On the action for a charged particle moving in a magnetic field

Manoj K.Harbola

REVIEW 2 major objections 3 minor 21 references

On the action for a charged particle moving in a magnetic field

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The action for a charged particle in a magnetic field must include the enclosed magnetic flux, and that flux term restores consistency with least action and the correct energy levels.

desk verdict A clear pedagogical re-derivation of the standard minimal-coupling action, but the universal statement of Eq. (24) gets the sign wrong for negative charges. read the letter →

arxiv 2506.16181 v1 pith:RFWBF7TV submitted 2025-06-19 physics.class-ph physics.chem-ph

Manoj K.Harbola This is my paper · ORCID

classification physics.class-phphysics.chem-ph

keywords MaupertuisprincipleleastactionchargedparticleinmagneticfieldfluxcanonicalmomentumvectorpotentialBohr-Wilson-SommerfeldquantizationLorentzforce

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the Maupertuis action for a charged particle in a uniform magnetic field is not the bare kinetic integral $m\int \vec v\cdot d\vec l$, since that integral is not stationary against nearby same-energy trajectories. Consistency with the circular orbit forced by the Lorentz force requires subtracting $q\Phi$, the product of charge and magnetic flux through the enclosed area, giving $S=m\int \vec v\cdot d\vec l-q\Phi$. With this action, the energy-action relation $E=\omega S/2\pi$ holds, the Bohr-Wilson-Sommerfeld condition gives $E_n=n\hbar\omega$, and rewriting the flux term via Stokes' theorem recovers the conventional canonical momentum $\vec p=m\vec v+q\vec A$ and Lagrangian. The pedagogical payoff the author seeks is a direct trajectory-based route to these results that does not require introducing the vector potential first.

What carries the argument

The carrying object is the trajectory-enclosed magnetic flux $\Phi=\int \vec B\cdot d\vec S$, whose first-order change under a virtual deformation matches the first-order change of the kinetic action. The mechanism is cancellation: subtracting $q\Phi$ from $m\int \vec v\cdot d\vec l$ makes the total first-order variation vanish for all nearby equal-energy paths. Stokes' theorem then converts the flux term into $\oint q\vec A\cdot d\vec l$, which is what identifies $\vec p=m\vec v+q\vec A$ and the standard Lagrangian.

What would settle it

Compute the first-order variation of $S=m\int \vec v\cdot d\vec l-q\Phi$ for a particle in a magnetic field with a small spatial gradient, using a virtual trajectory that is not a small radial deformation of a circle; if the variation is nonzero for some deformation, the flux-only action is not the general action. A second check: apply the Bohr-Wilson-Sommerfeld condition to the same physical orbit using two different gauges for $\vec A$; the action $\oint(m\vec v+q\vec A)\cdot d\vec l$ must be gauge-independent if Eq. (28) is the correct canonical action.

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Extended reading notes

Core claim

The central claim is that the action for a charged particle in a magnetic field must be defined as $S=m\int \vec v\cdot d\vec l-q\Phi$, where $\Phi$ is the magnetic flux through the area enclosed by the trajectory. The paper shows that the bare kinetic action has a first-order change $\Delta S_1=m v_0 a\int_0^{2\pi} f(\phi)d\phi$ between the circular orbit and a nearby equal-energy virtual orbit, and that this change equals $q\Delta\Phi$, the charge times the change in enclosed flux. Adding $-q\Phi$ therefore makes the total first-order variation vanish, restoring the principle of least action. The same action, written as $S=\int(m\vec v+q\vec A)\cdot d\vec l$ and extended to include energy variations, is shown in Appendix A to yield the Lorentz-force equation for general electromagnetic fields, so the uniform-field example is used as a lever to obtain the general canonical momentum $\vec p=m\vec v+q\vec A$ and the standard Lagrangian.

Load-bearing premise

The derivation infers the general action from first-order cancellation around circular orbits in a uniform field; the load-bearing premise is that the only needed correction is exactly $q\Phi$ and that this flux-term form survives for nonuniform fields and general virtual trajectories.

