Rayleigh Flow as a Constraint Optimization and Fundamental Mode Selection Mechanism in Dissipative Engineering Systems

Rayleigh Flow as a Constraint Optimization and Fundamental Mode Selection Mechanism in Dissipative Engineering Systems Mateus R. de Mariaa,∗ aUniversidade Federal do Ceará, Av. Humberto Monte, s/n, Campus do Pici, Bloco 914, CEP 60455-760, Fortaleza, CE, Brazil Abstract We present a short, self-contained technical note on a norm-preserving dissipative flow induced by the Rayleigh quotient of a positive self-adjoint operator. The nonlinear term acts as a global feedback that projects the dissipation onto a fixed-norm sphere, allowing the dynamics to be interpreted as a constrained gradient flow (continuous spectral normalization). In the typical engineering regime (after finite element discretization, finite volume schemes, or modal truncation), we establish global well-posedness, norm preservation, monotonicity of the Rayleigh quotient, and characterization of stationary states as eigenvectors. We further show that the flow selects the smallest eigenvalue compatible with the initial condition, corresponding to dynamic selection of the fundamental mode (low-frequency / low effective energy). As an applied example, we discuss vibration mode filtering in one-dimensional structures, interpreting the flow as a continuous algorithm for modal shape extraction and stabilization in discretized systems. Keywords: Rayleigh quotient, constrained gradient flow, spectral dynamics, dissipative systems, mode selection, applied analysis 2020 MSC: 35B40, 37N30, 47J30 ∗Corresponding author. Email address: [email protected] (Mateus R. de Maria) 1. Introduction In many engineering systems with distributed dynamics—such as mechanical vibrations in structures, thermal diffusion, distributed electrical circuits, and large-scale models obtained after finite element discretization—it is common for a natural modal decomposition to exist. In practice, one often seeks a robust procedure to extract or stabilize a dominant mode (typically the fundamental mode) while damping higher-frequency components that are more sensitive to noise and discretization artifacts. Classical tools include explicit spectral decomposition, Krylov-based eigenvalue solvers, and other discrete iterative schemes. In this note we discuss a continuous-time alternative driven by the Rayleigh quotient: a normpreserving, gradient-like flow that implements spectral normalization dynamically. The flow admits a transparent variational interpretation as a constrained descent of a quadratic energy on a fixed-norm sphere. Our goal is not to claim novelty of the underlying identities, but to present a clean self-contained analysis and an applied viewpoint. We show that, in the standard discretized regime, the flow is globally well posed, preserves the norm, makes the Rayleigh quotient strictly decrease unless an eigenstate is reached, and selects the smallest eigenvalue accessible from the initial condition—thereby acting as a continuous mechanism for fundamental mode selection. 2. Mathematical Model and Physical Interpretation Let Hbe a real Hilbert space with inner product ⟨·,·⟩ and norm ∥·∥. Let A:D(A)⊂H→Hbe a densely defined, positive, self-adjoint operator. For u= 0, define the Rayleigh quotient λ(u) = ⟨Au, u⟩ ⟨u, u⟩. We consider the flow ∂tu(t) = −Au(t) + λ(u(t)) u(t), u(0) = u0= 0.