# Is the magnetic Lorentz force $\vec{F} = q(\vec{v}\times\vec{B})$ a force of constraint?

I am currently studying the Lagrangian mechanics, and as far as I've understood, forces of constraint are the forces that are perpendicular to the surface of the movement of the object, thus do not cause any change in the velocity of, and constrain the trajectory of the object, (e.g. force due to the tension, the normal force). The magnetic Lorentz force fulfills all of those I mentioned above, so is it also a type of force of constraint?

No, it isn't. When you talk about a constraint force, you're talking about a force which constrains the motion of a particle to a particular spatial region, such as a curve or a surface. The Lorentz force does not do this; objects under the influence of the Lorentz force can explore all of space.

The confusion may lie in the fact that a constraint force is necessarily perpendicular to the object's velocity, but this is a consequence of its definition, not the definition itself. The fact that the Lorentz force is also perpendicular to the object's velocity does not make it a constraint force.

The Lorentz force is $$$$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})\tag{1}$$$$ I assume you're talking about the magnetic force $$$$\vec{F}_B = q(\vec{v}\times\vec{B}) \tag{2}$$$$ since electric fields can do work on particles.

The problem with using the magnetic force alone is that it's not Lorentz covariant. In other words, you might be able to use (2) to predict the motion of a charge in one inertial frame, but if you try to apply it to a different inertial frame moving relative to the first one, you'll predict the wrong motion for the charge. So if we want a consistent dynamical law for the motion of charges in (electro)magnetic fields, we have to use (1), and (1) can do work.

Furthermore, I would ask: if the magnetic force is a constraint force, what's the constraint?

• I agree everything you said here is factual, but what does the fact that (2) is not Lorentz covariant have anything to do with whether it's a constraint force or not? You didn't quite explain this imo. Commented Jun 27, 2023 at 19:32
• @MaximalIdeal OP requires of a constraint force that it doesn't change the speed of particles (the question says velocity, but I'm sure the OP meant speed). But (1) can change the speed of particles, so it doesn't satisfy this particular requirement (regardless of whether it may or may not satisfy others).
– d_b
Commented Jun 27, 2023 at 19:36
• Strictly speaking, $\vec{F}=q\vec{v}\times\vec{B}$ is the "Lorentz force." Lorentz added this magnetic force to the already known electric force to make the "Lorentz force law," so the force law includes both the electrical and Lorentz forces.
– Buzz
Commented Jun 28, 2023 at 3:14