Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

Impact of a Bumpy Nonuniform Electric Field on Oscillations of a Massive Point-Like Charged Particle

As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of a charged ring (being a localized charge distribution) along its symmetry axis perpendicular to the ring through its center rather than as expected being a diminishing field encounters a local maximum bump. It is the objective of this research-oriented study to analyze the impact of this bump on the characteristics of a massive point-like charged particle oscillating along the symmetry axis. Two scenarios with and without gravity along the symmetry axis are considered. In addition to standard kinematic diagrams, various phase diagrams conducive to a better understanding are constructed. Applying Computer Algebra System (CAS), [1] [2] most calculations are carried out symbolically. Finally, by assigning a set of reasonable numeric parameters to the symbolic quantities various 3D animations are crafted. All the CAS codes are included.

Three Dimensional Electric Circuits with Multiple Capacitors and Resistors

Two cubical 3D electric circuits with single and double capacitors and twelve ohmic resistors are considered. The resistors are the sides of the cube. The circuit is fed with a single internal emf. The charge on the capacitor(s) and the current distributions of all twelve sides of the circuit(s) vs. time are evaluated. The analysis requires solving twelve differential-algebraic intertwined symbolic equations. This is accomplished by applying a Computer Algebra System (CAS), specifically Mathematica. The needed codes are included. For a set of values assigned to the elements, the numeric results are depicted.

3D_Multi Resistor Electric Circuit

This report addresses the issues concerning the analysis of an electric circuit composed of multiple resistors configured in a 3-Dimension structure. Noting, all the standard textbooks of physics and engineering irrespective of the used components are circuits assembled in two dimensions. Here, by deviating from the “norm” we consider a case where the resistors are arranged in a 3D structure;e.g., a cube. Although, independent of the dimension of the design the same physics principles apply, transitioning from a 2D to a 3D makes the corresponding analysis considerably challenging. In general, with no exception, depending on the used components the analysis faces with solving a set of either algebraic or differential-algebraic equations. Practically, this interfaces with a Computer Algebra System (CAS). The main objective is symbolically to identify the current distributions and the equivalent resistor (s) of cubically assembled resistors.

Analyzing Electric Circuits with Computer Algebra

This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.

Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System

By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.

Impact of the Drag Force and the Magnus Effect on the Trajectory of a Baseball

We consider the impact of drag force and the Magnus effect on the motion of a baseball. Quantitatively we show how the speed-dependent drag coefficient alters the trajectory of the ball. For the Magnus effect we envision a scenario where the rotation of the ball confines the Magnus force to the vertical plane;gravity, drag force and the Magnus force make a trio-planar system. We investigate the interplay of these forces on the trajectories.

Quantum Standing Waves and Tunneling Through a Finite Range Potential

We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.

Negative Resistance and Its Impact on a RC-DC Driven Electric Circuit

It is a common misconception that electric “resistance” always is a positive defined electric element. i.e., the plot of the voltage across the resistor, V vs. its current, i is a slanted straight line with a positive slope. Esaki diode also known as tunnel diode is an exception to this character. For a certain voltage range, the current recedes resulting in a line with a negative slope;it is interpreted as negative resistance. In this research flavored report, we investigate the impact of the negative resistance in a typical classic electric circuit. E.g., a tunnel diode, D is inserted in a classic electric circuit that is composed of an ohmic resistor, R and a capacitor, C which are all in series with a DC power supply. The circuit equation for the RCD circuit is a nonlinear ordinary differential equation (NLODE). In line with the ever-growing popular Computer Algebra System (CAS), this is solved numerically utilizing two distinctly different CASs. The consistency of the solutions confidently leads to the understanding of the impact of the negative resistance. The circuit characteristics are compared to the classic analogous RC circuit. The report embodies an atlas of characteristics of the circuits making the analysis visually comprehensible.

Impact of Eccentricity on Nonlinear Oscillations of a Point-Like Charge in the Electric Field of a Curvature-Dependent Elliptic Charged Ellipse

Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly Mathematica [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.

Envelope of Family of Angled Projectiles and Its Universal Geometric Characteristics

Geometric properties of trajectories of angled projectiles under gravity pull are a popular common traditional theme discussed in introductory physics and engineering college courses. What is overlooked is the universal collective properties of the overarching specificities of families of such parabolas, the envelope. For instance [1] and references within explored the existence of one such envelope, however, even the most recent article [2] overlooked its global hidden properties. Here, we investigate exposing this hidden information. Having the equation of the envelope on hand we introduce its universal characteristics such as its: arc length, enclosed 2D surface area, surface area of the surface-of-revolution about the symmetry axis, and, the volume of the enclosure. Numeric values of these quantities are global as is e.g. the 45° projectile angle that maximizes the range of a projectile in vacuum irrespective, its initial speed. In our exploratory investigation, we utilize the popular Computer Algebra System (CAS) MathematicaTM [3] [4] [5].

Maxwell-Boltzmann Distribution Mathematica Based Practitioner’s Approach

Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut circling the usage of antiquated incomplete tabulated error functions given in the textbooks and professional literature replacing them with efficient upgrades. And, utilizing the animation features of Mathematica displaying the temperature-dependence of the distribution function assists in visualizing the character of the distribution.

Designing Physics Problems with Mathematica

We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem showing its development from the ground up. Following the objectives of this investigation first by applying the fundamentals of physics principles we solve the problem symbolically. Applying the solution we investigate the sensitivities of the quantities of interest for various scenarios generating feasible numeric parameters. Although a physics problem is investigated, the proposed methodology may as well be applied to other scientific fields. The codes needed for this particular project are included enabling the interested reader to duplicate the results, extend and modify them as needed to explore various extended scenarios.