(1) The interpretation is straightforward: the term −Au is dissipative (the gradient flow of a quadratic energy functional), while the term λ(u)uacts as a global feedback that removes the radial component of the dissipation, 2 forcing the evolution to remain on the sphere {u:∥u∥=∥u0∥}. Indeed, writing Pu=I−u⊗u ∥u∥2for the orthogonal projector onto the tangent space of the sphere, one has the identity ∂tu=−PuAu, showing that the Rayleigh Flow is precisely the tangential projection of the linear dissipative dynamics. Consequently, (1) can be interpreted as a constrained gradient flow: it performs descent of the energy functional E(u) = 1 2⟨Au, u⟩ subject to the constraint ∥u∥=∥u0∥. The critical points of this constrained optimization problem are exactly the eigenvectors of A. Remark 1 (Engineering regime).In many industrial applications, Aarises as a symmetric positive matrix (after discretization or modal truncation), and H=Rn. In this regime, (1) defines a smooth vector field on Rn\{0}and the analysis below is global and straightforward. In infinite dimensions, when A is unbounded (e.g., differential operators), well-posedness requires additional assumptions (mild solutions, semigroup theory, regularity in D(A1/2), etc.). As this is an applied note, we adopt the formulation covering the discretized regime and indicate where extra hypotheses enter in the infinite-dimensional case. 3. Analysis of the Flow We work in the typical engineering regime: H=Rnand A∈Rn×n symmetric positive semidefinite. The energy identities below remain valid in greater generality whenever the computations make sense. Proposition 1 (Global well-posedness in the discretized regime).Assume H=Rnand Ais symmetric positive semidefinite. Then, for any u0= 0, there exists a unique global classical solution u∈C1([0,∞); Rn)of (1). Proof. The vector field F(u) = −Au +λ(u)uis smooth on Rn\ {0}, yielding local existence and uniqueness. By Proposition 2, ∥u(t)∥=∥u0∥along the flow, so trajectories remain on a compact sphere and never reach the origin. Thus no finite-time blow-up occurs and the solution extends globally. 3 Proposition 2 (Norm preservation).Every sufficiently regular solution of (1) satisfies ∥u(t)∥=∥u0∥, t ≥0. Proof. Differentiating ∥u∥2along the flow, d dt∥u∥2= 2⟨∂tu, u⟩= 2⟨−Au +λ(u)u, u⟩=−2⟨Au, u⟩+ 2λ(u)∥u∥2= 0. Proposition 3 (Monotonicity of the Rayleigh quotient).Along any regular solution of (1), d dtλ(u(t)) = −2 ∥u(t)∥2∥(A−λ(u(t))I)u(t)∥2≤0. In particular, λ(u(t)) is non-increasing and satisfies inf σ(A)≤λ(u(t)) ≤λ(u0). Moreover, d dt λ(u(t)) = 0 if and only if u(t)is an eigenvector of A. Proof. Assume ∥u(t)∥ = 0 and differentiate λ(u) = ⟨Au, u⟩/∥u∥2along (1). Using self-adjointness, d dt⟨Au, u⟩= 2⟨Au, ˙u⟩= 2⟨Au, −Au +λ(u)u⟩=−2∥Au∥2+ 2λ(u)⟨Au, u⟩. By Proposition 2, ∥u(t)∥is constant, hence ˙ λ(u(t)) = 1 ∥u∥2 d dt⟨Au, u⟩=−2 ∥u∥2∥Au∥2−λ(u)⟨Au, u⟩. Finally, expand ∥(A−λ(u)I)u∥2=∥Au∥2−2λ(u)⟨Au, u⟩+λ(u)2∥u∥2=∥Au∥2−λ(u)⟨Au, u⟩, since λ(u)∥u∥2=⟨Au, u⟩. This gives the stated identity and shows that ˙ λ= 0 iff (A−λI)u= 0. 4 Proposition 4 (Characterization of the ω-limit set).In the discretized regime, there exists λ∗∈Rsuch that λ(u(t)) →λ∗as t→ ∞ and ∥(A−λ(u(t))I)u(t)∥ → 0. Any accumulation point u∗of the trajectory satisfies Au∗=λ∗u∗, that is, u∗belongs to an eigenspace of A. Theorem 1 (Selection of the accessible fundamental mode).Let 0≤µ1≤ · · · ≤ µnbe the eigenvalues of A(with multiplicities) and write u0=Pn j=1 cjϕj in an orthonormal eigenbasis. Define k= min{j:cj= 0}. Then λ∗=µk, and every accumulation point belongs to the eigenspace associated with µk. If µkis simple, then u(t)→ ±∥u0∥ϕkas t→ ∞. Proof. Write u(t) = Pn j=1 cj(t)ϕjin an orthonormal eigenbasis Aϕj=µjϕj. Taking inner products of (1) with ϕjyields ˙cj(t) = −(µj−λ(u(t))) cj(t). Let k= min{j:cj(0) = 0}. Then