拉格朗日形式体系中的耗散

Relation Between the Partial Derivatives of the Kinetic Energy in the Lagrangian and Hamiltonian Formalisms of Dynamics

The partial derivative of the kinetic energy of a dynamical system with respect to a generalized coordinate as it appears in the Lagrangian formalism is not equal to the derivative of the kinetic energy with respect to the same coordinate in the Hamiltonian formalism but differs by a sign. We find another exact relation between the two partial derivatives in the case of a conservative system. We also identify another form of kinetic energy whose partial derivative with respect to a generalized coordinate vanishes identically.

Structure of Constrained Systems in Lagrangian Formalism and Degree of Freedom Count

A detailed program is proposed in the Lagrangian formalism to investigate the dynamical behavior of a theory with singular Lagrangian. This program goes on, at different levels, parallel to the Hamiltonian analysis. In particular, we introduce the notions of first class and second class Lagrangian constraints. We show each sequence of first class constraints leads to a Neother identity and consequently to a gauge transformation. We give a general formula for counting the dynamical variables in Lagrangian formalism. As the main advantage of Lagrangian approach, we show the whole procedure can a…

Necessary criterion of choosing the energy-momentum tensor and the Lagrange formalism

It is shown that the necessary criterion for choosing the energy-momentum tensor of a physical system is the form of its linear invariant, which should be the lagrangian of this physical system. Examples of energy-momentum tensors that meet this criterion are given. For an electromagnetic field in vacuum, the linear invariant of the energy-momentum tensor must be the canonical lagrangian or the quadratic invariant of the electromagnetic field tensor. A mathematical method for obtaining the lagrangian that complements the Lagrange formalism is proposed.

An extended Lagrangian formalism

A simple formal procedure makes the main properties of the lagrangian binomial extendable to functions depending to any kind of order of the time--derivatives of the lagrangian coordinates. Such a broadly formulated binomial can provide the lagrangian components, in the classical sense of the Newton's law, for a quite general class of forces. At the same time, the generalized equations of motions recover some of the classical alternative formulations of the Lagrangian equations.

解析场论的拉格朗日形式化

Demystifying the Lagrangian formalism for field theories

本文通过三个步骤解析场论的拉格朗日形式化，推导作用量定义，证明欧拉-拉格朗日方程的坐标变换不变性，并以电磁学为例证明其导出麦克斯韦方程。

The geometry of Rayleigh dissipation

Geometric mechanics is a branch of mathematical physics that studies classical mechanics of particles and fields from the point of view of geometry. In a geometric language, symmetries can be expressed in a natural manner as vector fields that generate the corresponding symmetry group. Moreover, with the geometric machinery, the phase space of a mechanical system with symmetries can be reduced to a space with less dimensions. As it is well-known, non-conservative forces cannot be written as the gradient of a potential, so they cannot be absorbed in the Lagrangian or Hamiltonian. The descriptio…