The first-order Euler-Lagrange equations and some of their uses

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor g ij and its first two derivatives together with the first derivative of a vector field ψ i are investigated. In general, the Euler-Lagrange equations obtained by variation of g ij are of fourth order in g ij and third order in ψ i . It is shown that in a four dimensional space the only Euler-Lagrange equations which are of second order in g ij and first order in ψ i are the Einstein field equations (with an energy-momentum term). Various conditions are obtained under which the Einstein-Maxwell field equations are then an inevitable consequence.

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Implicit Regularization and Momentum Algorithms in Nonlinearly Parameterized Adaptive Control and Prediction

Neural Computation ◽

10.1162/neco_a_01360 ◽

2021 ◽

Vol 33 (3) ◽

pp. 590-673

Author(s):

Nicholas M. Boffi ◽

Jean-Jacques E. Slotine

Keyword(s):

Adaptive Control ◽

Nonlinear Control ◽

Adaptive Dynamics ◽

Observer Design ◽

Local Geometry ◽

Natural Gradient ◽

Lagrange Equations ◽

First Order ◽

Adaptive Nonlinear Control ◽

Mirror Descent

Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors—out of the many that will achieve perfect tracking or prediction—for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian. We show that its Euler Lagrange equations lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.

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Variational formulas for submanifolds of fixed degree

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02100-8 ◽

2021 ◽

Vol 60 (6) ◽

Author(s):

Giovanna Citti ◽

Gianmarco Giovannardi ◽

Manuel Ritoré

Keyword(s):

Vector Fields ◽

Riemannian Metric ◽

Curvature Operator ◽

Lagrange Equations ◽

Fixed Degree ◽

Third Order ◽

First Order ◽

Area Functional ◽

Mean Curvature Operator ◽

Graded Manifold

AbstractWe consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler–Lagrange equations. The resulting mean curvature operator can be of third order.

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On a Non-Newtonian Calculus of Variations

Axioms ◽

10.3390/axioms10030171 ◽

2021 ◽

Vol 10 (3) ◽

pp. 171

Author(s):

Delfim F. M. Torres

Keyword(s):

Calculus Of Variations ◽

Integral Functionals ◽

Lagrange Equations ◽

Real Numbers ◽

Positive Real ◽

First Order ◽

Independent Variable ◽

The Subject ◽

Minimal Resistance ◽

Admissible Functions

The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

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Calculation of the Ion Optical Properties of Inhomogeneous Magnetic Sector Fields, including the Second Order Aberrations in the Median Plane

Zeitschrift für Naturforschung A ◽

10.1515/zna-1959-0205 ◽

1959 ◽

Vol 14 (2) ◽

pp. 121-129 ◽

Cited By ~ 20

Author(s):

H. A. Tasman ◽

A. J. H. Boerboom

Keyword(s):

Optical Properties ◽

Order Approximation ◽

Median Plane ◽

Second Order ◽

Central Path ◽

Resolving Power ◽

Lagrange Equations ◽

Magnetic Sector ◽

First Order ◽

First Order Approximation

Investigation is made of the ion optical properties of inhomogeneous magnetic sector fields. In first order approximation the field is assumed to vary proportional to r—n (0 ≦ n < 1); the term in the magnetic field expansion which determines the second order aberrations is chosen independent of n, which makes the elimination possible of e. g. the second order angular aberration. From the EULER— LAGRANGE equations the second order approximation of the ion trajectories in the median plane and the first order approximation outside the median plane are derived for the case of normal incidence and exit of the central path in the sector field. An equation is presented giving the shape of the pole faces required to produce the desired field. The influence of stray fields is neglected. The object ana image distances are derived, as well as the mass dispersion, the angular, lateral and axial magnification, the resolving power, and the inclination of the plane of focus of the mass spectrum. The maximum transmitted angle in the z-direction is calculated. The resolving power proves to be proportional to (1—n) -1 whereas the length of the central path is proportional to (1—n) -½. An actual example is given of a 180° sector field with n=0.91, where the mass resolving power is increased by a factor 11 as compared with a homogeneous sector field of the same radius and slit widths.

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On a New Analytic Theory of the Moon's Motion II: Orbit and Length of Months

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-58-2020-13-54 ◽

2020 ◽

Vol 58 ◽

pp. 13-54

Author(s):

Ramon González Calvet ◽

Keyword(s):

Order Approximation ◽

Polar Coordinates ◽

The Sun ◽

The Moon ◽

Lagrange Equations ◽

Moon System ◽

First Order ◽

Relative Coordinates ◽

First Order Approximation ◽

Good Agreement

The differential equation in polar coordinates of the Moon's orbit is outlined from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed with relative coordinates and accelerations. The orbit of the Moon calculated this way is similar to Clairaut's modified orbit and has better parameters than those previously published. An improvement to this orbit is proposed based on theoretical arguments. With help of this new orbit, the variations in the draconic, synodic and anomalistic months are also computed showing very good agreement with observations.

