# Hamiltonian does NOT equal energy?

• nonequilibrium
In summary: This is due to the fact that the constraint forces do virtual work and thus contribute to the total energy of the system. This is different from holonomic systems where the constraint forces do not contribute to the total energy.
nonequilibrium
Hello,

Just to be sure: is the following correct?

Imagine a long rod rotating at a constant angular speed (driven by a little motor). Now say there's a small ring on the rod that can move on the rod without friction. The ring is then held onto the rod by an ideal binding force (I don't know the correct term in English, a force that doesn't deliver virtual work). This ring, free to move along the rod, can be described using the Lagrangian formalism.

Now is it true that the binding forces are non-holonomic? And this is the reason why $$H \neq E$$? It seems like it anyway, but I remember reading a thread on here a while ago looking for simple systems for which $$H \neq E$$ while all examples given were very mathematical and far-fetched in a physical sense, so this makes me a bit unsure about what I've written above, hence the double-check.

Oops, important EDIT:

"Now is it true that the binding forces are non-holonomic?"

I meant non-SCLERONOMIC, i.e. rheonomic.

mr. vodka said:
Hello,

Just to be sure: is the following correct?

Now is it true that the binding forces are non-holonomic? And this is the reason why $$H \neq E$$?

Yes,the constraint forces are rheonomic and this is whry $H\neq E=T+V$

## 1. What is the Hamiltonian?

The Hamiltonian is a mathematical function in classical mechanics that describes the total energy of a system, including its potential and kinetic energy.

## 2. How is the Hamiltonian different from energy?

The Hamiltonian is not the same as energy, as it takes into account both potential and kinetic energy, whereas energy typically refers to only one of these components.

## 3. Why is it important to differentiate between the Hamiltonian and energy?

Differentiating between the Hamiltonian and energy allows for a more comprehensive understanding of a system's dynamics and behavior. It also allows for the use of different mathematical tools and techniques to analyze and model the system.

## 4. Can the Hamiltonian ever equal energy?

Yes, in certain systems, such as a particle moving in a conservative force field, the Hamiltonian can equal the total energy of the system.

## 5. How is the Hamiltonian used in quantum mechanics?

In quantum mechanics, the Hamiltonian is used to describe the total energy of a quantum system and how it evolves over time. It is a fundamental component of the Schrödinger equation, which is used to calculate the behavior of quantum systems.

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