Volume 16, Issue 6
The Calculus of Generating Functions and the Formal Energy for Hamiltonian Algorithms

Kang Feng

DOI:

J. Comp. Math., 16 (1998), pp. 481-498

Published online: 1998-12

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  • Abstract

In [2--4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.

  • Keywords

Generating function calculus of generating functions Darboux transformation cotangent bundles Lagrangian submanifold invariance of generating function formal energy

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@Article{JCM-16-481, author = {}, title = {The Calculus of Generating Functions and the Formal Energy for Hamiltonian Algorithms}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {6}, pages = {481--498}, abstract = { In [2--4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9176.html} }
TY - JOUR T1 - The Calculus of Generating Functions and the Formal Energy for Hamiltonian Algorithms JO - Journal of Computational Mathematics VL - 6 SP - 481 EP - 498 PY - 1998 DA - 1998/12 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9176.html KW - Generating function KW - calculus of generating functions KW - Darboux transformation cotangent bundles KW - Lagrangian submanifold KW - invariance of generating function KW - formal energy AB - In [2--4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.
Kang Feng. (1970). The Calculus of Generating Functions and the Formal Energy for Hamiltonian Algorithms. Journal of Computational Mathematics. 16 (6). 481-498. doi:
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