MSP (Mathematical Sciences Publishers), Project Euclid, and Duke University Press have partnered to launch the **MSP on Euclid** collection, bringing seven mathematics journals published by MSP together with the strong functionality available through the Project Euclid platform.

Dedicated to providing alternatives in math publishing, MSP, Project Euclid, and Duke University Press are all not-for-profit and have similarly sized programs. The goal of the **MSP on Euclid** collection is to offer libraries new features as well as the option to consolidate their platforms. **MSP on Euclid** includes the same seven journals sold by MSP in their MSP package but now also hosted by Project Euclid and sold by Duke University Press.

“Project Euclid is pleased to welcome seven of MSP’s distinguished journals to our platform and to work with Duke University Press on increasing their dissemination,” said Leslie Eager, Project Euclid’s director of publishing services. “Our three organizations strive to provide the mathematics community with truly excellent not-for-profit publishing services, and we look forward to strengthening our impact through this collaboration.”

“MSP is committed to finding new ways to make mathematics publishing more sustainable and to bring our content to scholars around the world. We hope this new collaboration will benefit researchers as well as libraries,” said Rob Kirby, chief executive of MSP.

**MSP on Euclid** provides institutional subscribers with valuable enhancements of the MSP package still available from MSP. Project Euclid’s platform offers libraries a subscription management tool that stores librarian contact and IP information. Institutional subscribers gain COUNTER- and SUSHI-compliant usage statistics exclusively through the platform. Single sign-on authentication through Shibboleth is also available. Duke University Press provides institutions with all sales services and customer support for the collection.

“Duke University Press has been publishing mathematics scholarship for over 80 years. We hope that our experience in math sales and customer support will bring MSP’s well-regarded, high-quality content to a wider audience,” said Steve Cohn, director of Duke University Press.

For additional information and pricing for **MSP on Euclid**, visit dukeupress.edu/Libraries.

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here we can consider some aspects of mathematics as say prof dr mircea orasanu and prof horia orasanu

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LAGRANGIAN AND OPTIMIZATIONS CONSTRAINTS

Author Horia Orasanu

ABSTRACT

In order to begin to make a connection between the microscopic and macroscopic worlds, we need to better understand the microscopic world and the laws that govern it. We will begin placing Newton’s laws of motion in a formal framework which will be heavily used in our study of classical statistical mechanicsIn this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell).

1 INTRODUCTION

The accurate numerical simulation of advection dominated problems is an important problem in many scientific applications. However, due to the non-dissipative nature of the equations considered, care has to be taken to obtain a stable numerical scheme (see, for example, [1]). A large body of research has been accumulated that describes finite difference, finite volume, and finite element discretizations of such problems. However, for advection-dominated problems so-called semi-Lagrangian methods offer a competitive alternative. These methods integrate the characteristics back in time and consequently have to use some interpolation scheme to reconstruct the desired value at the grid points. Strictly speaking, semi-Lagrangian methods can only be applied to systems of first-order differential equations. However, in many instances, first-order systems arise from the splitting of more complicated equations or constitute the linear part of an evolution equation (which is then treated separately from the nonlinearity).

2 NEW ASPECTS IN THE THEORY OF STABILITY OF HAMILTONIAN

Basic Lagrangian Mechanics

This page contains an extremely simple but (hopefully!) informative introduction to Lagrangian mechanics.

“Lagrangian mechanics” is, fundamentally, just another way of looking at Newtonian mechanics. Newtonian mechanics, in a nutshell, says:

(1a)

I’ve labeled them with their common names: the second and third laws. The “first law”, which I didn’t show, can be derived from the other two laws, if we assume all forces arise from interactions between objects. The “Second law” as shown here assumes the mass of a body is constant (unless it ejects a second body or merges with a second body); that’s true for Newtonian mechanics but not in relativity theory. From these two (or three) laws one can derive conservation of energy, momentum, and angular momentum.

The fundamental forces in the universe are all conservative, and many forces we deal with in everyday life are conservative as well (friction being one obvious exception). A conservative force can be represented as the gradient of a potential; when an object is being affected only by conservative forces, we can rewrite the second law as:

(1b)

or, in vector form, using r as the object’s position vector,

(1c)

where φ is the potential function.

Since the argument of the inverse cosine must be in the range from -1 to +1 (in order for max to be real-valued), this implies that 0 E/(mgr) 2. In other words, the energy E must satisfy the inequalities 0 E 2mgr, which was to be expected, because 2mgr is the potential energy the particle would have at the “12 o’clock” position relative to the potential energy at the “6 o’clock” position. If the total energy exceeds this value, then it must have some kinetic energy when it reaches the top of the disk, so it will continue to roll, without ever reversing course.

We can consider two generate cases. First, if r = 0 and R > 0, the integrand in (1) is just a constant, so (setting 1 = 0) we immediately have

Since the vertical position is constant in this case, the total energy is just the kinetic energy E = K = mv2/2, so this implies vt = R. Second, if R = 0 and r > 0, the rolling disk reduces to a stationary pivot point, and the system is a simple pendulum. In this case equation (1) reduces to

For sufficiently small angular displacements , the value of cos() approaches 1 2/2, so we can make this substitution and divide through by mr/2 to give the energy equation of a pendulum for small displacements

Differentiating with respect to time gives

Dividing through by 2(d/dt), we arrive at the differential equation

with the solution

which shows that the frequency (for sufficiently small angular displacements) is independent of both the mass and the energy. The period of the oscillation is

On the other hand, if the angular displacement is not small, we must use equation (4), which can be integrated as shown in equation (3) with R = 0. For cases when the system’s total energy is close to the boundary with non-periodic configurations, the integral is very ill-conditioned (near the top of the disk), but the system can still be evaluated as described in Tilting Pencils.

In the general case, with R > 0 and r > 0, we can define = r/R and = E/(mgr), where 0 < < 2 in order for the system to be periodic. For a periodic system the energy parameter can be expressed in terms of the maximum angular displacement m as 1 cos(m), so equation (3) can be written as

Interestingly, this expression is invariant under replacement of with its reciprocal 1/, as can be verified by direct substitution. This implies that, for a disk of radius R, the time response with a particle of mass m attached at a radius R, when released from a maximum angle m, is identical to the response with the particle attached at a radius R/.

For a disk of unit radius (R = 1) with the mass released from an angle m = /2, the figure below shows the scaled integrand of (5) as a function of , so the time required to rock (and roll) from one angle to another is given by the area under this curve between those two angles.

For example, the angular displacement versus time would be identical for the two disks of radius R shown below in their maximum displacement positions.

It may seem surprising that these two systems will rock and roll identically, because the right hand system has four times the potential energy, which means the mass particle must have twice the speed at the "6 o'clock" configuration. The reason this may be confusing is that the radial arms are in the ration 4:1, so we might think for a given angular speed the right hand particle is moving four times as fast (rather than twice as fast, as required by the energy balance). However, this confusion is due to overlooking the rolling motion of the disks. When in the "6 o'clock" positions, the right hand mass particle will indeed be moving at four times the speed of the left hand particle with respect to the disk centers, but the disk centers will be moving in the opposite direction at the geometric mean of those two speeds. The net effect is that the left and right hand particles will be moving with the total speeds (R R) and (R/ R) where is the angular velocity, so the speed ratio is actually

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