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.