Editorial extensions

If this is right

For uniform magnetic fields the action per period becomes $\pi m v_0 R$, so $E=\omega S/2\pi$ and the Bohr-Wilson-Sommerfeld condition yields $E_n=n\hbar\omega$, matching the correspondence-principle frequency $\omega$.
Bohr-Wilson-Sommerfeld quantization of a charge in a magnetic field can be carried out directly from the trajectory, without first constructing a Lagrangian or choosing a vector potential.
Rewriting the flux term as a line integral of $\vec A$ gives the canonical momentum $\vec p=m\vec v+q\vec A$ and, through $L=\vec v\cdot\vec p-E$, the familiar Lagrangian of Eq. (2).
Appendix A extends the action to general electromagnetic fields: requiring its variation to vanish is equivalent to the Lorentz-force equation of motion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The nonlocality of the flux term hints that velocity-dependent forces generally add trajectory-enclosed geometric terms to the Maupertuis action, a pattern familiar from quantum geometric phases; this connection is left implicit in the paper.
One could test whether a similar enclosed-flux correction appears for other velocity-dependent forces, such as the Coriolis force, by repeating the first-order stationarity calculation for circular orbits.
The derivation fixes the correction by first-order stationarity on circular orbits, so a natural extension is to check whether the flux-only form remains sufficient for nonuniform fields and for virtual deformations that are not small radial variations of a circle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a short, readable paper that derives the standard action S = ∫(mv + qA)·dl for a charged particle in a magnetic field from Maupertuis' principle rather than from the usual Lagrangian construction. The derivation for a positive charge in a uniform field is correct: the first-order change of the kinetic action for nearby virtual paths is qΔΦ, and adding -qΦ cancels it. That is a legitimate, if reverse-engineered, way to motivate the flux term. The extension to general fields in Appendix A is standard and fine, and the identification p = mv + qA and the Lagrangian come out correctly.

The paper is honest about being a construction from the equation of motion, and it cites the standard references. For a pedagogical journal it has value: the flux-term route to the canonical momentum is simpler than the usual Lagrangian detour.

The soft spots are real but mostly fixable. Most important: Eq. (24) is stated as universal, S = m∫v·dl - qΦ, but the derivation assumes a positive charge moving clockwise. For a negative charge, the physical orbit is counterclockwise, the line integral ∮A·dl changes sign, and the correct action from your own Eq. (28) is S = m∫v·dl + qΦ. As written, Eq. (24) gives a non-stationary action for negative charges. The fix is to define Φ as the oriented flux or to carry the charge sign explicitly. This matters because the abstract and conclusion present Eq. (24) as the general result.

There is also a caption mismatch: the text says clockwise, Figure 2 caption says counterclockwise for the same setup. The clockwise sense is the physically correct one for a positive charge in B out of the page, so the caption needs correcting.

Minor: the claim that BWS gives 'the correct energy' should be qualified, since old quantum theory misses the zero-point offset. The paper already relies on the correspondence principle, so a footnote would suffice.

In sum: the central derivation checks out for the stated orientation; the negative-charge sign issue is a genuine gap in the claimed universality. A serious referee could ask for the sign qualification, a corrected caption, and a more careful energy statement. I would send it to review with the expectation of minor-to-moderate revision.

Referee Report

2 major / 3 minor

Summary. This paper proposes that the Maupertuis action for a charged particle in a uniform magnetic field must be modified from the kinetic expression S1 = m∫v·dl to S = m∫v·dl − qΦ, where Φ is the magnetic flux through the orbit. The modification is motivated by a first-order variation calculation for a positive charge moving clockwise in B = B ẑ (Section IV(a)), which shows that the kinetic action changes by qΔΦ under a nearby virtual trajectory; subtracting qΦ restores stationarity. Rewriting the flux via Stokes' theorem gives S = ∫(mv + qA)·dl, from which the canonical momentum p = mv + qA and the standard Lagrangian (2) follow. Appendix A shows that this vector-potential action yields the Lorentz force equation via Maupertuis' principle, and Appendix B applies the BWS condition to obtain E_n = nℏω.

