Conservation Laws for Dynamical System Via Generalized Differential Variational Principles of Jourdian and Gauss

2018 ◽  
Author(s):  
Amjad Hussain ◽  
Muhammad Asim
Keyword(s):  
Dynamical System ◽  
Conservation Laws ◽  
Variational Principles ◽  
Generalized Differential

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.

Variational principles of guiding centre motion

1983 ◽  
Vol 29 (1) ◽  
pp. 111-125 ◽  
Author(s):  
Robert G. Littlejohn
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Equations Of Motion ◽  
Phase Volume ◽  
Rigorous Derivation ◽  
Volume Energy ◽  
Symmetric Systems

An elementary but rigorous derivation is given for a variational principle for guiding centre motion. The equations of motion resulting from the variational principle (the drift equations) possess exact conservation laws for phase volume, energy (for time-independent systems), and angular momentum (for azimuthally symmetric systems). The results of carrying the variational principle to higher order in the adiabatic parameter are displayed. The behaviour of guiding centre motion in azimuthally symmetric fields is discussed, and the role of angular momentum is clarified. The application of variational principles in the derivation and solution of gyrokinetic equations is discussed.

Simplifying Primitive Equations: Rotating Shallow-Water Models and their Properties

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Variational Principles ◽  
Mean Field ◽  
Tangent Plane ◽  
Primitive Equations ◽  
Water Model ◽  
Shallow Water Model ◽  
Shallow Water Models ◽  
Rotating Shallow Water

In this chapter, one- and two-layer versions of the rotating shallow-water model on the tangent plane to the rotating, and on the whole rotating sphere, are derived from primitive equations by vertical averaging and columnar motion (mean-field) hypothesis. Main properties of the models including conservation laws and wave-vortex dichotomy are established. Potential vorticity conservation is derived, and the properties of inertia–gravity waves are exhibited. The model is then reformulated in Lagrangian coordinates, variational principles for its one- and two-layer version are established, and conservation laws are reinterpreted in these terms.

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