Symbolic Computation and Graphic Presentation of Magneto-Static Field and Its Associated Vector Potential for a Steady Looping Current

With the advent of Computer Algebra System (CAS) such as Mathematica [1], challenging symbolic longhand calcula-tions can effectively be performed free of error and at ease. Mathematica’s integrated features allow the investigator to combine the needed symbolic, numeric and graphic modules all in one interactive environment. This assists the author to focus on interpreting the output rather than exerting the efforts of relating the scattered separate modules. In this note the author, utilizing these three features, explores the magneto-static field and its associated vector potential of a steady looping current. In particular by deploying the numeric features of Mathematica the exact value of the vector potential of the looping current conducive to its 3D graph is presented.

Angular Impulse and Spinning Bouncing Ball

A solid ball of mass m, size r and spin ω about an axis through its center is dropped freely from a height h on a rough horizontal plane. Assuming its angular momentum is parallel to the horizontal plane upon impact it bounces repeatedly drifting on a vertical plane. We analyze the kinematics of the bouncing ball assuming the impacts are semi-elastic without slipping. By varying the spin and relevant parameters, a robust Mathematica [1] program enables simulating the trajectories.

Nonlinear Oscillations of a Magneto Static Spring-Mass

The Duffing equation describes the oscillations of a damped nonlinear oscillator [1]. Its non-linearity is confined to a one coordinate-dependent cubic term. Its applications describing a mechanical system is limited e.g. oscillations of a theoretical weightless-spring. We propose generalizing the mathematical features of the Duffing equation by including in addition to the cubic term unlimited number of odd powers of coordinate-dependent terms. The proposed generalization describes a true mass-less magneto static-spring capable of performing highly non-linear oscillations. The equation describing the motion is a super non-linear ODE. Utilizing Mathematica [2] we solve the equation numerically displaying its time series. We investigate the impact of the proposed generalization on a handful of kinematic quantities. For a comprehensive understanding utilizing Mathematica animation we bring to life the non-linear oscillations.

Legendre Polynomial and Nonlinear Oscillating Point-Like Charged Particle

In this article we explore the kinematics of a point-like charged particle placed within the interior plane of a charged ring. Analytically we formulate the electric field of the ring along a representative diagonal. Graph of the field as a function of the distance from the center of the ring assists foreseeing oscillating movement of the charged particle. We formulate the equation of motion;this is a nonlinear differential equation. Applying Computer Algebra System (CAS), specifically Mathematica [1] we solve the equation numerically. Utilizing the solution we quantify the kinematic quantities of interest including oscillations period. Although the equation of motion is nonlinear its period is regulated. For better understanding we take an advantage of Mathematica animation features animating the nonlinear oscillations.

Compression of Characteristics of Trajectories in Rotating Frames vs. Nonuniform Magnetic Fields

The equation of motion of an object moving in a frictionless horizontal rotating frame is somewhat comparable to the one describing the motion of a point-like charged particle projected in a magnetic field. We show that the impact of angular velocity in the former is equivalent to the impact of the magnetic field in the latter. We consider scenarios conducive to comparable trajectories for these two distinct areas of physics. We extend the analysis considering two separate routes. For the rotating frame we investigate the impact of friction and for the magnetic field the effect of field in-homogeneities. We utilize Mathematica [1] throughout, most notably for solving coupled partial differential equations.

Rotating Capacitor and a Transient Electric Network

This paper presents a rotating parallel-plate capacitor;one of the plates is assumed to turn about the common vertical axis through the centers of the square plates. Viewing the problem from a purely geometrical point of view, we evaluate the overlapping area of the plates as a function of the rotated angle. We then envision the rotation as being a mechanical continuous process. We consider two different rotation mechanisms: a uniform rotation with a constant angular velocity and, a rotation with a constant angular acceleration—we then evaluate the overlapping area as a continuous function of time. From the electrostatic point of view, the time-dependent overlapping area of the plates implies a time-dependent capacitor. Such a variable, a time-dependent capacitor has never been reported in literature. We insert this capacitor into a series with a resistor, forming a RC circuit. We analyze the characteristics of charging and discharging scenarios on two different parallel tracks. On the first track we drive the circuit with a DC power sup-ply. We study the implications of the rotation modes. We compare the response of each case to the corresponding tradi-tional constant capacitor of an equivalent RC circuit;the quantified results are intuitively just. On the second track, we drive the circuit with an AC source. Similar to the analysis of the first track, we generate the relevant electrical characteristics. In the latter case, we also analyze the sensitivity of the response of the circuit with respect to the fre-quency of the source. The analyses of the circuits encounter nontrivial differential equations. We utilize Mathematica [1] to solve these equations.

Characteristics of a Two-Body Holonomic Constraint Mechanical System

Two massive blocks are connected with a massless unstretchable line of 2l. One of the masses is placed on a horizontal frictionless table, l distance away from the edge of the table the other one is held horizontally equidistance from the edge along the extension of the line. The latter is released from rest. As it falls under gravity’s pull, it drags the one on the table. It is the interest of this investigation to analyze the kinematics of the system. Because of the holonomic constraint of the system, analysis of the problem encounters complicated super nonlinear coupled differential equations. Utilizing Mathematica we solve the equations numerically. Applying the solutions we quantify numerous kinematic quantities;most interestingly we evaluate the run-time, and the trajectory of the falling block. Analysis is robust allowing us to address the “what if” scenarios.