Physics

This paper provides a detailed introduction into the differential algebra (DA) based normal form algorithm using the example of the symplectic one dimensional system of the centrifugal governor. The intention of this paper is to make the single steps of the algorithm as transparent as possible in the hope that the understanding and use of DA normal form methods will spread throughout the scientific community.

The temperature of a mechanical body has a kinetic interpretation: it describes the relative motion of particles within the body. Since the relative velocity of two particles is a Lorentz invariant, so is the temperature. In statistical physics, the temperature is defined as the inverse of the partial derivative of the entropy with respect to the internal energy (the energy in the rest frame of reference). Since the internal energy is a Lorentz invariant, so is the temperature. The Lorentz invariance of the temperature is a consequence of the symmetry between two bodies with equal proper temperatures, moving relative to one another with a constant relative speed, and in thermal contact. We give an equivalent, covariant definition of the temperature in terms of the energy and momentum of the body. We also note contradictions in the earlier articles that derived various transformation laws for the internal energy and temperature.

In this paper we have chosen to work with two different approaches to solving the inverse problem of the calculus of variation. The first approach is based on an integral representation of the Lagrangian function that uses the first integral of the equation of motion while the second one relies on a generalization of the well known Noether's theorem and constructs the Lagrangian directly from the equation of motion. As an application of the integral representation of the Lagrangian function we first provide some useful remarks for the Lagrangian of the modified Emden-type equation and then obtain results for Lagrangian functions of (i) cubic-quintic Duffing oscillator, (ii) Liénard-type oscillator and (iii) Mathews-Lakshmanan oscillator. As with the modified Emden-type equation these oscillators were found to be characterized by nonstandard Lagrangians except that one could also assign a standard Lagrangian to the Duffing oscillator. We used the second approach to find indirect analytic (Lagrangian) representation for three velocity-dependent equations for (iv) Abraham-Lorentz oscillator, (v) Lorentz oscillator and (vi) Van der Pol oscillator. For each of the dynamical systems from (i)-(vi) we calculated the result for Jacobi integral and thereby provided a method to obtain the Hamiltonian function without taking recourse to the use of the so-called Legendre transformation.

We analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn't exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.

Helical torsion springs with tangential legs are used for many applications: from simple everyday use systems, like clothes peg, to advanced systems, as sectional doors. Torsion springs are usually exploited with an inner rod as a guide. The required space between the spring and the rod make the spring tilted. Unfortunately, as far as we are aware, no industrial software is able to determine the initial system angle made by the spring in its tilted position. All the torque/angle curves are presented with relative angles. It means that whatever the number of coils considered, the angle is null when the torque is null. For that reason, we have developed a methodology using both CATIA and ABAQUS to determine the system angle at the beginning of the behavior. The model exploits wires designed with CATIA and imported in ABAQUS for the center rod, the torsion spring and the rods of the system to apply the torque. Several options to model contact between the several parts have been investigated. They mostly use connector with an axial property and a stop criterion. The obtained results are very interesting and enable to plan further studies.

Kirchhoff's current law is thought to describe the translational movement of charged particles through resistors. But Kirchhoff's law is widely used to describe movements of current through resistors in high speed devices. Current at high frequencies/short times involves much more than the translation of particles. Transients abound. Augmentation of the resistors with ad hoc 'stray' capacitances is often used to introduce transients into models like those in real resistors. But augmentation hides the underlying problem, rather than solves it: the location, value and dielectric properties of the stray capacitances are not well determined. Here, we suggest a more general approach, that is well determined. If current is redefined as in Maxwell's equations, independent of the properties of dielectrics, Kirchhoff's law is exact and transients arise automatically without ambiguity. The transients in a particular real circuit-a high density integrated circuit for example-can then be described by measured constitutive equations together with Maxwell's equations without the introduction of arbitrary circuit elements.

Recently, it was shown that surface electromagnetic waves at interfaces between continuous homogeneous media (e.g., surface plasmon-polaritons at metal-dielectric interfaces) have a topological origin [K. Y. Bliokh et al., Nat. Commun. 10, 580 (2019)]. This is explained by the nontrivial topology of the non-Hermitian photon helicity operator in the Weyl-like representation of Maxwell equations. Here we analyze another type of classical waves: longitudinal acoustic waves corresponding to spinless phonons. We show that surface acoustic waves, which appear at interfaces between media with opposite-sign densities, can be explained by similar topological features and the bulk-boundary correspondence. However, in contrast to photons, the topological properties of sound waves originate from the non-Hermitian four-momentum operator in the Klein-Gordon representation of acoustic fields.

We extend Kramers-Kronig relations beyond the optical approximation, that is to dielectric functions ε(q,ω) that depend not only on the frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The results we derive do not apply exclusively to electrodynamics but also to other theories of isotropic linear response in which the response function depends on both wave number and frequency.

In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the potentials A μ are taken as the dynamical variables. In this paper I take the electric field E ??and the magnetic field B ??as the the dynamical variables. I find a Lagrangian that gives the dynamical Maxwell equations and include the constraint equations by using Lagrange multipliers. In passing to the Hamiltonian one finds that the canonical momenta ? ??E and ? ??B are constrained giving 6 second class constraints at each point in space. Gauss's law and ??????B ??=0 can than be added in as additional constraints. There are now 8 second class constraints, leaving 4 phase space degrees of freedom. The Dirac bracket is then introduced and is calculated for the field variables and their conjugate momenta.

Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyper\-elastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting.