Z. E. Musielak, T. B. Watson
A novel method to make Lagrangians Galilean invariant is developed. The method, based on null Lagrangians and their gauge functions, is used to demonstrate the Galilean invariance of the Lagrangian for Newton's law of inertia. It is suggested that this new solution of an old physics problem may have implications and potential applications to all gauge-based theories of physics.
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It is shown that the N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the free Lagrangian involving (N+1)/2-th order time derivative.
This paper undertakes a study of the nature of the force associated with the local U (1) gauge symmetry of a non-relativistic quantum particle. To ensure invariance under local U (1) symmetry, a matter field must couple to a gauge field. We show that such a gauge field necessarily satisfies the Maxwell equations, whether the matter field coupled to it is relativistic or non-relativistic. This result suggests that the structure of the Maxwell equations is determined by gauge symmetry rather than the symmetry transformation properties of space-time. In order to assess the validity of this notion…
We propose a new Eulerian turbulence theory to obtain a closed set of equations for homogeneous, isotropic turbulent velocity field correlations and propagator functions by incorporating constraints of random Galilean invariance. This incorporation generates a few different equations for propagator and the present theory suggests a way to utilize them into the closure solutions of two-time and single-time velocity correlations' equations so as to properly account for random sweeping phenomena. The present theory yields exact solutions when applied to simple model problem of random oscillator.
Galilean invariance leaves its imprint on the energy spectrum and eigenstates of $N$ quantum particles, bosons or fermions, confined in a bounded domain. It endows the spectrum with a recurrent structure which in capillaries or elongated traps of length $L$ and cross-section area $s_\perp$ leads to spectral gaps $n^2h^2s_\perpρ/(2mL)$ at wavenumbers $2nπs_\perpρ$, where $ρ$ is the number density and $m$ is the particle mass. In zero temperature superfluids, in toroidal geometries, it causes the quantization of the flow velocity with the quantum $h/(mL)$ or that of the circulation along the tor…
Galilean Relativity and the Path Integral Formalism in Quantum Mechanics
针对牛顿力学拉格朗日量不满足伽利略变换不变性、量子力学中伽利略 boost 需波函数非平凡变换的问题，采用费曼路径积分形式，证明量子力学中伽利略对称群的投影表示与拉格朗日量的非不变性等价，为理解经典与量子力学中的对称性表示提供了简洁路径。
The conformal transformations corresponding to $N$-Galilean conformal symmetries, previously defined as canonical symmetry transformations on phase space, are constructed as point transformations in coordinate space.
Isotropy and Galilean invariance of Lattice Boltzmann Method: Theoretical and numerical analysis using oblique dipole benchmark *
聚焦二维九速格子玻尔兹曼模型，分析D2Q9格式精度及寄生项消除，探讨平衡分布影响，用斜偶极子基准数值研究立方项及不同平衡分布选择。
Null Lagrangians and their gauge functions are derived for given standard and non-standard Lagrangians. The obtained standard null Lagrangians generalize those previously found but the non-standard null Lagrangians are new. The gauge functions are used to make the action invariant and introduce the exact null Lagrangians, which form a new family of null Lagrangians. The conditions required for the action to be invariant are derived for all null Lagrangians presented in this paper, thus, making them exact. As a specific application, the exact null Lagrangians are derived for a second-order ordi…
The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa <cite>38</cite> is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa <cite>38</cite>, we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass…
A general non-local point transformation for position-dependent mass Lagrangians and their mapping into a "constant unit-mass" Lagrangians in the generalized coordinates is introduced. The conditions on the invariance of the related Euler-Lagrange equations are reported. The harmonic oscillator linearization of the PDM Euler-Lagrange equations is discussed through some illustrative examples including harmonic oscillators, shifted harmonic oscillators, a quadratic nonlinear oscillator, and a Morse-type oscillator. The Mathews-Lakshmanan nonlinear oscillators are reproduced and some "shifted" Ma…
As is well known, there are different Lagrangians which lead to the same Euler-Lagrange operator. The gauge invariance of a Lagrangian guarantees that of the corresponding Euler-Lagrange operator, but not vice versa. We show that the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian results in Noether conservation laws.
We show that a well known for NLS equation scaling invariance and Galilean invariance property of its solutions can be extended with appropriate modifications to the whole AKNS hierahchy and its reduced and deformed versions.
