Lorentz-like forces and the Helmholtz conditions

In summary, it seems that acceleration independent forces that obey the Helmholtz condition are necessary for the Lorentz form, but not sufficient for a force to be derivable from a generalized potential.
  • #1
andresB
626
374
The question is:

Are acceleration independent forces that obey the Helmholtz condition necessary of the Lorentz form?

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According to the "On Feynman’s proof of the Maxwell equations" (Hughes, R. J. (1992) American Journal of Physics 60, 301) the answer seems to be YES.

However section 3.7 of "Foundations of Theoretical Mechanics I: The inverse problem in Newtonian mechanics" seems to be saying that the self adjointness of a given Newton equation (Aka that it obeys the Helmholtz conditions...I think) is not sufficient to guarantee that the force is derivable from a generalized potential.
 
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  • #2
So, it seems that acceleration independent forces that obey the Helmholtz condition are necessary for the Lorentz form, but they do not guarantee that a force is derivable from a generalized potential.
 

1. What are Lorentz-like forces?

Lorentz-like forces refer to any force that follows the same mathematical form as the Lorentz force law, which describes the force exerted on a charged particle by an electric and magnetic field. This includes forces such as the Coulomb force, which is the force between two charged particles, and the magnetic force, which is the force exerted on a moving charged particle in a magnetic field.

2. What are the Helmholtz conditions?

The Helmholtz conditions are a set of mathematical conditions that must be satisfied for a vector field to be considered a conservative field. This means that the work done by the field on a particle moving from one point to another is independent of the path taken. The Helmholtz conditions state that the field must be continuous, irrotational (curl-free), and solenoidal (divergence-free).

3. How are Lorentz-like forces related to the Helmholtz conditions?

The Helmholtz conditions are necessary for a vector field to be considered a conservative field, which means that it can be described by a scalar potential. In the case of Lorentz-like forces, the electric and magnetic fields can be described by scalar potentials, and thus they satisfy the Helmholtz conditions. This allows for the use of the scalar potential to calculate the work done by these forces.

4. What are some real-world applications of Lorentz-like forces and the Helmholtz conditions?

Lorentz-like forces and the Helmholtz conditions have many applications in physics and engineering. They are used to describe the behavior of charged particles in electric and magnetic fields, which is important in fields such as electromagnetism, plasma physics, and particle accelerators. They are also used in the study and design of electric motors, generators, and other electromechanical devices.

5. Are there any exceptions to the Helmholtz conditions?

While the Helmholtz conditions are necessary for a vector field to be considered conservative, there are some cases where they may not be satisfied. For example, in the presence of sources or sinks, the divergence of a vector field may not be zero, violating the solenoidal condition. In these cases, the field is not considered conservative and cannot be described by a scalar potential. Additionally, the Helmholtz conditions do not apply to non-conservative fields, such as friction or air resistance, which do not have a scalar potential.

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