Hamiltonian vs. Energy: Exploring the Relationship in Analytic Mechanics

In summary, the conversation discusses the topic of systems where the Hamiltonian is not equal to the total energy, particularly in the context of analytic mechanics and Lagrangian and Hamiltonian dynamics. The conversation also mentions the possibility of quantum mechanical systems where the Hamiltonian is not the energy. The participants are seeking examples and references for such systems and welcome any input or thoughts on the matter. One known example is mentioned in a book by Goldstein, Safko, and Poole.
  • #1
pmb_phy
2,952
1
I was wondering if anyone knows of systems for which the Hamiltonian is not equall to the total energy? This is an interesting problem in analytic mechanics (e.g. Lagrangian and Hamiltonian dynamics) but is rarely, if ever, mentioned in forums and newsgroups. I'd love to see a large set of examples for which this is true. I'd like to get an intuitive feeling for when the Hamiltonian equals the energy. I'm also very interested in whether there are quantum mechanical systems for which the Hamiltonian is not the energy. All input, references, thoughts and comments are welcome. There is an example of this in Classical Mechanics - Third Ed., by Goldstein, Safko and Poole page 345-346. Thank you.

Best wishes

Pete
 
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  • #2
My understanding is that all nonconservative (dissipative) forces are outside of the Hamiltonian. Friction, for example.
 
  • #3
Andy Resnick said:
My understanding is that all nonconservative (dissipative) forces are outside of the Hamiltonian. Friction, for example.
I neglected to say that I'm interested only in monogentic systems. Such systems have only forces which are the gradients of a potetial energy function. This does not mean that the Hamiltonian quals the energy though, hence the post, i.e. I'm seeking more examples than that in Goldstein's text. Thanks.

Best wishes

Pete
 

1. What is the difference between Hamiltonian and energy in analytic mechanics?

The Hamiltonian and energy are both important concepts in the field of analytic mechanics, but they are not the same thing. The Hamiltonian is a mathematical function that represents the total energy of a system, including kinetic and potential energy. On the other hand, energy refers to the ability of a system to do work, and it can take different forms such as mechanical, thermal, or electrical energy.

2. How are Hamiltonian and energy related in analytic mechanics?

The Hamiltonian and energy are closely related in analytic mechanics. In fact, the Hamiltonian is often referred to as the "energy function" because it represents the total energy of a system. Both Hamiltonian and energy are conserved quantities, meaning they do not change over time in a closed system. In other words, the Hamiltonian represents the total energy of a system at any given time.

3. Can the Hamiltonian and energy be used interchangeably in analytic mechanics?

No, the Hamiltonian and energy cannot be used interchangeably. While they are closely related, they represent different concepts in analytic mechanics. The Hamiltonian is a mathematical function that describes the energy of a system, while energy refers to the ability of a system to do work. Therefore, they cannot be used interchangeably.

4. How is the Hamiltonian used in analytic mechanics?

The Hamiltonian is a fundamental concept in analytic mechanics and is used to describe the behavior of physical systems. It is often used to derive equations of motion and understand the dynamics of a system. In addition, the Hamiltonian can also be used to calculate the total energy of a system and determine its stability.

5. Is the Hamiltonian always conserved in analytic mechanics?

In classical mechanics, the Hamiltonian is always conserved in a closed system where there is no external force acting on the system. However, in quantum mechanics, the Hamiltonian can change over time due to the Heisenberg uncertainty principle. Therefore, the conservation of the Hamiltonian depends on the context in which it is being used.

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