scholarly journals PHASE PORTRAITS OF THE HENON-HEILES POTENTIAL

2018 ◽  
pp. 5-9
Author(s):  
E. Malkov ◽  
S. Momynov
Keyword(s):  
Numerical Integration ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Analytical Form ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Poincaré Section ◽  
Dimensional Problem ◽  
Poincare Section

In this paper the Henon-Heiles potential is considered. In the second half of the 20th century, in astronomy the model of motion of stars in a cylindrically symmetric and time-independent potential was studied. Due to the symmetry of the potential, the three-dimensional problem reduces to a two-dimensional problem; nevertheless, finding the second integral of the obtained system in the analytical form turns out to be an unsolvable problem even for relatively simple polynomial potentials. In order to prove the existence of an unknown integral, the scientists Henon and Heiles carried out an analysis of research for trajectories in which the method of numerical integration of the equations of motion is used. The authors proposed the Hamiltonian of the system, which is fairly simple, which makes it easy to calculate trajectories, and is also complex enough that the resulting trajectories are far from trivial. At low energies, the Henon-Heiles system looks integrable, since independently of the initial conditions, the trajectories obtained with the help of numerical integration lie on two-dimensional surfaces, i.e. as if there existed a second independent integral. Equipotential curves, the momentum and coordinate dependences on time, and also the Poincaré section were obtained for this system. At the same time, with the increase in energy, many of these surfaces decay, which indicates the absence of the second integral. It is assumed that the obtained numerical results will serve as a basis for comparison with analytical solutions. Keywords: Henon-Heiles model, Poincaré section, numerical solutions.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

A rapidly convergent procedure for solving the equations of subsonic potential flow. I. Numerical solutions

1960 ◽  
Vol 255 (1280) ◽  
pp. 101-113 ◽  
Keyword(s):  
Potential Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Local Density ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Approximation Solution ◽  
Flow Equations ◽  
Bernoulli’S Equation ◽  
First Approximation

The subsonic potential flow equations for a perfect gas are transformed by means of dependent variables s = ( ρ / ρ 0 ) n q/ a 0 and σ = 1/2 In ( ρ 0 / ρ ), where q is the local velocity, ρ and a the local density and speed of sound, and the suffix 0 indicates stagnation conditions, n is a parameter which is to be chosen to optimize the approximations. Bernoulli’s equation then becomes a relation between s 2 and σ which is independent of initial conditions. A family of first-approximation solutions in terms of the incompressible solution is obtained on linearizing. It is shown that for two-dimensional flow, the choice n = 0∙5 gives results as accurate as those obtained with the Karman—Tsien solution. The exact equations are then transformed into the plane of the incompressible velocity potential and stream function and the first-approximation results substituted in the non ­linear terms. The resulting second-approximation equations can then be solved by a relaxation method and the error in this approximation estimated by carrying out the third-approximation solution. Results are given for a circular cylinder at a free-stream Mach number, M ∞ = 0∙4, and a sphere at M ∞ = 0∙5. The error in the velocity distribution is shown to be less than ±1 % in the two-dimensional case. A rough and ready compressibility rule is formulated for axisymmetric bodies, dependent on their thickness ratios.

Local dimension of ergodic measures for two-dimensional Lorenz transformations

2000 ◽  
Vol 20 (3) ◽  
pp. 911-923 ◽  
Author(s):  
THOMAS STEINBERGER
Keyword(s):  
Probability Measures ◽  
Two Dimensional ◽  
Poincaré Section ◽  
Poincare Section ◽  
Local Dimension ◽  
Lorenz Equation ◽  
Characteristic Exponents ◽  
Ergodic Measures

A class of transformations on $[0,1]^2$, which includes transformations obtained by a Poincaré section of the Lorenz equation, is considered. We prove a formula which connects local dimension, entropy and characteristic exponents of ergodic invariant probability measures.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

scholarly journals 2D TESTING OF A FINITE DIFFERENCE SCHEME FOR MODELLING OF A VISCOUS DIFFUSION PROCESS IN COMPRESSIBLE GAS

2020 ◽  
Vol 22 (5) ◽  
pp. 128-131
Author(s):  
V.V. Nikonov ◽  
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Initial Conditions ◽  
Stokes Problem ◽  
Velocity Fields ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Time Step ◽  
Acceptable Accuracy ◽  
Compressible Gas

Viscous subproblem of direct numerical simulation of compressible gas is solved. This subproblem is tested on the two-dimensional problem of impulse start of a flat plate (Stokes’ problem). Three calculations were made with the different initial conditions and velocity fields were obtained. The numerical results are compared with the solution of Stokes’ problem. Analyzing the results, we can conclude that in order to achieve acceptable accuracy, it suffices to choose a time step according to the rule that the author formulated in his earlier works.

scholarly journals Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Fixed Point ◽  
Angular Velocity ◽  
Periodic Solutions ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
First Integral ◽  
The Body ◽  
Large Parameter

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

scholarly journals Motion of a rolling sphere on an azimuthally symmetric surface

2019 ◽  
Vol 65 (2 Jul-Dec) ◽  
pp. 128
Author(s):  
D. M. Marín Quiroz
Keyword(s):  
Numerical Solutions ◽  
Normal Force ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Lagrangian Mechanics ◽  
Wolfram Mathematica ◽  
Symmetric Surface ◽  
Translational Speed ◽  
Insight Into

This paper analyzes the translational motion that a sphere rolling over an azimuthally symmetric surface, under the presence of a constant gravitational field, and with the rolling-without-slipping condition, exhibits in two different situations: with and without friction with air, where the latter is expressed as a power-series function of the sphere’s translational speed. In order to achieve this, the equations of motion for each case are obtained through the use of Lagrangian Mechanics and are subsequently solved by numerical computation in Wolfram Mathematica. For the frictionless case, periodic behavior and a conservation law for the angular coordinate have been found, along with the condition under which an effective potential energy can be approximated as well as the relationships between initial conditions that produce gravitational-like trajectories for the motion of the sphere. The equations of motion derived for the case with friction are found to predict the energy loss and general decay of the sphere’s motion. Likewise, the normal force over the sphere as a function of time is obtained through the method of Lagrange's Undetermined Multipliers, and thus, the general conditions that the motion must satisfy in order to be described by the obtained models. Overall, this research provides insight into the type and characteristics of the motion performed by the system in these two cases, both through equations and their numerical solutions for different surfaces and initial conditions.

scholarly journals The Problem of Balancing an Inverted Spherical Pendulum on an Omniwheel Platform

2021 ◽  
Vol 17 (4) ◽  
pp. 507-525
Author(s):  
A. S. Shaura ◽  
◽  
V. A. Tenenev ◽  
E. V. Vetchanin ◽  
◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Inverted Pendulum ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Hybrid Genetic Algorithm ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Spherical Pendulum ◽  
Pendulum System ◽  
End Point

This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.

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