The Jacobi Energy Function

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Conservation Laws ◽  
Total Energy ◽  
Time Dependence ◽  
Energy Function ◽  
Tangent Bundle ◽  
Implicit Time ◽  
Translational Invariance ◽  
Explicit Time Dependence ◽  
Generalised Coordinates ◽  
Lagrange Formalism

This chapter focuses on the Jacobi energy function, considering how the Lagrange formalism treats the energy of the system. This discussion leads nicely to conservation laws and symmetries, which are the focus of the next chapter. The Jacobi energy function associated with a Lagrangian is defined as a function on the tangent bundle. The chapter also discuss explicit vs implicit time dependence, and shows how time translational invariance ensures the generalised coordinates are inertial, meaning that the energy function is the total energy of the system. In addition, it examines the energy function using non-inertial coordinates and explicit time dependence.

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 New Results on the Coagulation Equation

1987 ◽  
Vol 115 ◽  
pp. 547-547
Author(s):  
Pierre Bastien ◽  
Claude Lejeune
Keyword(s):  
Mass Spectrum ◽  
Time Dependence ◽  
Star Clusters ◽  
Mass Function ◽  
Mass Dependence ◽  
Coagulation Rate ◽  
Precise Form ◽  
Explicit Time Dependence ◽  
Coagulation Equation ◽  
Free Parameters

In attempting to reproduce the initial stellar mass function, we solved analytically the coagulation equation with an explicit time dependence in the coagulation rate in order to simulate the gravitational collapse of the fragments upon themselves as they move within the progenitor cloud. Two separate cases have been studied, with and without a mass dependence in the coagulation rate. The solution show that (1) inclusion of self-gravitation is very important and can change the results to the point of preventing coalescence to work altogether, depending on the values of the two free parameters, (2) the precise form of the mass dependence of the coagulation rate is not of prime importance in most situations of astrophysical interest, and (3) coagulation alone is not sufficient to yield a realistic mass spectrum and fragmentation must also be taken into account. Coagulation is more efficient for massive fragments and fragmentation for the smaller ones. These results are applied to different regions: star clusters, associations, and starburst regions.

scholarly journals RESTORING TIME DEPENDENCE INTO QUANTUM COSMOLOGY

2012 ◽  
Vol 21 (11) ◽  
pp. 1242011 ◽  
Author(s):  
AHARON DAVIDSON ◽  
BEN YELLIN
Keyword(s):  
Quantum Theory ◽  
Time Dependence ◽  
Quantum Cosmology ◽  
Scale Factor ◽  
Time Dependent ◽  
Hamiltonian Constraint ◽  
Canonical Variables ◽  
Explicit Time Dependence ◽  
Dirac Brackets ◽  
Lapse Function

Mini superspace cosmology treats the scale factor a(t), the lapse function n(t) and an optional dilation field ϕ(t) as canonical variables. While pre-fixing n(t) means losing the Hamiltonian constraint, pre-fixing a(t) is serendipitously harmless at this level. This suggests an alternative to the Hartle–Hawking approach, where the pre-fixed a(t) and its derivatives are treated as explicit functions of time, leaving n(t) and a now mandatory ϕ(t) to serve as canonical variables. The naive gauge pre-fix a(t) = const . is clearly forbidden, causing evolution to freeze altogether; so pre-fixing the scale factor, say a(t) = t, necessarily introduces explicit time dependence into the Lagrangian. Invoking Dirac's prescription for dealing with constraints, we construct the corresponding mini superspace time-dependent total Hamiltonian and calculate the Dirac brackets, characterized by {n, ϕ}D ≠ 0, which are promoted to commutation relations in the quantum theory.

Towards Translational Invariance of Total Energy with Finite Element Methods for Kohn-Sham Equation

2016 ◽  
Vol 19 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Gang Bao ◽  
Guanghui Hu ◽  
Di Liu
Keyword(s):  
Finite Element ◽  
Total Energy ◽  
Finite Element Methods ◽  
Electronic System ◽  
Hamiltonian Operator ◽  
Real Space ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Translational Invariance ◽  
Sham Equation

AbstractNumerical oscillation of the total energy can be observed when the Kohn- Sham equation is solved by real-space methods to simulate the translational move of an electronic system. Effectively remove or reduce the unphysical oscillation is crucial not only for the optimization of the geometry of the electronic structure, but also for the study of molecular dynamics. In this paper, we study such unphysical oscillation based on the numerical framework in [G. Bao, G. H. Hu, and D. Liu, An h-adaptive finite element solver for the calculations of the electronic structures, Journal of Computational Physics, Volume 231, Issue 14, Pages 4967–4979, 2012], and deliver some numerical methods to constrain such unphysical effect for both pseudopotential and all-electron calculations, including a stabilized cubature strategy for Hamiltonian operator, and an a posteriori error estimator of the finite element methods for Kohn-Sham equation. The numerical results demonstrate the effectiveness of our method on restraining unphysical oscillation of the total energies.

A Conservation Theorem for Simple Nonholonomic Systems

1983 ◽  
Vol 50 (3) ◽  
pp. 647-651 ◽  
Author(s):  
T. R. Kane ◽  
A. K. Banerjee
Keyword(s):  
Time Dependence ◽  
Nonholonomic System ◽  
Nonholonomic Systems ◽  
Holonomic System ◽  
Explicit Time Dependence ◽  
Conservation Theorem

When the Hamiltonian of a holonomic system is free of explicit time dependence it remains constant throughout all motions of the system. In this paper, it is shown how, given a homogeneous simple nonholonomic system S, one can form a function E that remains constant throughout all motions of S, providing the forces acting on S fulfill certain requirements. An illustrative example is examined in detail.

Sign in / Sign up

Export Citation Format

Share Document