ck(t)= 0 for all tby uniqueness of solutions of scalar ODEs. For j > k consider the ratio rj(t) := cj(t) ck(t). Differentiating and using the equations for cj, ckgives ˙rj(t) = −(µj−µk)rj(t), hence rj(t) = rj(0) e−(µj−µk)t−−−→ t→∞ 0. Therefore, up to normalization, u(t)asymptotically aligns with the eigenspace of µk. Moreover, λ(u(t)) = Pn j=1 µjcj(t)2 Pn j=1 cj(t)2−−−→ t→∞ µk, since cj(t)/ck(t)→0for all j > k. If µkis simple, then u(t)/∥u0∥→±ϕk. 5 4. Applied Example: Vibration Mode Filtering in 1D Consider a one-dimensional bar of length L, clamped at both ends. A simple continuous model for spatial vibration shapes involves the elliptic operator A=−d2 dx2on (0, L), u(0) = u(L)=0, whose spectrum is discrete: ϕk(x) = sinkπx L, µk=kπ L2 , k ≥1. In numerical or industrial implementations, the system is discretized, yielding a stiffness matrix Kand (after normalization) a symmetric positive operator A∈Rn×n. The Rayleigh Flow ˙u=−Au +λ(u)u can then be interpreted as a continuous algorithm for modal shape extraction: it preserves global scale while continuously reducing the effective energy ⟨Au, u⟩. By the previous theorem, if the initial state has a component along the fundamental mode, the dynamics converges to a shape proportional to that mode. 5. A simple normalized Euler scheme In the discretized regime H=Rn, the Rayleigh Flow suggests a lightweight integration scheme that preserves the norm by construction. Given a time step ∆t > 0and um= 0, define ˜um+1 =um−∆t Aum, um+1 =∥um∥˜um+1 ∥˜um+1∥. This is the explicit Euler step for ˙u=−Au followed by a projection onto the sphere {u:∥u∥=∥u0∥}. A semi-implicit variant replaces ˜um+1 by the solution of (I+ ∆t A)˜um+1 =umbefore renormalization, improving stability for stiff operators. Such schemes are widely compatible with finite element stiffness matrices (and generalized eigenproblems Kϕ =µMϕ after mass normalization), making the Rayleigh Flow a convenient conceptual bridge between continuous mode selection and standard computational pipelines. 6 6. Discussion and Limitations The Rayleigh Flow offers a simple and transparent mechanism for modal selection. Its main limitations are: (i) the global nature of the feedback term λ(u), requiring aggregated state information; (ii) sensitivity to noise in practical measurements; (iii) the fact that real vibration dynamics are typically second order in time, so the flow should be interpreted as an optimization or control procedure rather than a physical trajectory. 7. Conclusion We have presented a concise applied analysis of the Rayleigh Flow, highlighting its interpretation as a constrained gradient flow and its role in dynamic selection of the fundamental mode. In the discretized engineering regime, we established global well-posedness, norm preservation, monotonicity of the Rayleigh quotient, and convergence to eigenstates. The onedimensional vibration example illustrates how the flow can be viewed as a continuous modal filtering and stabilization mechanism in real systems. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work the author used an AI language model (ChatGPT) to support drafting and language editing. After using this tool, the author reviewed, edited, and verified the manuscript as needed and takes full responsibility for the content of the submitted work. References [1] P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008. [2] R. Courant and D. Hilbert, Methods of Mathematical Physics, WileyInterscience, 1953. [3] L. Rayleigh, The Theory of Sound, Vol. I, Macmillan, 1877. [4] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997. 7