Conserved quantities and symmetry groups for the Kepler problem

1967 ◽  
Vol 50 (1) ◽  
pp. 95-105 ◽  
Author(s):  
A. Simoni ◽  
B. Vitale ◽  
F. Zaccaria
Keyword(s):  
Kepler Problem ◽  
Symmetry Groups ◽  
Conserved Quantities

scholarly journals On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Boussinesq Equations ◽  
Type Systems ◽  
Symmetry Groups ◽  
Conserved Quantities ◽  
Mathematical Properties ◽  
Different Types ◽  
Xxist Century ◽  
Boussinesq Models

© 2018 Elsevier B.V. The BOUSSINESQ equations are known since the end of the XIXst century. However, the proliferation of various BOUSSINESQ-type systems started only in the second half of the XXst century. Today they come under various flavors depending on the goals of the modeler. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the HAMILTONIAN. In the present paper a family of BOUSSINESQ-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known BOUSSINESQ models, the identification of those systems with additional HAMILTONIAN structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full EULER equations is also discussed.

Dynamical and symmetry groups of the SU(2) Kepler problem

2000 ◽  
Vol 33 (3-4) ◽  
pp. 326-355 ◽  
Author(s):  
Toshihiro Iwai ◽  
Takehiko Sunako
Keyword(s):  
Kepler Problem ◽  
Symmetry Groups

scholarly journals On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Boussinesq Equations ◽  
Type Systems ◽  
Symmetry Groups ◽  
Conserved Quantities ◽  
Mathematical Properties ◽  
Different Types ◽  
Xxist Century ◽  
Boussinesq Models

© 2018 Elsevier B.V. The BOUSSINESQ equations are known since the end of the XIXst century. However, the proliferation of various BOUSSINESQ-type systems started only in the second half of the XXst century. Today they come under various flavors depending on the goals of the modeler. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the HAMILTONIAN. In the present paper a family of BOUSSINESQ-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known BOUSSINESQ models, the identification of those systems with additional HAMILTONIAN structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full EULER equations is also discussed.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Dynamical System ◽  
Angular Momentum ◽  
Conservation Laws ◽  
Total Energy ◽  
Superposition Principle ◽  
Momentum Conservation ◽  
Conserved Quantities ◽  
Angular Momentum Conservation ◽  
Third Law ◽  
The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

scholarly journals On the accuracy of the binary-collision algorithm in particle-in-cell simulations of magnetically confined fusion plasmas

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Timo P. Kiviniemi ◽  
Eero Hirvijoki ◽  
Antti J. Virtanen
Keyword(s):  
Finite Difference ◽  
Electric Fields ◽  
Finite Size ◽  
Binary Collision ◽  
Conserved Quantities ◽  
Fusion Plasmas ◽  
Fusion Plasma ◽  
Particle In Cell ◽  
Magnetically Confined ◽  
Scale Length

Ideally, binary-collision algorithms conserve kinetic momentum and energy. In practice, the finite size of collision cells and the finite difference in the particle locations affect the conservation properties. In the present work, we investigate numerically how the accuracy of these algorithms is affected when the size of collision cells is large compared with gradient scale length of the background plasma, a parameter essential in full- $f$ fusion plasma simulations. Additionally, we discuss implications for the conserved quantities in drift-kinetic formulations when fluctuating magnetic and electric fields are present: we suggest how the accuracy of the algorithms could potentially be improved with minor modifications.

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