Standard and non-standard Lagrangians that give the same equation of motion are significantly different in their forms, as the latter do not have terms that clearly discernable energy-like expressions. A special family of these Lagrangians are null Lagrangians and their gauge functions. It is shown that non-standard null Lagrangians and their gauge functions can be used to introduce dissipative forces to dynamical systems. With standard null Lagrangians being known for introducing non-dissipative forces, the presented results allow for a complete picture of novel roles played by null Lagrangia…
Generalizing Parametrization Invariance in the Calculus of Variations
该研究在变分法经典框架下推广参数化不变性，提出T-Lagrangian与N-Lagrangian两种弱化不变性概念，证明一维下一阶、二阶T-Lagrangian对应模零拉格朗日量的参数化不变泛函，且一维N-Lagrangian均为零拉格朗日量。
Conservation laws related to the gauge invariance of Lagrangians and Euler-Lagrange operators in finite and infinite order Lagrangian formalisms are analyzed.
A Lagrangian formalism for variational second-order delay ordinary differential equations (DODEs) is developed. The Noether operator identity for a DODE is established, which relates the invariance of a Lagrangian function with the appropriate variational equations and the conserved quantities. The identity is used to formulate Noether-type theorems that give the first integrals for DODE with symmetries. Relations between the invariance of the variational second-order DODEs and the invariance of the Lagrangian functions are also analyzed. Several examples illustrate the theoretical results.
The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several particular cases of integrable quad-equations by Bazhanov, Mangazeev and Sergeev and by Lobb and Nijhoff. We provide a simple and case-independent proof for all integrable quad-equations. Moreover, we find a new relation for Lagrangians within one elementary quadrilateral which seems to be a fundamental building block of the various versions of flip invariance.
We show explicitly that a free Lagrangian expressed in terms of scalar, spinor, vector and Rarita-Schwinger (RS) fields is invariant under linear supersymmetry transformations generated by a global spinor-vector parameter. A (generalized) gauge invariance of the Lagrangian for the RS field is also discussed.
Notes on a paper by F. Génot and B. Brogliato on the Painlevé paradox
重新审视F. Génot和B. Brogliato对经典Painlevé问题的分析，发现其关于机械系统动力学演化的结论不具伽利略观察者不变性，运动状态分类依赖观察者，源于伽利略速度相加定理，还探讨了恢复观察者不变描述的模型 reformulations。
We prove the non-uniqueness theorem for the chronological products of a gauge model. We use a cohomological language where the cochains are chronological products, gauge invariance means a cocycle restriction and coboundaries are expressions producing zero sandwiched between physical states. Suppose that we have gauge invariance up to order n of the perturbation theory and we modify the first-order chronological products by a coboundary (a trivial Lagrangian). Then the chronological products up to order n get modified by a coboundary also.
Delay ordinary differential equations: from Lagrangian approach to Hamiltonian approach
本文针对延迟常微分方程提出哈密顿表述，建立哈密顿延迟算子恒等式并推导诺特定理型定理，探究相关不变性的关联，通过算例验证理论结果。
We consider perturbative quantum field theory in the causal framework. Gauge invariance is, in this framework, an identity involving chronological products of the interaction Lagrangian; it express the fact that the scattering matrix must leave invariant the sub-space of physical states. We are interested in generalizations of such identity involving Wick sub-monomials of the interaction Lagrangian. The analysis can be performed by direct computation in the lower orders of perturbation theory; guided by these computations we conjecture a generalization for arbitrary orders.
The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories -- general relativity, electromagnetism, Yang Mills, string theory, etc. The Dirac-Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac-Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this paper I analy…
A general affine connection has enough degrees of freedom to describe the classical gravitational and electromagnetic fields in the metric-affine formulation of gravity. The gravitational field is represented in the Lagrangian by the symmetric part of the Ricci tensor and the classical electromagnetic field can be represented by the tensor of homothetic curvature. The simplest metric-affine Lagrangian that depends on the tensor of homothetic curvature generates the Einstein-Maxwell equations for a massless vector. Metric-affine Lagrangians with matter fields depending on the connection are sub…
We study constraints imposed by the Galilean invariance on linear electromagnetic and elastic responses of two-dimensional gapped systems in background magnetic field. Exact relations between response functions following from the Ward identities are derived. In addition to viscosity-conductivity relations known in literature we find new relations between the density-curvature response and the chiral central charge.
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