CANONICALIZATION OF CONSTRAINED HAMILTONIAN EQUATIONS IN A SINGULAR SYSTEM CONSTRAINED HAMILTONIAN EQUATIONS IN A SINGULAR SYSTEM.”

: In this paper, the canonicalization of constrained Hamiltonian system is discussed. Because the constrained Hamiltonian equations are non-canonical, they lead to many limitations in the research. For this purpose, variable transformation is constructed that satisfies the condition of canonical equation, and the new variables can be obtained by a series of derivations. Finally, two examples are given to illustrate the applications of the result .


Introduction
Under the Legendre transformation, the singular Lagrangian system can be transformed into phase space and described by Hamiltonian canonical variables; there are inherent constraints between canonical variables, namely constrained Hamiltonian system. Many important physical systems belong to this system, such as, quantum electrodynamics (QED), quantum flavor dynamics (QFD), quantum chromodynamics (QCD) and general relativity (GR) which is used to describe the basic interaction of nature. The Lagrangian functions of supersymmetry, supergravity and string theory are singular as well. Therefore, the fundamental theory of constrained Hamiltonian system plays a significant role in theoretical physics, especially in modern quantum field theory [1][2][3][4] .
There are inherent constraints when the singular Lagrangian system transitions to phase space, but the quantization of the system is usually achieved by the canonical variables of phase space, at the point, the quantum method [5][6][7] in elementary quantum mechanics is in trouble. When the canonical variables are restrained, new problems arising from the quantization theory, have received extensive attention of people. Furthermore, if the canonicalization of variables ( , ) pq in constrained Hamiltonian system is realized, then we can express constrained Hamiltonian system to normal Hamiltonian system, the existing Hamiltonian system symmetry theory [8][9][10][11][12][13][14][15] can be [108] used in constrained Hamiltonian system conveniently. And at this time, the system has the symplectic structure [16] , the symplectic geometry method can be used for giving numerical solutions and numerical simulations of constrained Hamiltonian system. So many important physics systems can be simulated with this method. Therefore, the canonicalization of constrained Hamiltonian system is a major work. In previous studies, Yifa Tang has studied the canonicalization of gyrocenter system [17] . Some researches [18][19][20][21] only provide canonicalzation methods for some special cases, and the calculation process is complicated. On the other hand, according to Dardoux's theorem [22] , for a non-canonical Hamiltonian system, we can find theoretically the local canonical coordinates by solving differential equations, which is an uneconomic method for numerical purpose. To avoid solving differential equations, we do not follow the steps given in the standard proof of Dardoux's theorem. Instead, we explore another procedure to realize the canonicalization of the constrained Hamiltonian system.
In this paper, we define a variable transformation, new canonical variables for constrained Hamiltonian system can be given through a series of calculation, the purpose of canonicalization can be achieved. This method is more general, and makes the calculation simpler. In the end, two examples are given to illustrate the practicability of this method. After canonicalization, many useful theories and algorithms can be applied more conveniently to the constrained Hamiltonian system. Thus, there is important theoretical significance to carry out this research.

Hamiltonian Canonical Equations in a Singluar System
Supposing that the dynamics of a system with finite degrees of freedom is described by the Lagrangian ( , , ) ii L L t q q  & , ( And canonical Hamiltonian is Where repeated instructions represent summation. Hess matrix of function L is if its rank rn  . Using the Legendre transformation to convert the Lagrangian description to the Hamiltonian description, there are inherent constraints between the canonical variables in the phase space Then the canonical equations of singular system are given as If the singular systems with only second class constraints, that is, assume constraints (4) consist of second class primary constraints and secondary constraints, we have So all Lagrange multipliers j  in Eq. (6) can be completely determined by the compatibility condition of constraints

The Canonicalization of Constained Hamiltonian Equations
The motion equations for the constrained Hamiltonian system discussed above are noncanonical, in this section; we transform it into canonical form by means of the variable transformation.
The motion equations for constrained Hamiltonian system are Those can be rewritten as the following matrix form So 22 nn M  is an anti-symmetric matrix.
For the convenience of discussion, then the form of (11) can be simplified, expressed it in the form of vectors and gradients. Let Where  simulation of the constrained Hamilton system. After the results are obtained, the new variables are replaced with the old variables, that is, the results of the original system are obtained.

Example 1
The system's Lagrangian is 22 The generalized momenta of the system are The rank r of the Hess matrix of the Lagrangian can be obtained, 0 r  , this is a singular Lagrangian system, so that there are two constraints between canonical variables Canonical Hamiltonian of this system is 22 Total Hamiltonian is The Lagrange multipliers can be obtained by the compatibility condition of constraints, i.e., Eq.
Here, we define a new variable

Example 2
Next we consider an example with the non-potential generalized force.
The system's Lagrangian and generalized force are The motion equations of this system are derived as follows