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Winfried Lohmiller

Publications and source records attributed to Winfried Lohmiller.

10 recordsLinked to original sources
arXiv2026

Interpreting Bohm-like quantum potentials in "Computing quantum waves exactly from classical action"

In contrast to his earlier posting arXiv260502621 [6] commenting on the article rspa20250413 [5] and answered in arXiv260520443 [3] the same author of the new posting arXiv260605197 [7] does not seem to dispute any longer the following points i Assuming in [5] that along each extremal action path the action Laplacian or more generally the propagated density is space independent implies that the Bohm like potential terms are exactly zero a condition which is directly verified in all examples of [5] as well as in its relativistic Dirac and Maxwell extensions ii Just as in Feynmans well known results the standard polynomials in the treatment in [5] of the harmonic oscillator appear naturally from the Taylor expansion of the kernel and thus there is no circularity. He does not appear to dispute either that Bohm like terms along individual stationary action paths are indeed very different from the standard Bohm Madelung potential on the overall wave. This is illustrated e.g. in the detailed double slit example in [3] whose Bohm like terms are exactly zero while the usual Bohm Madelung potential can be very large. Also in that example the action behind the slits is a conic function so that exact Feynman or Van Vleck computations based on quadratic actions do not apply. His new posting [7] therefore concentrates on the claim in [3] that the space independence assumption on the Laplacian of the action if not directly verified can be achieved through time rescaling which is a fair discussion topic We all agree that the Schrodinger equation has to be fulfilled in the original x t coordinates. To this effect as in the hydrogen atom example of the original paper [5] we use a general result of Duru and Kleinert which the author of [6] and [7] may be unaware of. There independence of the constructed eigenwave on the rescaled time is key to the exact backmapping to the original x t coordinates.

quant-ph↗
arXiv2024

On computing quantum waves exactly from classical and relativistic action

We show that the Schr\"odinger equation can be solved exactly based only on classical least action. Fundamental postulates of quantum mechanics can in turn be derived directly from this construction. The results extend to the relativistic Klein-Gordon, Pauli, Dirac, and Maxwell equations, and suggest a smooth transition between physics across scales. Most quantum mechanics problems have classical versions which involve multiple least action solutions. The associated classical multipaths stem either from the initial position or momentum distribution, or from branch points, generated, e.g., by a multiply connected manifold (double slit experiment), by spatial inequality constraints (particle in a box), or by a singularity (Coulomb potential). We show that the exact Schr\"odinger wave function $\psi$ of the original quantum problem can be constructed by combining this classical multi-valued action $\phi$ with the density $\rho$ of the classical position dynamics, where a key point is that $\rho$ can be easily computed from $\phi$ along each extremal action path. The construction is general and does not involve any quasi-classical approximation. Examples illustrate how the quantum wave functions for the double-slit experiment or e.g., the hydrogen atom can be computed exactly from their classical least action counterparts. In a quantum measurement process, randomness originates from the determined forward mapping of an initial classical density distribution. In the Einstein-Podolsky-Rosen experiment, while Bell's inequalities are violated, from this perspective there is indeed a hidden variable in the form of a complex spinor. These results also provide a simpler computational alternative to Feynman path integrals, as they use only a minimal subset of classical paths and avoid zig-zag paths and time-slicing altogether.

quant-ph↗
arXiv2024

Constrained Least Action and Quantum Mechanics

Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action. The computation is based on solving a HamiltonJacobi equation for the action computing the classical density accordingly along all stationary action paths and finally constructing the exact wave function based on these classical quantities alone. The method requires that the action Laplacian or more generally the Laplacian of the propagated density be purely time-varying along stationary action paths. In the case of arbitrary nonlinear potentials this condition can still be verified without loss of generality by using a time rescaling. As the complexity of the wave computation is thus shifted to that of the action and the possible time rescaling this paper proposes a new analytical approach both to compute the action itself for a general nonlinear Hamilton-Jacobi pde and to concurrently construct a time rescaling as needed In contrast to solving the Schroedinger equation directly this computation of action and density extends naturally to systems with nonlinear potentials or position dependent inertia tensors. Hence in principle it can replace the approximations of quantum perturbation theory. For general nonlinear potentials, extending a result of Duru and Kleinert the time rescaling in a given metric is shown to correspond to a change of variables in the computed action density and wave with each eigenwave computed using that change of variables. We show that the approach makes it straightforward to construct the quantum wave for basic cases where no exact solution has been yet derived with the time rescaling unifying the computation. The approach first illustrated for the known three dimensional hyperbolic potential waves of the hydrogen atom is then used to compute the quantum waves for a quartic oscillator for which so far no direct exact solutions are known.

