Sources on explicitly time-dependent Hamiltonian formalism

In summary, the conversation was about finding a good source for the Hamiltonian formalism in the explicitly time-dependent case. A book titled "Geometric Formulation of Classical and Quantum Mechanics" was suggested as a possible resource, as well as searching for terms such as non-autonomous mechanics or control theory. The original poster also mentioned wanting a book that covers the topic without extending it, and inquired about Herbert Goldstein's book.
  • #1
TheoryofChaos
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Not sure I am posting this in the right subforum, if this is not the case, please feel free to move it.

Anyway, the title about sums it up - I need to find a good source which offers a thourough treatment of Hamiltonian formalism for the explicitly time-dependent case - could someone possibly suggest some?

Thanks in advance.
 
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  • #2
I haven't personally read this book, but it looks like it could be useful:

https://www.amazon.com/Geometric-Formulation-Classical-Quantum-Mechanics/dp/9814313726/ref=sr_1_3?s=books&ie=UTF8&qid=1339477934&sr=1-3&keywords=non+autonomous+mechanics

I found it surprisingly difficult to locate simple texts covering this topic! It might help to search for the phrases non-autonomous mechanics or non-autonomous dynamical systems as well. There's also a big literature on ODEs under the name control theory which might be relevant.
 
  • #3
Thanks for the post - This book, however, has "fiber bundles over R" as 1.1, not somewhat of a higher reach, then?

Also, theory of optimal control is, I think, supposed to be an extension of Hamiltonian formalism, I think I would be happy with a "non-extended" book for the time being.

Since Herbert Goldstein's book is considered to be a classic one in the field, and I haven't had a look at it, would anyone know if it treats the topics in any detail?

Thanks again.
 
  • #4
Noone, then?
 
  • #5


I would recommend looking into textbooks or articles on classical mechanics or quantum mechanics, as Hamiltonian formalism is often covered in these subjects. Some specific resources that may be helpful include "Classical Mechanics" by John R. Taylor, "Introduction to Quantum Mechanics" by David J. Griffiths, or "Theoretical Physics" by Georg Joos and Ira Freeman. Additionally, online resources such as lecture notes or videos from universities or research institutions may also provide a thorough treatment of explicitly time-dependent Hamiltonian formalism. It may also be helpful to consult with colleagues or experts in the field for recommendations on specific resources.
 

1. What is the Hamiltonian formalism?

The Hamiltonian formalism is a mathematical framework used to study the dynamics of physical systems. It is based on Hamilton's equations, which describe the evolution of a system in terms of its position and momentum.

2. What does it mean for a Hamiltonian to be explicitly time-dependent?

A Hamiltonian is explicitly time-dependent if it explicitly depends on time in its equations. This means that the equations of motion for the system will also have a time dependence, making the system's behavior more complex.

3. How is the explicitly time-dependent Hamiltonian formalism different from the traditional Hamiltonian formalism?

In the traditional Hamiltonian formalism, the Hamiltonian is assumed to be time-independent. This simplifies the equations of motion and allows for more straightforward analysis of the system's dynamics. However, in the explicitly time-dependent formalism, the Hamiltonian is allowed to explicitly depend on time, making the equations of motion more complex.

4. What types of systems are described by the explicitly time-dependent Hamiltonian formalism?

The explicitly time-dependent Hamiltonian formalism is often used to study systems that involve time-varying forces or time-dependent potentials. This includes systems such as electromagnetic fields, rotating reference frames, and systems with time-varying constraints.

5. What are some applications of the explicitly time-dependent Hamiltonian formalism?

The explicitly time-dependent Hamiltonian formalism has applications in various fields, including classical mechanics, quantum mechanics, and statistical mechanics. It is used to study a wide range of physical systems, from simple mechanical systems to more complex systems such as molecular dynamics and cosmology.

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