Related papers: Differential Equations of Mathematical Physics
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. As a key feature, the laws are formulated in terms of quantum events rather than of particle states. Temporal and spatial…
Course material for mathematical methods of theoretical physics intended for an undergraduate audience.
The paper shows the relationship between the major wave equations in quantum mechanics and electromagnetism, such as Schroedinger's equation, Dirac's equation and the Maxwell equations. It is shown that they can be derived in a striking…
This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretisation of such problems does not necessarily yield Hamiltonian dynamics and…
Differentiable programming allows for derivatives of functions implemented via computer code to be calculated automatically. These derivatives are calculated using automatic differentiation (AD). This thesis explores two applications of…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
This paper presents a brief account of the important milestones in the historical development of the theory of differential equations. The paper begins with a discussion on the date of birth of differential equations and then touches upon…
What forms will have an equations of modern physics if the dimensions of our time and space are fractional? The generalized equations enumerated by title are presented by help the generalized fractional derivatives of Riemann-Liouville.
The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known…
Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but…
The most common physical formalisms are the Lagrangian formalism and the Hamiltonian formalism. From the superficial point of view, they are one and the same, but rewritten in other terms. However, it seems that the Hamiltonian formalism…
These lectures are intended to provide the theoretical basis of describing high-energy particle collisions at a level appropriate to graduate students in experimental high energy physics. They are supposed to be familiar with quantum…
Quantum Physics-Informed Neural Networks (QPINNs) integrate quantum computing and machine learning to impose physical biases on the output of a quantum neural network, aiming to either solve or discover differential equations. The approach…
Variational principles are a unifying mathematical framework across many areas of physics, yet their instruction at the undergraduate level remains primarily analytical. This work presents a pedagogically oriented and computationally…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
Differential forms provide a coordinate-free way to express many quantities and relations in mathematical physics. In particular, they are useful in plasma physics. This tutorial gives a guide so that you can read the plasma physics…
We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra,…
The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In…
The indeterministic character of physical laws is generally considered to be the most important consequence of quantum physics. A deterministic point of view, however, together with the possibility of well defined Hamiltonian trajectories,…