Related papers: Making Sense of the Legendre Transform
Noether's Theorem is familiar to most physicists due its fundamental role in linking the existence of conservation laws to the underlying symmetries of a physical system. Typically the systems are described in the particle-based context of…
Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to…
Serious mathematical defect in the important kinematics theorem known in continuum mechanics as Convection (or Transport) Theorem is reported. We claim that the traditional demonstration does not take into account a special constraint on…
We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The usual functional Gamma does…
This note presents an attempt to provide a conceptual framework for variational formulations of classical physics. Variational principles of physics have all a common source in the {\it principle of virtual work} well known in statics of…
The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack…
It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate and Legendre's…
The main purpose of this paper is to propose some interesting number theory problems related to the Legendre's symbol and the two-term exponential sums.
The transition from quantum to classical statistics is studied in light of Huggett's finding that the empirical data do not support the usual claim that the distinction between classical and quantum objects consists in the capacity of…
The gap between classical mechanics and quantum mechanics has an important interpretive implication: the Universe must have an irreducible fundamental level, which determines the properties of matter at higher levels of organization. We…
We discuss several ways of illustrating fundamental concepts in statistical and thermal physics by considering various models and algorithms. We emphasize the importance of replacing students' incomplete mental images by models that are…
Most educational literature on conceptual change concerns the process by which introductory students acquire scientific knowledge. However, with modern developments in science and technology, the social significance of learning successive…
Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
One of the key challenges in quantum machine learning is finding relevant machine learning tasks with a provable quantum advantage. A natural candidate for this is learning unknown Hamiltonian dynamics. Here, we tackle the supervised…
It is demonstrated that any statistics can be represented by an attractor of the solution to a corresponding systen of ODE coupled with its Liouville equation. Such a non-Newtonian representation allows one to reduce foundations of…
Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, \textit{what is thermodynamics?}, has remained…
This is the material for two lectures given at Ecole Polytechnique in May 2011 for the math teachers of "classes pr\'eparatoires"(parallel to the undergraduate classes in universities). The introduction is a personal overview on Fourier…
Legendre transform between thermodynamic quantities such as the Helmholtz free energy and entropy plays a key role in the formulation of the canonical ensemble. In the standard treatment, the transform exchanges the independent variable…
The formalism of quantum mechanics is presented in a way that its interpretation as a classical field theory is emphasized. Two coupled real fields are defined with given equations of motion. Densities and currents associated to the fields…