Dear Bolivar,

Well, I see that the statement "There is no such thing as non-Hamiltonian systems" is incorrect. I was not familiar with this use of the term "non-Hamiltonian" (roughly speaking the systems where we do not have enough information to write down the Hamiltonian governing the system.).

Thanks.

There is no such thing as non-Hamiltonian systems - if you can write Newton's equations of motion for a system, you can also write Hamiltonian equations of motion for it. The use of Lagrangian or Hamiltonian functions to apply quantization conditions is because of the convenience. Lagrangian and Hamiltonian formulations lend themselves "naturally" for quantization.

Dear Sayeed,

Read please my paper

http://www.researchgate.net/publication/220000825_Quantization_of_non-Hamiltonian_physical_systems

Article Quantization of non-Hamiltonian physical systems

Dear Bolivar,

Well, I see that the statement "There is no such thing as non-Hamiltonian systems" is incorrect. I was not familiar with this use of the term "non-Hamiltonian" (roughly speaking the systems where we do not have enough information to write down the Hamiltonian governing the system.).

Thanks.

So basically, our dearest Bolivar was not genuinely asking a question but just trying to promote his papers.

Dear Matias,

Be sure that my goal isn't any self-promotion, Ok?

I intend only to discuss the concept of quantization.

Bolivar.

Dear A.O Bolivar,

Don't get me wrong, self-promotion might be a good thing in the right context.

But if as you say, you intend to discuss the concept of quantization, then there is one thing you may do: DISCUSS IT.

Until now, you have posted a question with an answer you already knew, you have waited for someone arguing you were wrong, and to show him right you have posted your paper.

Besides posting your paper, since you are an expert in non-Hamiltonian quantization, you could use Jaswin and Sayeed's seemingly wrong preconceptions to enlighten us with your knowledge.

In all, I am trying to encourage you to expand a little bit because I believe it will be interesting for me and for the readers of these posts in ResearchGate.

All the best!

Dear Matias and Colleagues,

Following Matias' recommendations I would like to suggest the reading of the introduction (the problem of quantization) of my paper http://www.researchgate.net/publication/220000829_The_Wigner_representation_of_classical_mechanics_quantization_and_classical_limit_Physica_A_301_(2001)_219240

I think this reading procedure will shed some light on the issue of the significance of the quantization phenomenon.

Article The Wigner representation of classical mechanics, quantizati...

Well, non-Hamiltonian systems not only exist, but there are also more of them around us than Hamiltonian ones. Roughly speaking, a Hamiltonian system is a special case of a system whose equations of motion have a first integral, i.e. an energy-like scalar function. The reason it is useful is normally because one can derive forces from such a function via gradients. Now, most dynamical systems in nature do not have such a scalar function. More generally, most dynamical system do not have a close form first integral from which the dynamics can be derived.

In this respect, energy s only a convenient concept rather than an actual entity, but this is a different discussion.

In light of this, I do find the question intriguing and I will look up the suggested paper. However, if the paper only refers to quantisation of systems that have a first integral then it is not the fundamental issue this question implies.

Thank you!

Dear Dr. Blumenfeld,

Read please my recent paper

http://www.researchgate.net/publication/220000766_The_dynamical-quantization_approach_to_open_quantum_systems?ev=prf_pub_p2

In this article I quantize Brownian motion, a typical non-Hamiltonian system!

Best regards,

Bolivar.

Article The dynamical-quantization approach to open quantum systems

The concept of a quantum is more fundamental than the mathematical apparatus that came later. Lagrangian and Hamiltonian are legacies of classical mechanics essentially. They are applicable to conservative systems. Nonconservative systems(that is those systems in which there is nonnegligible energy dissipation) need a different approach. It is good to remember that E= h nu and the well known deBroglie equation are more basic to QM than anything else. You can build much on these two. Heisenberg and Dirac and Schrodinger wrote their equations as equivalent ways to formulate problems. In fact to appreciate gut-level physics it is not necessary at all to think of Lagrangian and Hamiltonian to begin with.

Even in canonical quantization of field, we state with Lagrangian density and get Hamiltonian density and then use commutation relations to quantize the field.

I would like to ask a very naive question. I'm not a "differential equations business man" but, as far as i know (as well as my memory from my undergraduate course in mechanics) all mechanical systems that can be described in terms of "generalized coordinates", admit a hamiltonian description in the sense that they all have a symplectic form (that by definition is closed) associated with phase space. So my question its quite simple, why you claim that only a few systems are hamiltonian?

Greeting

Raúl

The answer lies in the word 'conservative'. Meaning energy conservation or momentum conservation. Once the system behaves such that there is dissipation, then one needs to look at the cause for dissipation and include

them in the equation. Newton's equation is an example of conservative system description. Langecin equation is an example of handling nonconservative system formulation. Bolliver's Brownian motion theory is an example. In fact coupled Langevin equations can handle Brownian motion. Most general approach to such systems is Fokker-Planck equation. In fact Kramer asked and answered a most fundamental question: How does a particle come out of a cage? This is the famous 'escape problem'. That is over 50 years ole. In all these cases one uses th probability approach. Another example of non-Hamiltonian approach is the Boltzmann equation. I hope I have convinced people that there is more to physics/chemistry than Hamiltonians and Lagrangians.

