Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

2007 ◽  
Vol 152 (2) ◽  
pp. 1111-1121 ◽  
Author(s):  
A. Yu. Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Dimensional Case ◽  
Finite Dimensional ◽  
Taylor Approximation ◽  
Finite Dimensional Case

scholarly journals From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 8 ◽  
Author(s):  
François Gay-Balmaz ◽  
Hiroaki Yoshimura
Keyword(s):  
Nonequilibrium Thermodynamics ◽  
Classical Mechanics ◽  
Discrete Systems ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Irreversible Processes ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Continuum Systems

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite-dimensional case of discrete systems, as well as for the infinite-dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton’s principle, we show, with the help of thermodynamic systems with gradually increasing complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat, and mass transfer in both the adiabatically closed cases and open cases. On the continuum side, we illustrate our theory using the example of multicomponent Navier–Stokes–Fourier systems.

ALTERNATIVE HAMILTONIAN DESCRIPTION FOR QUANTUM SYSTEMS

1990 ◽  
Vol 05 (15) ◽  
pp. 1229-1234 ◽  
Author(s):  
B. A. DUBROVIN ◽  
G. MARMO ◽  
A. SIMONI
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Dimensional Case ◽  
Hamiltonian Description ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Kähler Structures ◽  
Time Invariant ◽  
Classical And Quantum Mechanics

The existence of time-invariant Kähler structures is analyzed in both Classical and Quantum Mechanics. In Quantum Mechanics, a family of such Kähler structures is found, in the finite-dimensional case it is proven that this family is complete.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

scholarly journals Approximations of Variational Problems in Terms of Variational Convergence

2017 ◽  
Vol 20 (K2) ◽  
pp. 107-116
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Multiobjective Optimization ◽  
Metric Space ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Variational Problems ◽  
Dimensional Case ◽  
Variational Convergence ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Definition Of

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.

scholarly journals Examples of S—expansions of Lie Algebras

2021 ◽  
Vol 2090 (1) ◽  
pp. 012067
Author(s):  
G. Javier Rosales
Keyword(s):  
Lie Algebras ◽  
Three Dimensional ◽  
Infinite Dimension ◽  
Dimensional Case ◽  
Finite Dimensional ◽  
Finite Dimensional Case

Abstract In this note, we give examples of S—expansions of Lie algebras of finite and infinite dimension. For the finite dimensional case, we expand all real three-dimensional Lie algebras. In the case of infinite dimension, we perform contractions obtaining new Lie algebras of infinite dimension.

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