Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

On the Zakharov–Mikhailov action: $$4\hbox {d}$$ Chern–Simons origin and covariant Poisson algebra of the Lax connection

We derive the $$2\hbox {d}$$ 2 d Zakharov–Mikhailov action from $$4\hbox {d}$$ 4 d Chern–Simons theory. This $$2\hbox {d}$$ 2 d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the $$2\hbox {d}$$ 2 d level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “$$H={{\,\mathrm{Tr}\,}}L^2$$ H = Tr L 2 ”.

Acquiring the Symplectic Operator Based on Pure Mathematical Derivation then Verifying It in the Intrinsic Problem of Nanodevices

The symplectic algorithm can maintain the symplectic structure and intrinsic properties of the system, its cumulative error is small and suitable for multi-step calculation. At present, the widely accepted symplectic operators are obtained by solving the Hamilton equation based on artificial definitions and assumptions in advance. There are inevitable dispersion errors. We solve the equation by pure mathematical derivation without any artificial limitations and assumptions. The way to accurately obtain high-precision symplectic operators greatly reduces the dispersion error from the beginning. The numerical solution of the one-dimensional Schrödinger equation for describing the intrinsic problem of nanodevices is used as an application environment to compare the total energy distribution of the particle wave function in the box, thus verifying the properties of the Symplectic Operator based on Pure Mathematical Derivation by comparing with Finite-Difference Time-Domain (FDTD) and the widely accepted symplectic operator.

Generalization of Nambu–Hamilton Equation and Extension of Nambu–Poisson Bracket to Superspace

We propose a generalization of the Nambu–Hamilton equation in superspace R 3 | 2 with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace R 3 | 2 by means of the right-hand sides of the proposed generalization of the Nambu–Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of the Nambu–Hamilton equation in superspace leads to a family of ternary brackets of even degree functions defined with the help of a Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space R 3 . We study the structure of the ternary bracket in a more general case of a superspace R n | 2 with n real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is the usual Nambu–Poisson bracket, extended in a natural way to even degree functions in a superspace R n | 2 , and the second is a new ternary bracket, which we call the Ψ -bracket, where Ψ can be identified with an invertible second order functional matrix. We prove that the ternary Ψ -bracket as well as the whole ternary bracket (the sum of the Ψ -bracket with the usual Nambu–Poisson bracket) is totally skew-symmetric, and satisfies the Leibniz rule and the Filippov–Jacobi identity ( Fundamental Identity).

Generalization of Nambu-Hamilton Equation and Extension of Nambu-Poisson Bracket to Superspace

We propose a generalization of Nambu-Hamilton equation in superspace $\mathbb R^{3|2}$ with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace $\mathbb R^{3|2}$ by means of the right-hand sides of proposed generalization of Nambu-Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of Nambu-Hamilton equation in superspace leads to family of ternary brackets of even degree functions defined with the help of Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space $\mathbb R^{3}$. We study the structure of ternary bracket in a more general case of a superspace $\mathbb R^{n|2}$ with $n$ real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is usual Nambu-Poisson bracket, extended in a natural way to even degree functions in a superspace $\mathbb R^{n|2}$, and the second is a new ternary bracket, which we call $\Psi$-bracket, where $\Psi$ can be identified with invertible second order functional matrix. We prove that ternary $\Psi$-bracket as well as the whole ternary bracket (the sum of $\Psi$-bracket with usual Nambu-Poisson bracket) is totally skew-symmetric, satisfies the Leibniz rule and the Filippov-Jacobi identity (Fundamental Identity).