Dear colleagues,

The Newton’s motion equation is gained on the basis of the model bodies in the form of the material points (MP). Such idealization of models of real bodies leads us to the Second law of Newton. According to this law, the acceleration of MP is proportional to the potential force which acts on it. The work of this force is equal to their integral along the way. The energy conservation law of MP from here follows. In connection with this law the dynamics of MP is determined by two types of energy: the kinetic energy and potential energy. Along trajectory of MP the sum of these types of energy is constant. The MP motion is reversible. It is follows from the Newton’s second law.

But all bodies in the nature have a structure. They have the internal energy which is caused by relative motion of the body’s elements. Therefore the works of the external forces change not only the body’s motion energy but the internal energy also. However the Newton’s motion equation, which has been constructed on the basis of models of structureless bodies, does not include the terms responsible for the change an internal energy. In practice the change an internal energy are taken into account by addition to the Newton’s motion equation of the empirical force of a friction.

The work of the frictional forces defines the dissipative part of motion energy which goes to the body’s internal energy and dissipated in the environment. The friction coefficient is taken from the experiment. Thus, the rigorous description of the dynamics of bodies in the frame of classical mechanics is absent. It is due to the simplification of the bodies models. We have assumed from here that for description of a motion of real bodies, the MP should be replaced on a structural particle and the motion equation for the structural particles should be obtained.

Hamiltonian formalism is constructed based on the Newton’s motion equation and using the D’Alambert principal when all collective forces between bodies are potentially [Lanczos C., The variation principles of mechanics. 1962, Univer. of Toronto press.]. Therefore, the Hamiltonian formalism is convenient only for non-dissipative systems which are near equilibrium. But these systems are linear and integrable. So, if Hamiltonian then quantization!

V.M. Somsikov, Center of cosmic investigation. Institute of ionosphere. Almaty

Dear Somsikov, Wonderful. Evey sentence you have written motivates. You have- for instance- talked about Path Integrals in one brief sentence. Linearity and integrability are mandatory for analysing the behaviour of Hamiltonian systems.

Let me cite the 'history' of the path integral method from an entry in my monthly poster IMPERTINENT QUESTIONS vol 21 no 6 ,June12

Worlds Apart in Chemistry: A Personal Tribute to J. C. Slater

Richard F. W. Bader

Department of Chemistry, McMaster University, Hamilton, ON L7L 2T1, Canada

J. Phys. Chem. A, 2011, 115 (45), pp 12667–12676

DOI: 10.1021/jp203531x

(From Yahoo mail input 28 May 12)

Unfortunately I could not give you a nice flowchart.

A reading of the book of Dirac’s life entitled The Strangest Man is a most stirring experience, bringing one back to the beginnings of quantum mechanics where every attempt was made “to establish a basis for theoretical quantum mechanics founded exclusively on relationships between quantities which are in principle observable.” The prime movers in this quest were Heisenberg and Dirac. One of Dirac’s most important contributions in the passage from classical to quantum mechanics, a passage that consumed much of his early efforts, was unfortunately published in an obscure Russian journal where it remained largely unread until it was found by Feynman while a graduate student at Princeton. The paper posed the question, “what corresponds in the quantum theory to the Lagrangian method of classical mechanics?”, a method that, as Dirac pointed out, is clearly superior in the simplicity of its structure to that of the classical Hamiltonian approach. Dirac’s partial answer to this question provided the key to solving the problem of introducing the action integral into quantum mechanics that occupied Feynman’s mind, leading to his formulation of the path integral technique. His contribution was followed two years later by Schwinger’s independently derived statement of the quantum action principle, each contribution providing a complete formulation of quantum mechanics stated in terms of single principle. The present paper points out that the successful introduction of the action principle into quantum mechanics made possible by Dirac, enables one to proceed still further by extending Schwinger’s quantum action principle to an open system, to an atom in a molecule. Thus the quantum theory of an atom in a molecule has its roots in the question posed by Dirac in 1933. The paper proposes a return to a greater use of the theorems of quantum mechanics in interpretive chemistry from that begun by Slater in 1933, a staunch advocate of theory following in the footsteps of observation

If you are anyone would like to read the entire poster or wish to be in the mailing list please leave your e-mail ID.

Dear CS Sunandana

1. I think that the main issue in the development of quantum mechanics of open systems, taking into account randomness and non-equilibrium is: how to construct the mechanics of the systems basing on the mechanics of their element. As it may seem strange, the existing formalisms of classical mechanics are applicable only to closed systems in equilibrium. Without solving this issue it is impossible to explore non-equilibrium systems. Boltzmann was a first who collided with these questions. To solve this issue, I suggested using the mechanics of structured particles, which describes the dissipative processes in the framework of Newton's laws for the material points (see VM Somsikov. 1. The restrictions of classical mechanics in the description of dynamics of nonequilibrium systems and the way to get rid of them . New Advances in Physics, Vol. 2, No 2, September 2008, pp. 125-140, 2. About mechanics of the structured particles arXiv: 1103.2839v1 [physics.class-ph] 15 Mar 2011).

2. If we solved the problem of constructing the equations of the dynamics of systems based on the equations of the dynamics of their elements, then you can dispense with the description of statistical methods that are practically unacceptable for the description of non-equilibrium systems.

3. Unfortunately, the works on the relationship between classical mechanics and quantum mechanics, which was launched by E. Schrödinger , today is not very popular, although they are necessary for an understanding of wave-particle duality. (E. Schrodinger. Ann d Phiziks 70, 361, 489. 734, 80 437, SI, 109 in 1920, Naturwissenschaften, 14, 664. In 1926, Ε Schrodinger Vbhan llungon zur Wellenmechimk-J Barth Leipzig 1927).

I would like to read the poster about these problems. My E mail: [email protected]