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Generalization of Hamiltonian mechanics to a three-dimensional phase space
Progress of Theoretical and Experimental Physics Pub Date : 2021-05-19 , DOI: 10.1093/ptep/ptab066
Naoki Sato 1
Affiliation  

Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic $2$-form, a closed second-order differential form. Due to closure, the symplectic $2$-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic $2$-form is replaced by a symplectic $3$-form. We show that, using the closure of the symplectic $3$-form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic $3$-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic $3$-form.

中文翻译:

将哈密顿力学推广到三维相空间

经典哈密顿力学是通过泊松括号对哈密顿函数的作用来实现的。哈密​​顿函数是系统的运动(能量)常数。Poisson 括号的属性被封装在辛 $2$-form 中,这是一个封闭的二阶微分形式。由于闭包,哈密顿流保留了辛 $2$-形式,​​并通过李-达布定理在相空间上分配了一个不变量 (Liouville) 测度。在本文中,我们提出将经典哈密顿力学推广到三维相空间:将经典泊松括号替换为作用于一对哈密顿函数的广义泊松括号,而辛$2$-形式由辛替换3 美元的形式。我们证明,使用辛 $3$-form 的闭包,与经典李-达布定理类似的结果成立:局部存在光滑坐标,使得辛 $3$ 形式的分量是常数,相空间具有保留的体积元素。此外,与经典理论一样,广义泊松括号的雅可比恒等式在数学上表达了相关辛形式的闭包。因此,常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。存在光滑坐标,使得辛$3$-形式的分量是常数,并且相空间具有保留的体积元素。此外,与经典理论一样,广义泊松括号的雅可比恒等式在数学上表达了相关辛形式的闭包。因此,常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。存在光滑坐标,使得辛$3$-形式的分量是常数,并且相空间具有保留的体积元素。此外,与经典理论一样,广义泊松括号的雅可比恒等式在数学上表达了相关辛形式的闭包。因此,常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。相空间具有保留的体积元。此外,与经典理论一样,广义泊松括号的雅可比恒等式在数学上表达了相关辛形式的闭包。因此,常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。相空间具有保留的体积元。此外,与经典理论一样,广义泊松括号的雅可比恒等式在数学上表达了相关辛形式的闭包。因此,常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。常数斜对称三阶逆变张量总是定义广义泊松括号。这与假设基本恒等式替代雅可比恒等式的哈密顿力学的推广形成对比。特别是,我们发现基本恒等式代表了比辛 $3$-form 的闭包更强的要求。
更新日期:2021-05-19
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