31 research outputs found
Theoretical reconstruction of Galileo's two-bucket experiment
In the present work, we address the solution of a problem extracted from a
historical context, in which Galileo supposedly conducted an experiment to
measure the percussion force of a water jet. To this end, the conservation
equations of fluid mechanics in unsteady state are employed in the theoretical
reconstruction of the experiment. The experimental apparatus consists of a
balance, in which a counterweight hangs on to one of its extremities, and two
buckets, in the same vertical, hang on to the other extremity. The water jet
issuing from an orifice in the bottom of the upper bucket strikes the lower
bucket. The objective is to find the jet percussion force on the lower bucket.
The result of the analysis revealed that the method proposed by Galileo for the
calculation of the jet percussion force is incorrect. The analysis also
revealed that the resultant force during the process is practically null, which
would make Galileo's account of the major movements of the balance credible,
despite his having not identified all the forces acting on the system.Comment: 16 pages, 5 figure
Euler\u27s three-body problem
In physics and astronomy, Euler\u27s three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented
On a New Class of Oscillations
In this publication, Euler derived for the first time, the differential equation of the (undamped) simple harmonic oscillator under harmonic excitation, namely, the motion of an object subjected to two acting forces, one proportional to the distance travelled, the other one varying sinusoidally with time
Principles for Determining The Motion of Blood Through Arteries
Translation of Principia pro motu sanguinis per arterias determinando (E855). This work of 1775 by L. Euler is considered to be the first mathematical treatment of circulatory physiology and hemodynamics
Euler's variational approach to the elastica
The history of the elastica is examined through the works of various
contributors, including those of Jacob and Daniel Bernoulli, since its first
appearance in a 1690 contest on finding the profile of a hanging flexible cord.
Emphasis will be given to Leonhard Euler's variational approach to the
elastica, laid out in his landmark 1744 book on variational techniques. Euler's
variational approach based on the concept of differential value is highlighted,
including the derivation of the general equation for the elastica from the
differential value of the first kind, from which nine shapes adopted by a
flexed lamina under different end conditions are obtained. To show the
potential of Euler's variational method, the development of the unequal
curvature of elastic bands based on the differential value of the second kind
is also examined. We also revisited some of Euler's examples of application,
including the derivation of the Euler-Bernoulli equation for the bending of a
beam from the Euler-Poisson equation, the pillar critical load before buckling,
and the vibration of elastic laminas, including the derivation of the equations
for the mode shapes and the corresponding natural frequencies. Finally, the
pervasiveness of Euler's elastica solution found in various studies over the
years as given on recent reviews by third parties is highlighted, which also
includes its major role in the development of the theory of elliptic functions.Comment: 18 pages, 7 figure
Lagrangians for variational formulations of the Navier-Stokes equation
Variational formulations for viscous flows which lead to the Navier-Stokes
equation are examined. Since viscosity leads to dissipation and, therefore, to
the irreversible transfer of mechanical energy to heat, thermal degrees of
freedom have been included in the construction of viscous dissipative
Lagrangians, by embedding of thermodynamics aspects of the flow, such as
thermasy and flow exergy. Another approach is based on the presumption that the
pressure gradient force is a constrained force, whose sole role is to maintain
the continuity constraint, with a magnitude that is minimum at every instant.
From these considerations, Lagrangians based on the minimal energy dissipation
principal have been constructed from which the application of the
Euler-Lagrange equation leads to the standard form of the Navier-Stokes
equation directly, or at least they are capable of generating the same
equations of motion for simple steady and unsteady 1 D viscous flows. These
efforts show that there is equivalence between Lagrangian, Hamiltonian, and
Newtonian mechanics as far as the derivation of the Navier-Stokes equation is
concerned. However, one of the conclusions is that the attractiveness of the
variational approach in more complex situations is still an open question for
the applied fluid mechanician.Comment: 13 page
Euler\u27s Navigation Variational Problem
In a 1747 publication, De motu cymbarum remis propulsarum in fluviis (“On the motion of boats propelled by oars in rivers”), Leonhard Euler (1707-1783) works out various instances of a boat moving at constant speed across a stream flowing in straight streamlines at assigned speeds, in which one of these gives rise to a variational problem consisting of finding the quickest crossing path between two points on opposite side of the river banks, which is generally known as the navigation variational problem. This problem together with the well-known catenary and brachistochrone problems, are considered classical examples in the calculus of variations. Here, we shall present a brief account on Euler’s recurrent interests in calculus of variations, mainly laid out in three publications that span between 1738 and 1744. Particular focus will be given to Euler’s navigation variational problem. A brief account on Lagrange’s contributions to variational calculus is also presented
On the Rectilinear Motion of Three Bodies Mutually Attracting Each Other
This is an annotated translation from Latin of E327 -- De motu rectilineo trium corporum se mutuo attrahentium (“On the rectilinear motion of three bodies mutually attracting each other”). In this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). Here Euler finds that the parameter that controls the relative distances among the bodies is given by a quintic function