R. E. Gamboa Saraví, J. E. Solomin
The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.
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We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\R$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The cor…
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.
The purpose of this paper is to announce some new results on the structure of the higher order Euler-Lagrange mapping of the multiple-integral variational calculus on fibered manifolds,namely a description of its kernel and its image,and an explicit characterization of the conditions under which a system of partial differential equations (of arbitrary order)is a system of the Euler--Lagrange equations.
Invariantised Euler-Lagrange equations and conserved quantities for nonconservative Herglotz variational problems
本文利用活动标架和不变量变分法，导出了一维 n 阶 Herglotz 变分问题的广义 Euler-Lagrange 方程及其守恒量的结构，并应用于受限 Lorentz 群 SO^+(1,2) 不变的问题。
We shall introduce the principal symbol for Euler-Lagrange operators and use them to charac- terise well-posed initial value problems. We shall clarify how constraints can arise in Lagrangian covariant theories by extending the standard treatment in GR. Finally, we sketch a quantization procedure based on what done in LQG.
A unified formulation of rigid body dynamics based on Gauss principle is proposed. The Lagrange, Kirchhoff and Newton-Euler equations are seen to arise from different choices of the quasicoordinates in the velocity space. The group-theoretical aspects of the method are discussed.
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…
Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to Lagrangian mechanics and gravitational waves are given.
Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, ω)$, their properties and a kind of classification of vector fields on the manifold, we generalize Liouville's theorem in classical mechanics to two sequences, the symplectic(-like) and the Hamiltonian-(like) Liouville's theorems. This also generalizes Noether's theorem, since the sequence of symplectic(-like) Liouville's theorems link to the cohomology directly.
We obtain several Euler-Lagrange equations for variational functionals defined on a set of Hölder curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.
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…
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.
General solution to the Euler-Poisson equations of a free Lagrange top directly for the rotation matrix
本文针对自由拉格朗日陀螺，直接求解欧拉-泊松方程得到旋转矩阵的一般解析解，并与庞索运动图像比较。
Euler-Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in the flow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models; which include power law, Bingham, Herschel-Bulkley, Carreau and Cross; are used for demonstration.
We argue that, under multidimensional position-dependent mass (PDM) settings, the Euler-Lagrange textbook invariance falls short and turned out to be vividly incomplete and/or insecure for a set of PDM-Lagrangians. We show that the transition from Euler-Lagrange component presentation to Newtonian vector presentation is necessary and vital to guarantee invariance. The totality of the Newtonian vector equations of motion is shown to be more comprehensive and instructive than the Euler-Lagrange component equations of motion (they do not run into conflict with each other though). We have successf…
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian $L(x(t)$, where $_a^cD_t^αx(t))$ and $0<α< 1$, such that the following is the corresponding Euler-Lagrange % \begin{equation}_tD_b^α(_a^cD_t^α) x(t)+ b(t,x(t))(_a^cD_t^αx(t))+f(t,x(t))=0. \end{equation} % At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations % \begin{equation}_tD_b^α(_a^cD_t^αx(t))=λx(t), (λ\in R) \end{equation} % \begin{equation}_tD_b^α(_a^cD_t^αx(t)…
Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the undefined functions correspond to isentropic flows of an ideal gas, different forms of the hyperbolic shallow water equations. Complete group classification of the equation with respect to these functions is performed. Using Noether's theorem, all conservation laws are obtained. Their analogs in Eulerian coordinates are given.
It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the …
Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the equivalent Lagrangians play an important role because they admit the same Euler-Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant the Lagrangian we obtain the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivat…
The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include as particular cases the classic…
Difference Discrete Variational Principle,EULER-Lagrange Cohomology and Symplectic, Multisymplectic Structures
本文通过多参数微分方法研究差分离散变分原理，导出离散欧拉-拉格朗日方程和正则方程，并应用于辛/多辛几何及其保结构算法。
In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler-Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler-Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector de…
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.
