论文标题
受限的动态：广义的谎言对称性，奇异的Lagrangians和Hamiltonian机械师的通道
Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics
论文作者
论文摘要
在Euler-Lagrange运动方程的对称性的指导下，提出了对奇异Lagrangians受约束动力学的研究。我们发现，这些运动方程式允许广义的谎言对称性，在拉格朗日相空间上，这种对称的发电机位于拉格朗日两种形式的核中。能量方程\ TexteMdash的解决方案称为二阶Euler-Lagrange Vector Fields（SOELVFS）\ TextEmdash具有具有此对称性的积分流。重要的是，虽然二阶拉格朗日向量字段不是这样的解决方案，但始终有可能向他们构造一个soelvf。我们发现，所有SOELVF都可以投影到汉密尔顿相位空间，而Lagrangian相位空间中的所有动力结构也是如此。特别是，可以从位于拉格朗日两种形式的核的向量构建的主要汉密尔顿约束，并且通过这种构建，我们表明SOELVF的Lagrangian约束算法等于总汉密尔顿的稳定性分析。重要的是，这种稳定性分析的最终结果给出了一个哈密顿矢量场，这是从拉格朗日约束算法获得的SOELVF的投影。奇异拉格朗日机械的拉格朗日式和哈密顿的表述相当于这种方式。
Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.
