On the fundamental formulae of dynamics. Offprint from: American Journal of Mathematics, vol. 2. THE "GIBBS-APPELL EQUATIONS" FOR CONSTRAINED DYNAMICAL SYSTEMS

GIBBS, Josiah Willard

Date d'édition : 1879

A propos de cet article

First edition, extremely rare offprint, of the 'Gibbs-Appell equations' (rediscovered by Paul Émile Appell in 1900). They describe the dynamics of constrained mechanical systems: 'constrained' means that the coordinates and velocities of the parts of the mechanical system are not free to take arbitrary values but must satisfy one or more equations. "The concepts and methods of analytical mechanics are at the foundation of most of modern-day physics . . . The general equations of motion for constrained mechanical systems were discovered independently by Gibbs [in the present work] and by Appell. They are often referred to as the celebrated Gibbs-Appell equations and are considered by most to represent the pinnacle of our understanding of the time evolution of constrained mechanical systems. Although the formulations of analytical mechanics by Newton, Lagrange, Gauss, and Hamilton are considered to be equivalent, they each have a slightly different perspective on the understanding of the evolution of a dynamical system. Besides adding to our understanding of the motion of mechanical systems, each perspective has certain advantages when dealing with specific situations. Though the problem of determining the evolution of a constrained mechanical system was first formulated at least as far back as Lagrange, the determination of the explicit equations of motion for such a system, especially when the constraints are non-integrable [i.e., when they cannot be reduced to equations involving the coordinates alone], has since been a major hurdle in mechanics. The Lagrange multiplier method that Lagrange devised to handle this problem is very difficult, if not impossible, to use (from both an analytical and a computational viewpoint) when the system has several tens of dgerees of freedom [i.e., number of coordinates] and is subject to many non-integrable constraints. More than 100 years after Lagrange, Gauss and Appell came up with a general approach for handling constrained motion by expanding the set of coordinates to include quasi-coordinates. The equations they proposed broke a major impasse in our ability to obtain the explicit equations of motion for constrained systems; many consider the Gibbs-Appell equations to be probably the simplest and most comprehensive equations of motion so far discovered" (Udwadia & Kalaba, p. 64). Gibbs (1839-1903) "was one of the greatest scientists in the United States in the 19th century. His application of thermodynamic theory converted a large part of physical chemistry from an empirical into a deductive science" (Britannica). OCLC lists only three copies worldwide (Huntington, Yale and Trinity College, Cambridge). No copies in auction records. Udwadia & Kalaba,'Equations of motion for mechanical systems,' Journal of Aerospace Engineering 9 (1996), pp. 64-69. 4to, pp. 49 64. Original printed wrappers (small repair to upper left corner of front wrapper).

N° de réf. du vendeur ABE-1635682217774

Détails bibliographiques

• Titre: On the fundamental formulae of dynamics. Offprint from: American Journal of Mathematics, vol. 2. THE "GIBBS-APPELL EQUATIONS" FOR CONSTRAINED DYNAMICAL SYSTEMS
• Date de parution: 1879
• Reliure: Soft cover
• Édition: 1st Edition
• État de l'article: Very Good