Least constraint approach to non-relativistic quantum mechanics

Abstract: We formulate a variational principle for non-relativistic quantum mechanics inspired by Gauss's principle of least constraint. We define a quantum constraint functional as the probability-weighted square deviation between the actual motion and the unconstrained motion that would arise from external forces alone. In this functional, the quantum potential plays the role of an intrinsic constraint that modifies the acceleration. Minimizing this quantum constraint functional with respect to the acceleration field yields the quantum Euler equations, which together with the continuity equation are equivalent to the Schrödinger equation. The principle is instantaneous and provides a differential characterization of quantum evolution. We demonstrate that this formulation is not a mere rewriting of existing dynamics: it provides a unified and technically economical treatment of geometric constraints and velocity-dependent dissipative forces, neither of which admits a straightforward global variational formulation. Potential applications to a broad range of quantum phenomena are also indicated.

• The paper introduces a novel local variational framework that re-derives the Schrödinger equation by minimizing a quantum constraint functional.
• It employs a hydrodynamic (Madelung) formulation to effectively handle nonholonomic constraints, dissipative forces, and geometric restrictions.
• The approach recovers known dynamics on curved manifolds and for dissipative systems, simplifying formulations without relying on global Lagrangian structures.

Least Constraint Principle in Non-Relativistic Quantum Mechanics

Introduction

The paper "Least constraint approach to non-relativistic quantum mechanics" (2604.26642) introduces a novel variational framework for quantum dynamics, formulated in analogy to Gauss’s principle of least constraint from classical mechanics. Traditionally, quantum evolution is derived from global variational principles (e.g., Hamilton’s principle, the Feynman path integral), which depend on the existence of a global action functional and a Lagrangian framework. Such global formulations are often cumbersome or inapplicable for systems with nonholonomic constraints, dissipative forces, or geometric restrictions. The present work circumvents these limitations by constructing an instantaneous, local-in-time variational principle for quantum mechanics, leveraging the hydrodynamic (Madelung) representation and providing an alternative route to the Schrödinger equation that can accommodate constrained and dissipative dynamics.

Formulation of the Quantum Least Constraint Principle

The core proposal is a quantum constraint functional $Z$, defined as the probability-weighted squared deviation between the actual acceleration of the Madelung probability fluid and the "unconstrained" acceleration dictated by external and quantum potentials. The main object is

$$Z\left[\frac{\partial\mathbf{v}}{\partial t}\right] = \int \rho(\mathbf{r}) \left\| \frac{D\mathbf{v}}{Dt} + \frac{1}{m}\nabla V + \frac{1}{m}\nabla Q \right\|^2 d^3r,$$

where $\rho$ and $\mathbf{v}$ are the probability density and velocity field, $V$ is the external potential, and $Q$ is the quantum potential.

Minimization of $Z$ with respect to the acceleration field (i.e., the time derivative of the velocity) under the constraint of probability conservation (the continuity equation) leads to the quantum Euler equations. These, combined with the continuity equation, are shown to be equivalent to the Schrödinger equation (up to the assumed single-valued phase of the wavefunction, with multi-valuedness discussed as an extension). The formal construction provides a local, differential variational characterization of quantum evolution, in direct contrast to the traditional time-integrated action extremization.

Theoretical Advantages and Geometric Interpretation

A central claim, supported by explicit derivations, is that this principle is technically and conceptually distinct from Lagrangian-based approaches and not a trivial reformulation of quantum dynamics. Two primary advantages are highlighted:

• Instantaneity and Locality: The least-constraint functional is formulated and minimized at each instant in time. This allows it to seamlessly incorporate velocity-dependent (dissipative) forces and geometric (holonomic or nonholonomic) constraints without the need for a global Lagrangian or Hamiltonian structure.
• Geometric Transparency: In the absence of external potentials, the principle reduces quantum evolution to the minimization of the deviation between the true material acceleration and the reference acceleration induced by the quantum potential—a measure of the curvature of the probability amplitude. Thus, quantum acceleration is dictated by local variations ("curvature") of the probability density, creating a direct analog to Hertz’s principle of least curvature from classical mechanics. This geometric viewpoint clarifies the non-dispersive character of free quantum evolution and the role of the quantum potential as encoding intrinsic curvature.

Applications

Constrained Quantum Dynamics on Curved Surfaces

The framework is applied to the case of a particle constrained to a curved surface (specifically a sphere), where traditional treatments require elaborate operator-ordering arguments and thin-layer quantization (e.g., da Costa’s method). Within the least constraint formalism, the constraint is naturally imposed by projecting the reference acceleration onto the tangent bundle of the surface, yielding—without the need for coordinate expansions—the geometric potential associated with the curvature (mean and Gaussian curvatures) of the surface. For a sphere, this recovers the standard surface Schrödinger equation with the da Costa geometric potential. The key insight is that the geometric potential emerges as a dynamical necessity from the force projection and not as an artifact of coordinate manipulations.

Dissipative Quantum Systems

For dissipative systems, the approach is demonstrated using the damped quantum harmonic oscillator. By including velocity-dependent damping directly in the unconstrained acceleration within $Z$, and minimizing as before, the resulting wave equation is the Kostin nonlinear Schrödinger equation, which incorporates a logarithmic nonlinearity to enforce proper dissipative decay of expectation values. Crucially, this variational construction does not require postulating dissipative effects at the level of observable evolution; instead, they arise naturally from the structure of the constrained acceleration.

Implications and Future Directions

The immediate implications of this work lie in the unification and simplification of hydrodynamic formulations of quantum mechanics, particularly for systems with geometric or dissipative complexities. This local variational structure paves the way for:

• Unified treatment of constrained, dissipative, and unconstrained quantum systems within a single variational framework.
• Technically efficient derivations of effective dynamics in curved manifolds without resorting to thin-layer expansions or operator ordering calculations.
• Seamless extension to velocity-dependent nonconservative forces—typically excluded from traditional global variational approaches.
• A geometric paradigm highlighting the quantum potential as an effective curvature and suggesting new interpretations of quantum fluid phenomena.

Potential future developments include extensions to flows with multi-valued phases, applications to multi-particle and quantum field systems, and the study of tunneling, caustics, and complex interference via this local approach. The local nature of the variational structure also suggests possible utility in the analysis or simulation of emergent quantum phenomena such as Bose–Einstein condensates in complex trapping geometries.

Conclusion

The paper establishes an instantaneous, quantum analog of Gauss’s principle of least constraint that yields the Schrödinger equation via differential variational minimization. This approach transcends mere reformulation, providing a technically economical and conceptually transparent route to handling constrained and dissipative quantum dynamics. Applications to quantum motion on curved manifolds and dissipative systems demonstrate the versatility and power of the framework, with further extensions poised to impact both theoretical and applied quantum mechanics.