math-ph↗
arXiv2023

MinMax Networks

While much progress has been achieved over the last decades in neuro-inspired machine learning, there are still fundamental theoretical problems in gradient-based learning using combinations of neurons. These problems, such as saddle points and suboptimal plateaus of the cost function, can lead in theory and practice to failures of learning. In addition, the discrete step size selection of the gradient is problematic since too large steps can lead to instability and too small steps slow down the learning. This paper describes an alternative discrete MinMax learning approach for continuous piece-wise linear functions. Global exponential convergence of the algorithm is established using Contraction Theory with Inequality Constraints, which is extended from the continuous to the discrete case in this paper: The parametrization of each linear function piece is, in contrast to deep learning, linear in the proposed MinMax network. This allows a linear regression stability proof as long as measurements do not transit from one linear region to its neighbouring linear region. The step size of the discrete gradient descent is Lagrangian limited orthogonal to the edge of two neighbouring linear functions. It will be shown that this Lagrangian step limitation does not decrease the convergence of the unconstrained system dynamics in contrast to a step size limitation in the direction of the gradient. We show that the convergence rate of a constrained piece-wise linear function learning is equivalent to the exponential convergence rates of the individual local linear regions.

cs.LG↗
arXiv2023

Natural Metrics in Contraction Analysis

Contraction analysis establishes exponential incremental convergence of a nonlinear system by solving a linear matrix inequality for a contraction metric, and has become a standard resource for solving problems in nonlinear control and estimation. This paper shows that, for a general nonlinear system, a contraction metric can be systematically derived by rewriting the system dynamics as a complex natural gradient dynamics. In this form, the variational dynamics can be modally decomposed with quadratic geodesic coordinates, and exact exponential convergence rates can be computed analytically. Specializing the results above to Hamiltonian systems shows that differential lengths of general Hamiltonian dynamics correspond to exact complex analytic exponential functions, whose eigenvalues can be analytically computed from the metric, damping, curvature, second covariant derivative of the potential energy, and first covariant derivative of the vector potential, a result which applies to both classical and relativistic systems. Incorporating nonlinear inequality constraints is also discussed. All derivations are tensor-based, and the computed eigenvalues themselves are coordinate-invariant, i.e., the contraction rates are independent of the chosen coordinate system. Simple examples including a gravity pendulum, gradient descent with non-convex cost, Schuler dynamics, and a two-link manipulator, illustrate that the computation of the decomposed convergence rates is straightforward. The role of inequality constraints is illustrated for a controller confined to an operational envelope.

math.DS↗
arXiv2016

Analytical SLAM Without Linearization

This paper solves the classical problem of simultaneous localization and mapping (SLAM) in a fashion which avoids linearized approximations altogether. Based on creating virtual synthetic measurements, the algorithm uses a linear time- varying (LTV) Kalman observer, bypassing errors and approximations brought by the linearization process in traditional extended Kalman filtering (EKF) SLAM. Convergence rates of the algorithm are established using contraction analysis. Different combinations of sensor information can be exploited, such as bearing measurements, range measurements, optical flow, or time-to-contact. As illustrated in simulations, the proposed algorithm can solve SLAM problems in both 2D and 3D scenarios with guaranteed convergence rates in a full nonlinear context.

cs.RO↗
arXiv2010

Shaping state and time-dependent convergence rates in non-linear control and observer design

This paper derives for non-linear, time-varying and feedback linearizable systems simple controller designs to achieve specified state-and timedependent complex convergence rates. This approach can be regarded as a general gain-scheduling technique with global exponential stability guarantee. Typical applications include the transonic control of an aircraft with strongly Mach or time-dependent eigenvalues or the state-dependent complex eigenvalue placement of the inverted pendulum. As a generalization of the LTI Luenberger observer a dual observer design technique is derived for a broad set of non-linear and time-varying systems, where so far straightforward observer techniques were not known. The resulting observer design is illustrated for non-linear chemical plants, the Van-der-Pol oscillator, the discrete logarithmic map series prediction and the lighthouse navigation problem. These results [23] allow one to shape globally the state- and time-dependent convergence behaviour ideally suited to the non-linear or time-varying system. The technique can also be used to provide analytic robustness guarantees against modelling uncertainties. The derivations are based on non-linear contraction theory [18], a comparatively recent dynamic system analysis tool whose results will be reviewed and extended.

nlin.CD↗
arXiv2005

Higher-Order Nonlinear Contraction Analysis

Nonlinear contraction theory is a comparatively recent dynamic control system design tool based on an exact differential analysis of convergence, in essence converting a nonlinear stability problem into a linear time-varying stability problem. Contraction analysis relies on finding a suitable metric to study a generally nonlinear and time-varying system. This paper shows that the computation of the metric may be largely simplified or indeed avoided altogether by extending the exact differential analysis to the higher-order dynamics of the nonlinear system. Simple applications in economics, classical mechanics, and process control are described.

nlin.PS↗
arXiv2004

Contraction Analysis of Nonlinear Distributed Systems

Contraction theory is a recently developed dynamic analysis and nonlinear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of nonlinear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton-Jacobi dynamics, and optimal controllers and observers. The Hamilton-Jacobi-Bellman controller and a similar optimal nonlinear observer design are studied. Explicit stability conditions are given, which extend the well-known conditions on controllability and observability Grammians for linear time-varying systems. Stability of the Hamilton-Jacobi dynamics is assessed by evaluating the Hessian of the system state along system trajectories. In contrast to stability proofs based on energy dissipation,this principle allows to conclude on stability of energy-based systems that are excited by time-varying inputs. In this context, contraction can be regarded as describing new variational conservation laws and the stability of entropy producing processes.

math-ph↗