Discrete Spinning Tops -- Difference equations for Euler, Lagrange, and Kowalevski tops
本文研究 Euler、Lagrange 和 Kowalevski 陀螺等可积刚体模型的时间离散化方法，并为 Kowalevski 陀螺提出一种能精确保持 $\oldsymbolγ^2=1$ 和 Kowalevski 积分守恒的新离散格式，数值实验验证有效。
On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion
本文研究了 Euler、Lagrange 和 Kovalevskaya 刚体运动可积情形的全局张量不变量，通过将含三次项的张量场代入不变性方程并用计算机代数系统求解，基于 Poincaré-Cartan 理论探讨了利用这些不变几何结构研究动力学的问题。
Lagrange points of Euler's solutions of the 3-body problem
本文针对三个主天体处于三体共线构型的圆型限制性四体问题开展中心构型分类研究，补充了Llibre 2021年的相关研究，相关结果可通过指定视频查看。
The mechanics of a flexible membrane decorated with a nematic liquid-crystal texture is considered in a variational framework. The variations on the splay, twist and the bend energy of the nematics are obtained from the local deformations leading to changes in the shape membrane. The Euler-Lagrange derivatives and the Noether charges are identified from the variational equations. The nematic stress tensor is obtained as a consequence of translational invariance. Likewise, the rotational invariance implies the torque nematic tensor. The corresponding boundary conditions are obtained for free ed…
We present some novel equilibrium shapes of a clamped Euler beam (Elastica from now on) under uniformly distributed dead load orthogonal to the straight reference configuration. We characterize the properties of the minimizers of total energy, determine the corresponding Euler-Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers. Moreover, we prove some sufficient conditions for stability and instability of particular solutions of the Euler-Lagrange conditions that can be applied to numerically found curled shapes.
Variational Dissipative Mechanics on Lie Algebroids
本文提出了一种在李代数垛上的变分耗散力学框架，推导了耗散系统的欧拉-拉格朗日-赫尔格洛兹方程，并建立了能量平衡定律和耗散不变量理论。
This work presents the variational principles and the intrinsic versions of several equations in field theories, in particular, for the Classical Euler-Lagrange field equations, the implicit Euler-Lagrange field equations and the non-holonomic implicit Euler-Lagrange field equations. The advantages of the variational and intrinsic versions of these equations is that the Lagrangians functions are not necessary regular Lagrangians. We present two examples of this situation: Navier's equations and the non-holonomic Cosserat rod. Finally we comment the Hamiltonian case when the Lagrangian is a hyp…
Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus f…
Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler-Lagrange equations are derived. Fractional equations of motion are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives.
Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a globa…
Collision-Free Bearing-Driven Formation Tracking for Euler-Lagrange Systems
针对多移动领导者下含参数不确定性的异构Euler-Lagrange系统，提出基于方位定位的分布式观测器与自适应控制律实现无碰撞编队跟踪，并给出确保避免碰撞的初始配置充分条件。
Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
本文研究具有仿射诺特荷的不定拉格朗日系统，利用对称性将固定能量连接问题化为两点边界问题，并建立推广稳态时空费马原理的变分原理。
Hamel's Equations and Geometric Mechanics of Constrained and Floating Multibody and Space Systems
本文通过Hamel原始形式体系推导约化Euler-Lagrange方程与联络曲率，利用锁定速度实现内蕴分解与惯性解耦，填补了Hamel表述与几何方法在局部坐标下联系的空白。
The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal's necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using Agrawal's Euler-Lagrange equation and the Lagrange multiplier technique, we obtain here a Noether-like theorem for fractional optimal control problems in the sense of Caputo.
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。
Weak Form of Differential Equations and Differential Identities
受量子力学启发，提出微分方程与恒等式的弱形式，并证明薛定谔方程是经典欧拉-拉格朗日方程的弱形式。
Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained and the recovery of the classical results was discussed.
A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of $G$. In this context the reduction of the discrete Euler-Lagrange equations is shown to lead to the so called discrete Euler-Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler-Poincaré equations leads to discrete Hamiltoni…
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