Is the Hamiltonian always the total energy?

In summary, the Hamiltonian is not always the total energy in classical mechanics. In some cases, such as in optics, a Hamiltonian derived from Fermat's principle is used instead. In other cases, involving non-holonomic constraints, the Hamiltonian may involve velocity-dependent terms and may not represent the total energy of the system. However, the important role of the Hamiltonian is its ability to generate equations of motion for any function of the canonical coordinates.
  • #1
alivedude
58
5
I'm working on some classical mechanics and just got a question stated:

Is the Hamiltonian for this system conserved? Is it the total energy?

In my problem it was indeed the total energy and it was conserved but it got me thinking, isn't the Hamiltonian always the total energy of a system when you are working with classical dynamics? My lecture notes tell me that "this and that is known as the Hamiltonian and it is usually identified with the total energy of the system". Could anyone give me an example when this is not the case?
 
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  • #2
There's a few different situations in which the Hamiltonian isn't the total energy. Some are reasonable, others are gnarly. The simplest examples of Hamiltonians that aren't the total energy arise in optics where it makes more sense to use a Hamiltonian derived from Fermat's principle treating optical path length as the action. The Hamiltonian is obtained by taking the Legendre transform of the Lagrangian (which is just the rate at which the optical path length grows). I'd recommend you check out Wikipedia's page on Hamiltonian optics for more details.

The messier cases of Hamiltonians that aren't total energies arise when you invoke a non-holonomic constraint when you define the momentum and Hamiltonian. The most famous example of this is the classical Hamiltonian for a particle moving in an arbitrary magnetic field. The canonical momentum isn't the kinetic momentum of the particle, instead it's P = mv + qA where A is the magnetic vector potential at that point. Additionally, the potential term involves a velocity-dependent integral, so you end up with a "kinetic energy" that depends on space and a "potential energy" that depends on the path taken and its velocity.

## H = \frac{1}{2m} (m \vec{v} + q \vec{A})^{2} + q \Phi - \int \nabla (q\vec{v} \cdot \vec{A}) \cdot d\vec{r} ##

I may have messed up the last term. Point is that it's velocity-dependent and depends on the A-field, which has gauge freedoms that make this Hamiltonian somewhat non-physical. But, physical or not, Hamiltonians have a very important role in calculations. The point of a Hamiltonian isn't to tell us about energy, the point is that a Hamiltonian is a function you can stick into a Poisson bracket to generate equations of motion for any function of the canonical coordinates. It's a single function that tells you how the whole system moves. So long as it meets that criterion, you can use whatever Hamiltonian you want.
 
  • #3
Thanks for taking the time and explain all this! It answers my questions very well. :)
 

1. Is the Hamiltonian always the total energy?

Not necessarily. The Hamiltonian is a mathematical operator used in quantum mechanics to represent the total energy of a system. However, in classical mechanics, the Hamiltonian may not always be equal to the total energy due to the inclusion of potential energy and the presence of external forces.

2. What is the difference between the Hamiltonian and total energy?

The Hamiltonian is a mathematical operator that represents the total energy of a system in quantum mechanics, while total energy is a physical quantity that represents the sum of kinetic and potential energy in classical mechanics. In quantum mechanics, the Hamiltonian operator also includes the effects of external forces and potential energy, while in classical mechanics, these are separate quantities.

3. Can the Hamiltonian be negative?

Yes, the Hamiltonian can be negative. In quantum mechanics, the Hamiltonian is a mathematical operator that represents the total energy of a system, which can include negative values due to the inclusion of potential energy. In classical mechanics, the Hamiltonian is not used, but the total energy can also be negative depending on the system's configuration.

4. Is the Hamiltonian always conserved?

No, the Hamiltonian is not always conserved. In classical mechanics, the total energy of a system may be conserved, but in quantum mechanics, the Hamiltonian operator may change over time due to the effects of external forces or potential energy. However, in certain systems, such as those with time-independent Hamiltonians, the Hamiltonian may be conserved.

5. How is the Hamiltonian related to time evolution?

The Hamiltonian is related to time evolution through the Schrödinger equation, which describes how the state of a quantum system changes over time. The Hamiltonian operator acts on the wave function of a system to produce the time derivative of the wave function, which represents the system's evolution in time. In classical mechanics, the Hamiltonian is also related to time evolution through the Hamilton's equations of motion, which describe the system's trajectory in phase space over time.

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