The invariance theorems obtained in analytical mechanics and derived from Noether's theorems can be adapted to fluid mechanics. For this purpose, it is useful to give a functional representation of the fluid motion and to interpret the invariance group with respect to time in the quadri-dimensional reference space of Lagrangian variables. A powerful method of calculation uses Lie's derivative, and many invariance theorems and conservation laws can be obtained in fluid mechanics.
We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions $\Sc(\mathbb R)$, and then we extend it to its dual set, $\Sc'(\mathbb R)$, the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.
Higher-derivative analogues of multidimensional conformal particle and many-body conformal mechanics are constructed. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.
We formulate the equations which determine a potential function in an $\mathcal{N}=1$ higher derivative supersymmetric mechanics compatible with the $osp(2|1)\oplus so(d)$ symmetry and provide a few explicit examples.
Based on the results in [Nucl. Phys. B 866 (2013) 212], we consider a way to construct a higher-derivative mechanical model which possesses the $l$-conformal Galilei symmetry. The dynamical system describes generalized Pais-Uhlenbeck oscillator in the presence of an effective external field.
Remarks related to the paper of Rafael de la Madrid: "On the inconsistency of the Bohm-Gadella theory with quantum mechanics", JPhysA 39, No. 29, 9255-9268 (2006)
(Draft 3) A generalized differential operator on the real line is defined by means of a limiting process. These generalized derivatives include, as a special case, the classical derivative and current studies of fractional differential operators. All such operators satisfy properties such as the sum, product/quotient rules, chain rule, etc. We study a Sturm-Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm-Liouville Theory. As an application of the developments herein we find the general solution of a generalize…
On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative
分数阶力学描述了保守和非保守系统。分数阶变分原理在研究分数阶力学中具有重要意义，并提出了几种不同的变分原理。在经典力学中，等效拉格朗日量起着重要的作用，因为它们承认相同的欧拉-拉格朗日方程。通过将适当函数的总时间导数加到给定的经典拉格朗日量上或乘以拉格朗日量的常数，我们得到相同的运动方程。在这项研究中，分数阶离散拉格朗日函数的分数阶导数不同的黎曼-刘维分数阶导数进行了分析。作为应用这一程序的结果，经典的结果作为一种特殊情况重新获得。通过对$Fa\grave{a}$ di Bruno公式的分数阶推广，得到了与给定分数阶拉格朗日量不同的分数阶拉格朗日量的具体表达式.研究了所得到的分数阶拉格朗日量所对应的分数阶欧拉-拉格朗日方程和分数阶Hamilton方程，并对两个算例进行了详细分析。
This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration we review several functional integration approaches to theory of turbulence. We notice in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation.
An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the actio…
The description of bumps in scattering cross-sections by Breit-Wigner amplitudes led in the framework of the mathematical Physics to its formulation as the so-called Resonance-Decay Problem. It consists of a spectraltheoretical component and the connection of this component with the construction of decaying states. First the note quotes a solution for scattering systems, where the absolutely continuous parts of the Hamiltonians are semi-bounded and the scattering matrix is holomorphic in the upper half plane. This result uses the approach developed by Lax and Phillips, where the energy scale i…
Working within the framework of parity-time-symmetric quantum mechanics we look into the possibility of entanglement generation and demonstrate that the feature of non-violation of no-signaling principle may hold for the simplest non-trivial case of bipartite systems. Basically our arguments are based on the computation of the reduced density matrix of one party to justify that the entropy of the other does not change.
The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated …
The method of nonlinear realizations is used to clarify some conceptual and technical issues related to the Schwarzian mechanics. It is shown that the Schwarzian derivative arises naturally, if one applies the method to SL(2,R) times R group and decides to keep the number of the independent Goldstone fields to a minimum. The Schwarzian derivative is linked to the invariant Maurer-Cartan one-forms, which make its SL(2,R)-invariance manifest. A Lagrangian formulation for a variant of the Schwarzian mechanics studied recently in [Nucl. Phys. B 936 (2018) 661] is built and its geometric descriptio…
