into the underlying symmetry and structure of the system, which can be useful in various fields such as mechanics, physics, and engineering.

4. How is Liouville integrability determined in a Hamiltonian system?

Liouville integrability is determined by analyzing the Hamiltonian function and identifying a set of integrals of motion, which are functions of the position and momentum variables that are conserved throughout the system's motion. If the number of integrals of motion is equal to the number of degrees of freedom in the system, then the system is Liouville integrable.

5. What are some real-world applications of Hamiltonian systems and Liouville integrability?

Hamiltonian systems and Liouville integrability have numerous applications in physics, mechanics, and engineering. Some examples include celestial mechanics, quantum mechanics, fluid dynamics, and control theory. They are also used in the design of spacecraft trajectories, particle accelerators, and other complex systems.

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