Abstract
Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.
Highlights
Birkhoffian mechanics can be traced back to Birkhoff’s monograph Dynamical Systems, which gave a new class of dynamic equations more common than Hamilton’s canonical equations and a new class of integral variational principles more common than Hamilton’s principle [1]
If the old variables as and the new variables as ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F4ðt, as, asÞ, the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system
It is because Birkhoff systems have many good properties, such as autonomous and semiautonomous Birkhoff systems have a Lie algebraic structure and proper symplectic form and Birkhoff’s equations have self-adjoint form, that Birkhoff systems are widely used in physics, mechanics, engineering, and other fields
Summary
Birkhoffian mechanics can be traced back to Birkhoff’s monograph Dynamical Systems, which gave a new class of dynamic equations more common than Hamilton’s canonical equations and a new class of integral variational principles more common than Hamilton’s principle [1]. The classical Hamilton canonical transformation theory plays an important role in solving dynamic equations. Santilli first proposed and preliminarily studied the transformation theory of Birkhoff’s equations and only gave one kind of generating function and its transformation [2]. In this paper, based on the basic identity of Birkhoff system’s generalized canonical transformation, we derive six generalized canonical transformation formulas by selecting different forms of generating functions and give the transformation relations between the old and new variables in each case. The canonical transformation formulas of classical Hamilton’s equations are the special cases of the generalized canonical transformation formulas of Birkhoff systems.
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