Significance. The paper offers a genuinely different route to a standard result: instead of postulating the Lagrangian, it derives the need for a flux (or vector-potential) term from the requirement of stationary action for circular orbits. The calculation is explicit, the use of Stokes' theorem is transparent, and Appendix A contains a useful general stationary-action derivation of the Lorentz force from S = ∫(mv + qA)·dl. These are real pedagogical strengths. The central idea is, however, presented in the abstract and Eq. (24) without the sign/orientation restrictions under which it was derived; for negative charges the unqualified formula fails. Because this is a local fix rather than a defect in the Appendix A derivation, the paper is a viable major-revision candidate.

major comments (2)

[IV(a), Eqs. (21)-(25)] Equation (24) is derived for a positive charge and clockwise motion, but it is presented in the abstract and in the text as the action for 'a charged particle' with no such restriction. The sign of the flux term is tied to the orientation of the orbit. For q<0, the physical circular orbit is counterclockwise for B = B ẑ, so dl is parallel to φ̂ rather than anti-parallel. Repeating the first-order calculation gives ΔS1 = +|q|BR a ∫ f(φ)dφ = +|q|ΔΦ, whereas Eq. (21) would give qBR a ∫ f(φ)dφ = -|q|ΔΦ. The replacement m v0 = qBR used to obtain Eq. (21) is valid only for q>0. Adding -qΦ then changes the action by ΔS = ΔS1 - qΔΦ = 2|q|ΔΦ ≠ 0, so the true orbit is not stationary for the action (24) when q<0. The paper should either restrict Eq. (24) to q>0 with the stated orientation, replace q by |q| in the flux term, or define Φ as an oriented flux whose sign depends on the direction of traversal and state that convention in the abstract.
[Appendix B and Section IV(a)] Appendix B states that BWS with S = n h gives E_n = nℏω and calls this 'the correct energy.' The correspondence-principle argument used there fixes only the level spacing ΔE = ℏω, not the absolute ground-state energy, and the quantum-mechanical Landau spectrum is E_n = (n+1/2)ℏω. The paper should either restrict the claim to the old-quantum-theory context ('the BWS condition gives ...') or add the standard caveat that the zero-point energy is not obtained in this scheme. As written, the phrase 'the correct energy' overstates the result.

minor comments (3)

[Figure 2 caption] The caption says the particle moves counterclockwise, while the text explicitly states the direction is clockwise; for q>0 and B out of the page, clockwise is the correct direction, so the caption should be corrected.
[Section IV(a), Eq. (24)] After the sign issue is fixed, the paper should add a sentence clarifying that Eq. (24) applies to closed periodic orbits; for an open segment between two points P and Q, the area flux Φ is not defined without an arbitrary closing curve, so the action of Eq. (24) is not directly usable for the open-path statement of Maupertuis' principle in Section III.
[Appendix B] The sentence 'The answer lies in the relationship between the energy and the action' compares Eqs. (4) and (16) and says the action in the first example is larger by a factor of two; this is correct, but the wording is easy to misread as comparing energies, so a small clarifying phrase would help.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the flux-term action is explicitly constructed from the stationarity condition and verified in Appendix A; the canonical momentum is a definitional read-off, not a hidden fit.

full rationale

The paper does not invoke any load-bearing self-citation. Section IV(a) computes the first-order variation of the kinetic action on the known circular orbit and finds ΔS1 = qΔΦ (Eqs. 20-23). It then explicitly defines the action with a compensating flux term (Eq. 24) so that ΔS = 0 (Eq. 25). This is a transparent inverse-problem construction, not a use of the conclusion as a premise: the coefficient q is forced by the Lorentz-force orbit, and the paper openly states it is adding a term to cancel ΔS1. The subsequent identification of canonical momentum p = mv + qA (Eqs. 27-29) is a direct read-off after applying Stokes' theorem to the already-constructed action; it is not an independent empirical prediction, but it is also not disguised circularity because the action was not assumed to contain qA. Appendix A independently verifies that the action yields the Lorentz equation of motion. The negative-charge sign issue in Eq. (24) is a correctness gap regarding oriented flux, not a circularity. Overall the derivation is self-contained and its central claim has independent variational content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no invented physical entities. It relies on standard vector calculus, the Lorentz force as the equation of motion, Maupertuis' principle with fixed energy, and the old quantum theory postulates. The coefficient of the flux term is fixed by the cancellation condition, not fitted to data.