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ON COSMOLOGICAL-TYPE SOLUTIONS IN MULTI-DIMENSIONAL MODEL WITH GAUSS–BONNET TERM

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887810004555 ◽

2010 ◽

Vol 07 (05) ◽

pp. 797-819 ◽

Cited By ~ 41

Author(s):

V. D. IVASHCHUK

Keyword(s):

Exact Solutions ◽

Power Law ◽

Equations Of Motion ◽

Exponential Dependence ◽

Dimensional Model ◽

Lagrange Equations ◽

First Order ◽

Scale Factors ◽

Effective Lagrangian ◽

Bonnet Term

A (n + 1)-dimensional Einstein–Gauss–Bonnet (EGB) model is considered. For diagonal cosmological-type metrics, the equations of motion are reduced to a set of Lagrange equations. The effective Lagrangian contains two "minisuperspace" metrics on ℝn. The first one is the well-known 2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric that is proportional to n-dimensional Berwald–Moor 4-metric. When a "synchronous-like" time gauge is considered, the equations of motion are reduced to an autonomous system of first-order differential equations. For the case of the "pure" Gauss–Bonnet model, two exact solutions with power-law and exponential dependence of scale factors (with respect to "synchronous-like" variable) are obtained. (In the cosmological case, the power-law solution was considered earlier in papers of N. Deruelle, A. Toporensky, P. Tretyakov and S. Pavluchenko.) A generalization of the effective Lagrangian to the Lowelock case is conjectured. This hypothesis implies existence of exact solutions with power-law and exponential dependence of scale factors for the "pure" Lowelock model of mth order.

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On a New Analytic Theory of the Moon's Motion III: Further Corrections

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-59-2021-67-99 ◽

2021 ◽

Vol 59 ◽

pp. 67-99

Author(s):

Ramon Gonzalez Calvet

Keyword(s):

Order Approximation ◽

Gravity Data ◽

Analytic Theory ◽

Lagrange Equations ◽

First Order ◽

A Value ◽

Spheroidal Shape ◽

Relative Coordinates ◽

Lunar Motion ◽

First Order Approximation

Further corrections to the analytic theory of the lunar motion deduced from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed in relative coordinates and accelerations are evaluated. Those terms arising from the second-order approximation, the planetary perturbations and Earth's spheroidal shape are calculated and bounded, all of them being very small. Finally, Earth's gravitational parameter is calculated from gravity data finding a value slightly higher than that provided by Jet Propulsion Laboratory.

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POINCARÉ GAUGE THEORY WITH COUPLED EVEN AND ODD PARITY DYNAMIC SPIN-0 MODES: DYNAMICAL EQUATIONS FOR ISOTROPIC BIANCHI COSMOLOGIES

International Journal of Modern Physics D ◽

10.1142/s0218271811020391 ◽

2011 ◽

Vol 20 (11) ◽

pp. 2125-2138 ◽

Cited By ~ 8

Author(s):

FEI-HUNG HO ◽

JAMES M. NESTER

Keyword(s):

Gauge Theory ◽

Cross Coupling ◽

Lagrange Equations ◽

Dynamical Equations ◽

First Order ◽

Class A ◽

Effective Lagrangian ◽

Dynamic Spin ◽

Bianchi Cosmologies ◽

Gauge Theory Of Gravity

We are investigating the dynamics of a new Poincaré gauge theory of gravity model, which has cross coupling between the spin-0+ and spin-0- modes. To this end we here consider a very appropriate situation — homogeneous-isotropic cosmologies — which is relatively simple, and yet all the modes have nontrivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies; for this case we find an effective Lagrangian and Hamiltonian for the dynamical system. The Lagrange equations for these models lead to a set of first-order equations that are compatible with those found for the FLRW models and provide a foundation for further investigations. Typical numerical evolution of these equations shows the expected effects of the cross parity coupling.

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An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action

Mathematics ◽

10.3390/math9182178 ◽

2021 ◽

Vol 9 (18) ◽

pp. 2178

Author(s):

Ivano Colombaro ◽

Josep Font-Segura ◽

Alfonso Martinez

Keyword(s):

Maxwell Equations ◽

Lagrangian Density ◽

Exterior Algebra ◽

Field Theories ◽

Lagrange Equations ◽

First Order ◽

Vector Calculus ◽

Stationary Action ◽

Vector Derivative ◽

Derivatives Of

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 derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

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