$\bf{S^1}$-index theory for the Lorentz force equation

Abstract: In this paper we prove that the $S<sup>1$-invariance of the Poincaré action functional associated to the Lorentz force equation gives the existence of multiple critical points which are periodic solutions with a fixed period. To do this, we prove an abstract multiplicity result which is based upon the Lusternik-Schnirelman method with the $S<sup>1$-index. The corresponding result in the context of the Fadell-Rabinowitz index is proved in Ekeland and Lasry (Ann. Math., 112 (1980)). The main feature of our abstract result is that it allows us to consider nonsmooth functionals satisfying only a weak compactness condition well adapted to the Poincaré functional.

• The paper develops a Lusternik–Schnirelman framework for nonsmooth S¹-invariant functionals by combining Ekeland–Lasry regularization, Benci’s S¹-index, and weak Palais–Smale compactness.
• The method applies to the relativistic Lorentz force equation, where critical points of the Poincaré action correspond exactly to periodic solutions despite the action’s singular velocity constraint.
• Under stated assumptions and sufficiently large local potential growth λ, the action has at least 3m distinct critical orbits, yielding 3m negative-level 2π-periodic solutions; the count grows with the Fourier-mode parameter m.

Overview and main contribution

This paper by Bereanu and Pîrvuceanu develops a Lusternik–Schnirelman theory with Benci's $S^1$-index for nonsmooth functionals of the form $\mathcal{I} = \Psi + \mathcal{F}$, where $\Psi$ is convex, lower semicontinuous, and extended-valued, and applies it to the Poincaré action functional of the relativistic Lorentz force equation (LFE). The central application establishes multiplicity of fixed-period periodic solutions: under suitable assumptions on autonomous electric and magnetic potentials, the action functional possesses at least $3m$ critical orbits at negative levels, which are $2\pi$-periodic solutions of the LFE. The abstract framework is deliberately adapted to functionals satisfying only a weak Palais–Smale condition $(wPS)$ — compactness in a weaker norm via a compact embedding — rather than the full $(PS)$ condition, which is essential because the natural setting for the LFE action is $H^1_{2\pi} \subset C_{2\pi}$ with Arzelà–Ascoli compactness.

The smooth analogue of the multiplicity theorem corresponds to the classical result of Ekeland and Lasry (Ann. Math., 1980), obtained there with the Fadell–Rabinowitz index; the present work extends that scheme to the nonsmooth relativistic setting.

Abstract multiplicity theorem for $S^1$-invariant nonsmooth functionals

The functional $\mathcal{I}: X \to (-\infty,\infty]$ on a Hilbert space $\mathcal{I} = \Psi + \mathcal{F}$0, compactly embedded in a Banach space $\mathcal{I} = \Psi + \mathcal{F}$1, satisfies hypothesis $\mathcal{I} = \Psi + \mathcal{F}$2: $\mathcal{I} = \Psi + \mathcal{F}$3 with $\mathcal{I} = \Psi + \mathcal{F}$4 and $\mathcal{I} = \Psi + \mathcal{F}$5 convex l.s.c. Critical points are defined via the subdifferential, i.e., solutions of the variational inequality $\mathcal{I} = \Psi + \mathcal{F}$6. The paper first proves the equivalence of Szulkin-type $\mathcal{I} = \Psi + \mathcal{F}$7-sequences and $\mathcal{I} = \Psi + \mathcal{F}$8-sequences using a separation lemma for convex functions bounded below by $\mathcal{I} = \Psi + \mathcal{F}$9.

Two structural tools carry the argument. First, the Ekeland–Lasry regularization: if $\Psi$0 is bounded below and satisfies condition $\Psi$1 — convexity of $\Psi$2 on $\Psi$3 for some $\Psi$4 — then the infimal convolution $\Psi$5 is $\Psi$6, invariant whenever $\Psi$7 is, preserves critical points ($\Psi$8 iff $\Psi$9), and inherits the weak Palais–Smale property. This reduction converts the nonsmooth problem into a smooth one without losing critical points.

Second, Benci's $3m$0-index together with Ghoussoub's location theorem. The authors establish an index computation (Lemma indexorbit): for finitely many points outside $3m$1, a small invariant neighborhood of their union has index exactly $3m$2. The construction uses averaging of a Tietze extension against the character $3m$3, where $3m$4 is the product of minimal periods of the individual orbits.

