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Register a new userUsing Mathematica software to solve ordinary differential equations and applying it to the graphical representation of trajectories
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We discuss the great importance of using mathematical software in solving problems in todays society. In particular, we show how to use Mathematica software to solve ordinary differential equations exactly and numerically. We also show how to represent these solutions graphically. We treat the particular case of a charged particle subject to an oscillating electric field in the xy plane and a constant magnetic field. We show how to construct the equations of motion, defined by the vectors position, velocity, electric and magnetic fields. We show how to solve these equations of Lorentz force, and graphically represent the possible trajectories. We end by showing how to build a video simulation for an oscillating electric field trajectory particle case.
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We propose a physical analogy between finding the solution of an ordinary differential equation (ODE) and a $N$ particle problem in statistical mechanics. It uses the fact that the solution of an ODE is equivalent to obtain the minimum of a functional. Then, we link these two notions, proposing this functional to be the interaction potential energy or thermodynamic potential of an equivalent particle problem. Therefore, solving this statistical mechanics problem amounts to solve the ODE. If only one solution exists, our method provides the unique solution of the ODE. In case we treat an eigenvalue equation, where infinite solutions exist, we obtain the absolute minimum of the corresponding functional or fundamental mode. As a result, it is possible to establish a general relationship between statistical mechanics and ODEs which allows not only to solve them from a physical perspective but also to obtain all relevant thermodynamical equilibrium variables of that particle system related to the differential equation.
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Waveform Relaxation method (WR) is a beautiful algorithm to solve Ordinary Differential Equations (ODEs). However, because of its poor convergence capability, it was rarely used. In this paper, we propose a new distributed algorithm, named Waveform Transmission Method (WTM), by virtually inserting waveform transmission lines into the dynamical system to achieve distributed computing of extremely large ODEs. WTM has better convergence capability than the traditional WR algorithms.
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