assumptions (5)

standard math Stokes' theorem and standard vector calculus identities
Used in Eq. (27) to convert the enclosed magnetic flux into a line integral of the vector potential, and in Appendix A to evaluate variations.
domain assumption Maupertuis' principle of least (stationary) action with fixed energy is a valid variational principle equivalent to the equation of motion
The whole derivation assumes fixed-energy virtual trajectories between the same endpoints; Appendix A derives the equation of motion from the action, which supports rather than proves the principle for velocity-dependent forces.
domain assumption The Lorentz force law m dv/dt = -∇U + qE + q v × B is the correct equation of motion
The action is constructed so that its stationarity reproduces this equation (Appendix A, Eq. A14).
domain assumption Bohr-Wilson-Sommerfeld quantization ∮p·dl = nh and Bohr's correspondence principle
Used in Appendix B to obtain energy levels; these are postulates of the old quantum theory and are not derived.
standard math The relation L = v·p - E between Lagrangian, canonical momentum, and energy
Used in Section IV(b) and Appendix A (Eq. A17) to recover the standard Lagrangian; cited to Goldstein, ref. 24.

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Cite this review

Pith. "Pith review of On the action for a charged particle moving in a magnetic field." pith.science (2026). https://pith.science/paper/RFWBF7TV

@misc{pith2026250616181,
  author       = {Pith},
  title        = {Pith review of: On the action for a charged particle moving in a magnetic field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFWBF7TV}},
  note         = {Machine review of arXiv:2506.16181}
}

read the original abstract

Application of the Bohr-Wilson-Sommerfeld quantization condition to a charged particle in a uniform magnetic field requires knowledge of the canonical momentum of such a particle, which in turn requires students to know about the vector potential. The canonical momentum in this situation is conventionally obtained by introducing an appropriate additional term involving the corresponding vector potential in the Lagrangian. In this work, we take a different approach to this problem by analyzing it with Maupertuis principle of least action. We show that satisfying this principle for a charged particle moving in a uniform magnetic field requires that a term proportional to the flux passing through the area enclosed by the trajectory of the particle be included in its action. This additional term accomplishes two tasks. First, it facilitates applying the Bohr-Wilson-Sommerfeld quantum condition for a charged particle moving in a uniform magnetic field without using the Lagrangian approach, and, secondly, it gives the appropriate canonical momentum and Lagrangian for a charged particle in a general electromagnetic field in a straightforward manner.

Figures

Figures reproduced from arXiv: 2506.16181 by the authors.

**Figure 1.** Circular trajectory of a particle, shown by red dot, under the influence of the force 𝐹(𝑟) = −𝑘𝑟. The trajectory, shown in black, is centered at the origin and has radius 𝑅. Also shown in blue is a nearby virtual trajectory of arbitrary shape. Both trajectories start and end at the same points 𝑃 and 𝑄, respectively. 𝑅 O P, Q x y [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Nahin, When Least is Best (Princeton University Press, Princeton, NJ, 2004), p 102

P .J. Nahin, When Least is Best (Princeton University Press, Princeton, NJ, 2004), p 102

work page 2004

[2] [2]

Dugas, A History of Mechanics (Dover Publications, Mineola, New York 2011), pp

R. Dugas, A History of Mechanics (Dover Publications, Mineola, New York 2011), pp. 254-257

work page 2011

[3] [3]

Hecht, Optics, 5th Ed

E. Hecht, Optics, 5th Ed. (Pearson, London, 2016), pp. 106-111

work page 2016

[4] [4]

Yourgrau and S

W. Yourgrau and S. Mandelstam, Variational principles in dynamics and quantum theory (Dover publications, Mineola, New York, 1979), pp. 11-18

work page 1979

[5] [7]

Goldstein, C

H. Goldstein, C. Poole and J. Safko, Classical Mechanics (Pearson, London Third edition 2001), pp. 365-368

work page 2001

[6] [8]

Eisberg and R

R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, New York, Second edition 2006), pp. 110-114

work page 2006

[7] [9]

Sin-itiro Tomonaga, Quantum Mechanics volume I: old quantum theory (North-Holland publishing, Amsterdam, 1962), pp. 118-135

work page 1962

[8] [10]

Heisenberg, ‘‘Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,’’ Z

W. Heisenberg, ‘‘Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,’’ Z. Phys. 33, 879 (1925). Its English translation is available in ref. 11

work page 1925