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A least action principle for damping motion has been previously proposed with a Hamiltonian and a Lagrangian containing the energy dissipated by friction. Due to the space-time nonlocality of the Lagrangian, mathematical uncertainties…
This work is an analytical calculation of the path probability for random dynamics of mechanical system described by Langevin equation with Gaussian noise. The result shows an exponential dependence of the probability on the action. In the…
The Principle of Least Action has evolved and established itself as the most basic law of physics. This allows us to see how this fundamental law of nature determines the development of the system towards states with less action, i.e.,…
We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly…
Despite the importance of the variational principles of physics, there have been relatively few attempts to consider them for a realistic framework. In addition to the old teleological question, this paper continues the recent discussion…
Formulating the equations of motion for cosmological bodies (such as galaxies) in an integral, rather than differential, form has several advantages. Using an integral the mathematical instability at early times is avoided and the boundary…
Machine Learning algorithms are typically regarded as appropriate optimization schemes for minimizing risk functions that are constructed on the training set, which conveys statistical flavor to the corresponding learning problem. When the…
The principle of least action provides a holistic worldview in which nature in its entirety and every detail is pictured in terms of actions. Each and every action is ultimately composed of one or multiples of the most elementary action…
We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles…
In this note we review the basic mathematical ideas used in finance in the language of modern physics. We focus on discrete time formalism, derive path integral and Green's function formulas for pricing. We also discuss various risk…
Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the…
Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none…
In numerical studies of diffusive dynamics, two different action functionals are often used to specify the probability distribution of trajectories, one of which requiring the evaluation of the second derivative of the potential in addition…
In this work, we propose an interesting method that aims to approximate an activation function over some domain by polynomials of the presupposing low degree. The main idea behind this method can be seen as an extension of the ordinary…
Nature provides a way to understand physics with reinforcement learning since nature favors the economical way for an object to propagate. In the case of classical mechanics, nature favors the object to move along the path according to the…
The least action principle occupies a central part in contemporary physics. Yet, as far as classical field theory is concerned, it may not be as essential as generally thought. We show with three detailed examples of classical interacting…
In spirit of the principle of least action, which means that when a perturbation is applied to a physical system its reaction is such that it modifies its state to "agree" with the perturbation by "minimal" change of its initial state. In…
Optimization is a major part of human effort. While being mathematical, optimization is also built into physics. For example, physics has the principle of Least Action, the principle of Minimum Entropy Generation, and the Variational…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and…