The resulting abstract theorem states: if $3m$5 is invariant, bounded below, satisfies $3m$6, $3m$7, $3m$8, has $3m$9 for all $2\pi$0, and $2\pi$1 (where $2\pi$2), then every level $2\pi$3 carries critical points of $2\pi$4 itself; moreover, coalescence $2\pi$5 at levels below $2\pi$6 forces infinitely many distinct critical orbits. Combined with the standard finite-dimensional computation $2\pi$7 for invariant subspaces $2\pi$8 with trivial fixed-point part, this yields the corollary used in the application: a $2\pi$9-dimensional invariant subspace whose boundary lies below level $(wPS)$0 guarantees at least $(wPS)$1 negative-level critical orbits.

A remark notes that the smooth version of the theorem holds verbatim in Banach spaces; Hilbert structure is needed only for the Ekeland–Lasry regularization.

The Poincaré action functional

For potentials $(wPS)$2 and $(wPS)$3, the LFE reads

$(wPS)$4

with $(wPS)$5 and $(wPS)$6. On $(wPS)$7 the action splits as $(wPS)$8, where $(wPS)$9 on the closed convex set $(PS)$0 (extended by $(PS)$1), and $(PS)$2.

The key equivalence is Theorem criticsol: $(PS)$3 is a $(PS)$4-periodic solution of the LFE if and only if $(PS)$5 is a critical point of $(PS)$6. This relies on Mawhin's lemma producing, for each $(PS)$7, a unique solution of the singular $(PS)$8-Laplacian boundary value problem with $(PS)$9.

Compactness is handled by Proposition pslfe: any $H^1_{2\pi} \subset C_{2\pi}$0-sequence with bounded averages $H^1_{2\pi} \subset C_{2\pi}$1 converges uniformly to a critical point preserving the level. The proof exploits the Lipschitz constraint $H^1_{2\pi} \subset C_{2\pi}$2 to pass uniform convergence through both $H^1_{2\pi} \subset C_{2\pi}$3 (via weak-$H^1_{2\pi} \subset C_{2\pi}$4 convergence of derivatives and l.s.c.) and $H^1_{2\pi} \subset C_{2\pi}$5 (dominated convergence for the $H^1_{2\pi} \subset C_{2\pi}$6 term). This weak form of compactness — convergence in $H^1_{2\pi} \subset C_{2\pi}$7 only, not in $H^1_{2\pi} \subset C_{2\pi}$8 — is precisely what the abstract theory was designed to accommodate.

Multiplicity of $H^1_{2\pi} \subset C_{2\pi}$9-periodic solutions

Specializing to period $S^1$0 with time translations acting as $S^1$1, autonomy of $S^1$2 and $S^1$3 makes both parts of $S^1$4 invariant, and $S^1$5 (constant loops).

Three lemmas assemble the hypotheses of the abstract corollary:

Convexity condition $S^1$6: Under $S^1$7 — $S^1$8, $S^1$9, $\mathcal{I}: X \to (-\infty,\infty]$0, $\mathcal{I}: X \to (-\infty,\infty]$1 off the origin, $\mathcal{I}: X \to (-\infty,\infty]$2 bounded, $\mathcal{I}: X \to (-\infty,\infty]$3 at infinity — plus $\mathcal{I}: X \to (-\infty,\infty]$4 — $\mathcal{I}: X \to (-\infty,\infty]$5 with $\mathcal{I}: X \to (-\infty,\infty]$6 bounded — the perturbed functional $\mathcal{I}: X \to (-\infty,\infty]$7 becomes convex on $\mathcal{I}: X \to (-\infty,\infty]$8 for large $\mathcal{I}: X \to (-\infty,\infty]$9. Convexity of $\mathcal{I} = \Psi + \mathcal{F}$00 handles the potential term; boundedness of $\mathcal{I} = \Psi + \mathcal{F}$01 and $\mathcal{I} = \Psi + \mathcal{F}$02 allows domination of the Hessian of the magnetic contribution by $\mathcal{I} = \Psi + \mathcal{F}$03.

Compactness below a threshold: $\mathcal{I} = \Psi + \mathcal{F}$04 is bounded below and satisfies $\mathcal{I} = \Psi + \mathcal{F}$05 for all

$\mathcal{I} = \Psi + \mathcal{F}$06

The proof shows that unbounded averages would force $\mathcal{I} = \Psi + \mathcal{F}$07 while $\mathcal{I} = \Psi + \mathcal{F}$08, contradicting the level bound. Note that compactness holds only strictly below this threshold — the threshold itself is not covered.

