Vol.4, No 17, 2005 pp. 209-224
UDC 531/534(045)=20

ON THE RELATIONS BETWEEN TWO PARAMETRIC
AND USUAL FORMULATIONS OF MECHANICS
Djordje Mušicki
Faculty of Phyisics, University of Belgrade, and
Mathematical Institute, Serbian Academy of Sciences and Arts
Belgrade, Serbia and Montenegro

Abstract. In this paper the relations between two parametric and usual formulation of analytical mechanics have been examined. The first, so-called homogeneous formalism represents the parametric formulation of the Hamilton's variational principle, founded in the physical sense by P. Dirac (1933). The second of these ones is the extended Lagrangian formalism for the rheonomic systems, given by the author himself (Dj. Mušicki, 1992, 2004), which is based on the extension of the set of generalized coordinates by the quantities, suggested by the form of nonstationary constraints, by which the influence of these constrains is included.
Firstly, the main ideas and results of these two parametric formulations have been presented and their principal characteristics emphasized. It was shown, in contrast to the homogeneous formalism, which is equivalent to the usual Lagrangian formulation, that this extended Lagrangian formalism is more general and natural than the usual formulation, including the influence of nonstationary constraints. Particularly, the connections between the corresponding energy change laws in these parametric formulations and the usual one have been established, from where the conditions for the validity and the form of energy conservation law in each of these formulations followed.
Key words: homogeneous formalism, extended Lagrangian formalism

O RELACIJAMA IZMEDJU DVE PARAMETARSKE I
UOBIČAJENE FORMULACIJE MEHANIKE
U ovom radu proučavane su relacije izmedju dve parametarske i uobičajene formulacije analitičke mehanike. Prva , tyv. Homogeni formalizam predstavlja parametarsku formulaciju Hamilton'ovog varijacionog principa, koji je u fizičkom smislu zasnovao P. Dirac )1933=. Drugi od njih je prošireni Lagrange'ov formalizam za reonomne sisteme, koji je formulisao sam autor (Đ. Musšcki, 1992, 2004) i koji se zasniva na proširenju skupa generalisanih koordinata veličinama, sugerisanim oblikom nestacionarnih veza, čime je uključen i uticaj ovih veza.
U prvom delu prikazane su glavne ideje i rezultati dveju parametarskih formulacija i istaknute njihove bitne karakteristike. Pokazano je, za razliku od homogenog formalizma koji je ekvivalentan uobičajenoj Lagrange-ovoj formulaciji, da je ovaj prošireni Lagrange'ov formalizam opštiji i prirodniji od uobičajene formulacije, uključujući i uticaj nestacionarnih veza. Posebno, ustanovljene su relacije izmedju odgovarajućih zakona promene energije u ovim parametarskimformulacijama i uobičajenoj, odakle slede uslovi za važenje i oblik zakona održanja energije u svakoj od ovih formulacija mehanike.
Ključne reči: homoheni formalizam,Lagrange-ove proširene jednačine.