Hamiltonian systems, integrability, chaos and MATH

In summary, the speaker's objective is to study Hamiltonian systems, integrable and non-integrable systems, and chaos. They are struggling to understand the concepts, particularly in the book "V.I. Arnold - Mathematical Methods of Classical Mechanics," and are wondering if there are other resources available. They are willing to spend a year or more to fully understand the subject. However, they have found some helpful materials, such as Ott E. Chaos in Dynamical Systems.
  • #1
diegzumillo
173
18
Hi there,

My objective is to study Hamiltonian systems, integrable and non integrable systems, where there will be chaos, etc. I have a general idea of everything.. the destroyed tori, the symplectic structure of hamilton's equations, etc. But nothing is very clear to me! And the most frustrating part of all this is that everything I want to know seems to be described in rich details in this book: V.I. Arnold - Mathematical Methods of Classical Mechanics. But I don't think it's written in any terrestrial language. :confused:

Is there a other way to approach this subject? Don't get me wrong, I love math but to master this book's content I would need a whole year, if not more, probably.

Diego
 
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  • #2
Diego Floor said:
Don't get me wrong, I love math but to master this book's content I would need a whole year, if not more, probably.

Which is a very reasonable amount of time for this subject. If you try to do it any faster than that, you'll just end up being frustrated, and make no progress.
 
  • #3
Agreed. But I just want to learn some basic concepts in this area, as this is not the main objective (I mean, it's only a "local" objective). I've already found some books and articles that are being really helpful (Ott E. Chaos in Dynamical Systems, for example, has a great discussion in it's final chapters)
 

1. What is a Hamiltonian system?

A Hamiltonian system is a mathematical model used to describe the motion of a physical system over time. It is based on the principles of classical mechanics and is defined by a set of equations known as Hamilton's equations. These equations describe the evolution of a system in terms of its position and momentum, and are used to predict the future behavior of the system.

2. What is integrability in the context of Hamiltonian systems?

Integrability in the context of Hamiltonian systems refers to the ability of a system to be solved exactly using analytical methods. This means that the system can be described completely by a set of equations without the need for numerical approximations. Integrable systems are important because they allow for a deeper understanding of the behavior of complex systems and can lead to the discovery of new physical phenomena.

3. How does chaos arise in Hamiltonian systems?

Chaos arises in Hamiltonian systems when small changes in initial conditions lead to drastically different outcomes. This is known as the butterfly effect and is a characteristic of chaotic systems. In Hamiltonian systems, chaos can arise due to the nonlinearity of the equations and the presence of multiple solutions or bifurcations.

4. What is the role of mathematics in studying Hamiltonian systems?

Mathematics plays a crucial role in the study of Hamiltonian systems. The equations that describe these systems are highly complex and require advanced mathematical techniques to solve and analyze. Mathematicians use a variety of mathematical tools, such as differential equations, symplectic geometry, and topology, to better understand the behavior of Hamiltonian systems.

5. How are Hamiltonian systems used in real-world applications?

Hamiltonian systems have a wide range of real-world applications, including in physics, engineering, and economics. They are used to model and predict the behavior of physical systems such as celestial bodies, fluids, and chemical reactions. In engineering, Hamiltonian systems are used to design control systems and optimize processes. They also play a role in economic models and financial forecasting.

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