scholarly journals The Jacobi-Maupertuis Principle in Variational Integrators

Author(s):  
Sujit Nair ◽  
Sina Ober-Blöbaum ◽  
Jerrold E. Marsden ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
2021 ◽  
Vol 2 (3) ◽  
pp. 431-441
Author(s):  
Odysseas Kosmas

In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.


1999 ◽  
Vol 40 (7) ◽  
pp. 3353-3371 ◽  
Author(s):  
C. Kane ◽  
J. E. Marsden ◽  
M. Ortiz

PAMM ◽  
2013 ◽  
Vol 13 (1) ◽  
pp. 33-34
Author(s):  
Odysseas Kosmas ◽  
Sigrid Leyendecker

2014 ◽  
Vol 11 (08) ◽  
pp. 1450066 ◽  
Author(s):  
Antonia Karamatskou ◽  
Hagen Kleinert

In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.


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