Explanation of Lorentz-Force In high-school level books (for example the german standard text: "Dorn-Bader") I have often seen an explanation of the Lorentz force as on the following picture:

The textbooks consider the superposition of the circular field of the wire and the homogenous field of the magnet (sure the homogenity doesn't matter here). Then the net field as you can see on the picture above on the right is larger on one side of the wire (here on the right) and smalle on the other one. So far so good. 
However why does this explain the occurence and direction of the Lorentz force. Do do so, one would need another principle for example that the wire always wants to go to the weaker field regions or something like this. And this principle should be somehow more evident than the Lorentz-force itself (which you can "see" experimentally).
But how is this needed principle exacly forumlated? Why it is correct? Is there any good reason that it is more evident than just taking the Lorentz-force as an experimental fact?
Would be great if someone could clarify the logic of this, evaluate the soundness of the argument and embedd it conceptually and mathematically in the big picture of electromagnetic theory.
Additionally I want to know if the above cited "explanation" has any common name and if there are university level textbooks which proceed in a similar way. I feel that this argument goes back to Michael Faraday (just by the style of reasoning) - so if someone has a reference to the orgin of the argument I would be interested in it, too.
By the way: The magnetic field in the above cited book ("Dorn Bader") is introduced by the interaction of permament magnets...
 A: This is an interesting question.  I did a bit of research and I think the principle at work here is that there is a pressure associated with an energy density.
From the Wikipedia article "Energy density":

Energy per unit volume has the same physical units as pressure, and in
  many circumstances is a synonym: for example, the energy density of a
  magnetic field may be expressed as (and behaves as) a physical
  pressure

To the right of the wire, the magnetic fields add while to the left, they subtract.  Thus, the magnetic energy density is greater to the right than to the left.
From the article "Magnetic pressure and tension":

The magnetic force (per unit volume) in the equation for fluid motion
  may be re-expressed as 
$$\mathbf J \times \mathbf B = \frac{1}{\mu_0}(\nabla \times \mathbf
 B) \times \mathbf B  = -\nabla \frac{B^2}{2\mu_0} +
 \frac{1}{\mu_0}(\mathbf B \cdot \nabla)\mathbf B$$

$\frac{B^2}{2\mu_0}$ is the magnetic pressure and the term $-\nabla \frac{B^2}{2\mu_0}$ is the magnetic pressure gradient or magnetic pressure force.
The term $\frac{1}{\mu_0}(\mathbf B \cdot \nabla)\mathbf B$ has a component that cancels the magnetic pressure force in the direction parallel to the magnetic field lines so the magnetic pressure force acts perpendicular to the field lines.  The remaining component is the magnetic tension force.
A: While the field is $\vec F = q\vec v \times \vec B$, it is not a terribly intuitive process.  The closest one comes to a vector-product in the real world is the Coralis force, where the wind goes clockwise around a low, and anticlockwise around a high.  
A moving charge sets up a circular magnetic field, which is one direction or the other, depending on the sign of the charge.  When this moves into a magnetic field, the fields add on one side, and subtract on the other side, so that charge is pushed perpendicular both to the field, and to the direction of travel.
In the case of a conductor, like this example, the charge is bound to a rod, and the rod is moved outwards (or inwards) to the magnet, as the current is flowing one way or the other.
Where the charge is not restricted to a mechanical device (like a peice of wire), the charge travels in a circle, and such devices are known as cyclotrons.  
Permanent magnets are used, because these are able to produce a constant $\vec B$.  An electromagnet is necessarily an changing-flux and changing-current thing, and thus can not make a constant flux-field.
The vector-product is not symmetric, ie $\vec A \times \vec B = -\vec B \times \vec A$, and since it is a parity thing in both 2d and 3d, one can use either a left-hand rule or a right hand rule.  But it must be used consistantly, like it's ok for everyone to drive on the left or on the right of the road, but everyone's got to do the same.  So we ram the right-hand rule, for rotating A onto B gives A×B.  It's just a way of keeping tabs on the symmetry-breaking process.
Jeffimenko's "Electricity and Magnetism" gives the formula alone, but vector relations occupy the full first chapter.  Like Heaviside, J supposes a healthy dose of vector arithmetic is needed before electricity is mentioned.
The text i used at uni in the seventies (Grant and Phillips 'Electromagnetism'), shows essentially the diagram in two parts (the poles, and then some pages later the wire), but both indicate three orthogonal vectors.  No name is attached to it, but it pretty much represents the notional way (legal metrology), that one would define this event: bereft of the complexities where the geometry adds additional factors to the product.
A: The deflection of a current-carrying wire, the deflection of the electron and also the Hall effects are all based on the same principle. One can break down both the Lorentz force and the Hall effects to the movement of an electron in the magnetic field as the easiest process to analyze.
It has been pointed out many times in this forum that the external magnetic field does not spend any energy to deflect the charges. However, since the deflection is (equivalent to) an acceleration, energy is required for this. Where does this energy come from?
A static charge in a magnetic field is not deflected. Although, something happens anyway. Every charge is at the same time also a magnetic dipole. This is not mentioned very often, but for example the electron has an unique value for its magnetic moment. The magnetic dipole of the electron and the external magnetic field interact and the electron is aligned with its poles at the external magnetic field.
Since only a moving electron (as for the Lorentz force phenomenon the current in the wire) is deflected in the magnetic field, the thought is obvious that the kinetic energy of the charge is involved in the energy turnover for the deflection.
In this article a connection is made between the radiation of electromagnetic energy and the magnetic dipole moment. (Thanks to anna v for pointing this out here).
If one wants, one can find an explanation for the cause of the Lorentz force. It is the emission of EM radiation which causes the deflection of the charges. For the Lorentz force this is not noticed, but should be detectable by a loss of energy at the wire (temperature increase and increase of the electrical resistance).
However, in the synchrotron or for free-electron laser, this EM radiation is obvious or even desired for the latter device.
A: Suppose that you knew about the Coulomb interaction between charges. Now imagine a wire with a current, and let the wire be at rest. This means that the ions in the wire are at rest, but the conduction electrons have a net velocity. The wire is electrically neutral. Then if there is a charge in motion outside the wire, there should be no force on it, you would say. 
But suppose that you also know about special relativity. Then in the rest frame of the conduction electrons, because of length contraction, the distance between conduction electrons will be greater and that between ions smaller. (The ion-ion distance in the wire rest frame is the proper length, so it is shorter in any other frame.) So in a different frame the wire is charged. In this frame there must be an electric force between the wire and the charge outside! There was no electric force in the lab frame, so in the the lab frame this force would be something else: the magnetic force. 
If you do this quantitatively you can find that the magnetic force is exactly what you expect it to be (I assume you have met with the field from a thin wire.)
This argument also shows that in relativistic theory you should not think about the electric force and the magnetic force, but rather the electromagnetic force, since different observers will disagree on what is electric and what is magnetic. In fact it was this situation that led Einstein to discover special relativity. Einstein's argument is sort of the reverse though: he showed that given that the equations of electromagnetism should be the same for all
observers, we should expect special relativity. The argument above is that given that our universe is described by Einstein's relativity, it is not consistent to have only an electric force. This is historically backwards as mentioned, but I think it's conceptually the better way to think. 
