An intrinsic definition of hamiltonian system

In summary: Your Name]In summary, there are various ways to define a Hamiltonian system, each with its own unique properties and advantages. These include the use of canonical transformations and symplectic geometry, which can provide a deeper understanding of the system. The best approach may depend on the specific problem and desired properties.
  • #1
BarbaraDav
15
0
(Sorry for my poor English, Please, forgive mistakes, if any.)

Dear Friends

A system of second order, in normal form, differential equations can be rewritten as a similar first order one, in infinite way (usually that's done introducing simply auxiliaries variables, but it can also be accomplished by means of auxiliaries functions).

Now, each analytical mechanics exposure gives a "recipes" (the Legendre transform) to turn a lagrangian system into the associated hamiltonian; rather, I would know if a property exists such that, among the infinite first order systems equivalent to the given lagrangian one, only the associated hamiltonian enjoys it.

This could allow to express an intrinsic definition of the associated hamiltonian system, instead of merely declaring the way to get it.

Warmest regards.

Barabara Da Vinci
(Italy)
 
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  • #2


Dear Barabara Da Vinci,

Thank you for your interesting question. I understand your concern about finding an intrinsic definition of the associated Hamiltonian system instead of simply following a recipe to obtain it. While the Legendre transform is a useful tool in converting a Lagrangian system into a Hamiltonian one, it is not the only way to do so. In fact, there are many different ways to define a Hamiltonian system, and each one may have its own unique properties and advantages.

One possible approach is through the use of canonical transformations. These transformations preserve the Hamiltonian structure of a system and can be used to generate new Hamiltonian systems with different properties. In this way, one could potentially define a Hamiltonian system that has a specific desired property.

Another approach is through the use of symplectic geometry. This mathematical framework allows for the study of Hamiltonian systems in a more general and abstract way, and can provide a deeper understanding of their properties and relationships.

Ultimately, the choice of which definition or approach to use may depend on the specific problem at hand and the properties that are most relevant for its solution. I hope this helps to answer your question and provides some insight into the different ways of defining a Hamiltonian system.


 
  • #3


Dear Barbara Da Vinci,

Thank you for your inquiry about an intrinsic definition of a Hamiltonian system. I understand your question about whether there is a specific property that only the associated Hamiltonian system has among the infinite first order systems equivalent to a given Lagrangian system.

To answer your question, the Hamiltonian system is defined as a system of equations that describe the dynamics of a physical system in terms of its position and momentum variables. These equations are derived from the Hamiltonian function, which is obtained through the Legendre transform from the Lagrangian function. Therefore, the Hamiltonian system is uniquely defined by the Hamiltonian function, and it is the only first order system that has this specific function.

In other words, the Hamiltonian system is the only one that can be obtained from the Lagrangian system through the Legendre transform. This is a fundamental property of the Hamiltonian system that sets it apart from other first order systems.

I hope this clarifies your question. If you have any further inquiries, please do not hesitate to reach out.

Best regards,

Scientist
 

1. What is the Hamiltonian system?

The Hamiltonian system is a mathematical framework used to describe the dynamics of a physical system. It is based on the concept of a Hamiltonian function, which represents the total energy of the system, and a set of equations known as Hamilton's equations, which govern the evolution of the system over time.

2. What does it mean for a system to be "intrinsic" in the context of Hamiltonian systems?

Intrinsic in the context of Hamiltonian systems refers to the fact that the system's behavior is determined solely by its internal properties and not influenced by external factors. In other words, the system is self-contained and its dynamics are determined by its own inherent characteristics.

3. How is the Hamiltonian system different from other mathematical frameworks used in physics?

The Hamiltonian system differs from other mathematical frameworks, such as the Lagrangian system, in the way it describes the dynamics of a physical system. While the Lagrangian system focuses on the system's motion and forces, the Hamiltonian system emphasizes the system's total energy and its evolution over time.

4. What are some examples of physical systems that can be described using the Hamiltonian framework?

The Hamiltonian system can be applied to a wide range of physical systems, including classical mechanics, thermodynamics, electromagnetism, and quantum mechanics. Examples of specific systems include a pendulum, a simple harmonic oscillator, a gas in a container, or a particle moving in a magnetic field.

5. How is the Hamiltonian system used in practical applications?

The Hamiltonian system has many practical applications, particularly in the fields of physics and engineering. It is used to model and analyze various physical systems, such as celestial mechanics, control systems, and quantum computers. Additionally, the Hamiltonian system is the basis for many numerical methods used in simulations and scientific computing.

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