# zbMATH — the first resource for mathematics

Variational metrics on $$\mathbb{R}\times TM$$ and the geometry of nonconservative mechanics. (English) Zbl 0818.53026
The paper under review presents a nice and compact exposition of several mathematical aspects of the time-dependent non-conservative classical mechanical systems. The recent extensive research in this area has shown that the setting of the variational calculus on fibered manifolds, together with certain generalizations of the concept of a connection, provide the mathematical tools which clarify and simplify the theory. Based on the results by J. Grifone, D. Krupka, M. de Léon, the author herself, and others, the paper offers a unified and clear picture of the relations between the variationality of metrics on $$\mathbb{R}\times TM$$, the variationality and metrizability of semispray connections on $$\mathbb{R}\times TM$$, and the global dynamics of a non-conservative system. The exposition starts with a detailed survey of the basic concepts, then several original results are achieved, and finally examples are presented which show in particular that a series of classical situations is covered, inclusive manifolds with Riemannian and Finslerian metrics, and manifolds with time-dependent variational metrics, manifolds with metrizable linear connections on $$TM$$.
Reviewer: J.Slovák (Brno)
##### MSC:
 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics) 70H03 Lagrange’s equations 53C05 Connections (general theory) 53Z05 Applications of differential geometry to physics 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
Full Text:
##### References:
 [1] GRIFONE J.: Structure presque-tangente et connexions. I, Ann. Inst. Fourier (Grenoble) 22 (1972), 287-334. · Zbl 0301.53020 [2] KAWAGUCHI H.: The d-connections in Lagrange geometry. Differential Geometry and Its Applications. Proc. Conf., Sept. 1989, Brno, Czechoslovakia, World Sci., Singapore. 1990, pp. 230-235. · Zbl 0791.53019 [3] KRUPKA D.: Lepagean forms in higher order variational theory. Proc. IUTAM-ISIMM Symp. on Modern Developments in Analytical Mechanics, June 1982, Turin; Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., Suppl. al Vol. 117 (1983), 197-238. [4] KRUPKA D.: Geometry of Lagrangean structures 2, 3. Proc. 14th Winter School on Abstract Analysis, Jan. 1986, Srní (Czech Rep.), Rend. Circ. Mat. Palermo (2) Suppl. 14 (1987), 178-224; Arch. Math. (Brno) 22 (1986), 211-228. [5] KRUPKA D., SATTAROV A. E.: The inverse problem of the calculus of variations for Finsler structures. Math. Slovaca 35 (1985), 217-222. · Zbl 0585.53019 [6] KRUPKOVÁ O.: Lepagean 2-forms in higher order Hamiltonian mechanics. II. Inverse problem. Arch. Math. (Brno) 23 (1987), 155-170. · Zbl 0651.58010 [7] KRUPKOVÁ O.: A note on the Helmholtz conditions. Differential Geometry and Its Applications. Proc. Conf., August 1986, Brno, Czechoslovakia, J.E. Purkyně University. Brno, 1986, pp. 181-188. [8] KRUPKOVÁ O.: Variational metrics on RxTM. Preprint, Silesian University at Opava, Opava (Czech Republic) (1991), 1-15. [9] de LEÓN M., RODRIGUES P. R.: Generalized Classical Mechanics and Field Theory. North-Holland, Amsterdam, 1985. · Zbl 0581.58015 [10] MANGIAROTTI L., MODUGNO M.: Fibered spaces, jet spaces and connections for field theories. Proc. of the Meeting ”Geometry and Physics”, Oct. 1982, Florence, Pitagora, Bologna, 1982, pp. 135-165. · Zbl 0539.53026 [11] MODUGNO M.: Torsion and Ricci tensor for non-linear connections. Differential Geom. Appl. 1 (1991), 177-192. · Zbl 0784.53008 [12] SAUNDERS D. J.: Jet fields, connections and second-order differential equations. J. Phys. A 20 (1987), 3261-3270. · Zbl 0627.70013 [13] SHIMADA H.: Cartan-like connections of special generalized Finsler spaces. Differential Geometry and Its Applications. Proc. Conf., Sept. 1989, Brno, Czechoslovakia, World Sci., Singapore, 1990, pp. 270-275. · Zbl 0791.53030 [14] VONDRA A.: Connections in the Geometry of Non-Autonomous Regular Higher-Order Dynamics. Thesis, Masaryk University, Brno (Czech Republic), 1991. [15] VONDRA A.: Semisprays, connections and regular equations in higher order mechanics. Differential Geometry and Its Applications. Proc. Conf., Brno, Czechoslovakia, 1989, World Scientific, Singapore, 1990, pp. 276-287. · Zbl 0809.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.