Comment on “A magnetic velocity Verlet method” [Am. J. Phys. 88, 1075 (2020)]

The implicit midpoint algorithm where $r ′ = r ( t + Δ t ) , v ′ = v ( t + Δ t )$⁠, and $r = r ( t ) , v = v ( t )$⁠, corresponding to Chambliss and Franklin's¹ Eqs. (5) and (6), takes the form of for a charged particle q in a general magnetic field $B ( r ) = B ( r ) B ̂ ( r )$⁠, with cyclotron frequency $ω ( r ) = q B ( r ) / m$⁠. This algorithm is known not to conserve energy. As noted by Boris,² dotting both sides of Eq. (3) with $( v ′ + v )$ gives In a spatially varying magnetic field, there is no reason why the triple product on the RHS of Eq. (4) should vanish, and therefore, the kinetic energy is generally not conserved. However, for a constant magnetic field where $ω ′ = ω$ and $B ̂ ′ = B ̂$⁠, it would vanish, which is why Chambliss and Franklin's Fig. 1 showed zero energy error.

This energy conserving feature can be generalized to the case of a nonconstant magnetic field, since as long as the same magnetic field is used to evaluate the acceleration at r′ and r, the RHS of Eq. (4) will still vanish as Eq. (5). This suggests that one should use the common magnetic field at the midpoint, resulting in the Boris solver²  widely used in plasma physics.³ 

Also, it should be further noted that, in a constant magnetic field, both velocity vectors (3) and (6) are rotated the same way. However, the Boris position (7) will remain on the gyrocircle orbit of radius $r g = v / ω$ regardless of the value of $Δ t$⁠. As a consequence, it also exactly conserves angular momentum, $r ′ × v ′ = r × v$⁠, at all $Δ t$⁠. This is because for motion perpendicular to $B ̂$⁠, since the constancy of r² implies the conservation of angular momentum. The implicit midpoint position (2), on the other hand, will yield positions on a larger circle tangent to the gyrocircle at the initial position. As a consequence, its angular momentum $r ( t ) × v ( t )$ will oscillate sinusoidally above the initial value as a function of time. This is clearly shown in Chambliss and Franklin's Fig. 2. In contrast, Boris's algorithm would have produced zero error to machine precision.