Index estimate near zero: Under the additional coercivity-at-zero assumption $\mathcal{I} = \Psi + \mathcal{F}$09, $\mathcal{I} = \Psi + \mathcal{F}$10 for $\mathcal{I} = \Psi + \mathcal{F}$11, the trigonometric subspace $\mathcal{I} = \Psi + \mathcal{F}$12 spanned by modes up to frequency $\mathcal{I} = \Psi + \mathcal{F}$13 is $\mathcal{I} = \Psi + \mathcal{F}$14-dimensional, invariant, and disjoint from $\mathcal{I} = \Psi + \mathcal{F}$15. Using $\mathcal{I} = \Psi + \mathcal{F}$16 and the spectral bound $\mathcal{I} = \Psi + \mathcal{F}$17 on $\mathcal{I} = \Psi + \mathcal{F}$18, one obtains

$\mathcal{I} = \Psi + \mathcal{F}$19

so for $\mathcal{I} = \Psi + \mathcal{F}$20 the boundary of the ball $\mathcal{I} = \Psi + \mathcal{F}$21 lies below the compactness threshold. Finally, no constant loop can be a negative-level critical point, since such a point must satisfy $\mathcal{I} = \Psi + \mathcal{F}$22, contradicting $\mathcal{I} = \Psi + \mathcal{F}$23 off the origin.

Assembling these pieces yields the main result:

For each $\mathcal{I} = \Psi + \mathcal{F}$24 there exists $\mathcal{I} = \Psi + \mathcal{F}$25 such that if $\mathcal{I} = \Psi + \mathcal{F}$26, then $\mathcal{I} = \Psi + \mathcal{F}$27 has at least $\mathcal{I} = \Psi + \mathcal{F}$28 critical orbits at negative levels, which are $\mathcal{I} = \Psi + \mathcal{F}$29-periodic solutions of the Lorentz force equation.

The count $\mathcal{I} = \Psi + \mathcal{F}$30 reflects $\mathcal{I} = \Psi + \mathcal{F}$31. An explicit example illustrates the mechanism: $\mathcal{I} = \Psi + \mathcal{F}$32 satisfies $\mathcal{I} = \Psi + \mathcal{F}$33–$\mathcal{I} = \Psi + \mathcal{F}$34 with $\mathcal{I} = \Psi + \mathcal{F}$35, so the number of $\mathcal{I} = \Psi + \mathcal{F}$36-periodic solutions diverges as $\mathcal{I} = \Psi + \mathcal{F}$37.

A normalization remark clarifies physical applicability: restoring units gives $\mathcal{I} = \Psi + \mathcal{F}$38 with prescribed $\mathcal{I} = \Psi + \mathcal{F}$39; hence the sign convention $\mathcal{I} = \Psi + \mathcal{F}$40 corresponds to attractive interaction, appropriate for positively charged particles, and $\mathcal{I} = \Psi + \mathcal{F}$41 for negatively charged ones.

Limitations and open questions

Several restrictions are inherent to the method and are acknowledged implicitly or explicitly. The compactness threshold $\mathcal{I} = \Psi + \mathcal{F}$42 depends on the sup-norm of the magnetic potential, so strong magnetic fields push the admissible energy window downward and inflate $\mathcal{I} = \Psi + \mathcal{F}$43; the dependence of $\mathcal{I} = \Psi + \mathcal{F}$44 on $\mathcal{I} = \Psi + \mathcal{F}$45 beyond its bounds is not quantified. The growth condition $\mathcal{I} = \Psi + \mathcal{F}$46 is required only locally around the origin but interacts with the asymptotic level $\mathcal{I} = \Psi + \mathcal{F}$47 through $\mathcal{I} = \Psi + \mathcal{F}$48, which is defined non-constructively. The Ekeland–Lasry regularization requires Hilbert space structure and condition $\mathcal{I} = \Psi + \mathcal{F}$49, excluding more general nonsmooth settings. Whether the multiplicity count can be sharpened from orbit counting to Morse-theoretic refinements, or extended to nonautonomous potentials (where the $\mathcal{I} = \Psi + \mathcal{F}$50-symmetry is lost and one must work with other group actions or none), remains open within this framework.

Conclusion

The paper provides a Lusternik–Schnirelman scheme with the $\mathcal{I} = \Psi + \mathcal{F}$51-index valid for nonsmooth functionals satisfying only a weak compactness condition, built on the Ekeland–Lasry regularization and Ghoussoub's location principle. Applied to the Poincaré action of the relativistic Lorentz force equation with autonomous potentials, it yields at least $\mathcal{I} = \Psi + \mathcal{F}$52 distinct critical orbits — equivalently, $\mathcal{I} = \Psi + \mathcal{F}$53-periodic solutions — at negative energy levels whenever the quadratic growth rate $\mathcal{I} = \Psi + \mathcal{F}$54 of the electric potential near the origin exceeds an explicit threshold $\mathcal{I} = \Psi + \mathcal{F}$55. The result parallels the Ekeland–Lasry theorem in the smooth Hamiltonian setting and demonstrates that the variational machinery for singular $\mathcal{I} = \Psi + \mathcal{F}$56-Laplacian problems supports full equivariant minimax theory.