ERMAT. We shall then propose a synthesis of thesetwo, which, perhaps, can be disputed, but which has incontestable elegance. Moreover,we shall find a solution to the problem we have posed.2.2. Two principles of least action in classical dynamicsIn classical dynamics, the principle of least action is introduced as follows:The equations of dynamics can be deduced from the fact that the integralt2t1dt,between fixed time limits,t1andt2and specified by parametersqiwhich give the state ofthe system,has a stationalry value.By definition,, known as Lagrange’s function, orLagrgian, depends onqiand ˙qidqidtThus, one has:(2.2.1)δt2t1dt0From this one deduces the equations of motion using the calculus of variations givenby LAGRANGE:(2.2.2)ddt∂∂˙qi∂∂qiwhere there are as many equations as there areqi.It remains now only to define. Classical dynamics calls for:(2.2.3)EkinEpoti.e., the difference in kinetic and potential energy. We shall see below that relativisticdynamics uses a different form for.Let us now proceed to the principle of least action of MAUPERTUIS. To begin, wenote that LAGRANGE’s equations in the general form given above, admit a first integralcalled the “system energy” which equals:(2.2.4)W∑i∂∂˙qi˙qiunder the condition that the functiondoes not depend explicitely on time, which weshall take to be the case below.dWdt∑i∂∂qi˙qi∂∂˙qi¨qi∂∂˙qi¨qiddt∂∂˙qi˙qi∑i˙qiddt∂∂˙qi∂∂˙qi(2.2.5)
2.2. TWO PRINCIPLES OF LEAST ACTION IN CLASSICAL DYNAMICS17which according to LAGRANGE, is null. Therefore:(2.2.6)WconstWe now apply HAMILTON’s principle to all “variable” trajectories constrained toinitial positionaand final positionbfor which energy is a constant. One may write, asW,t1andt2are all constant:(2.2.7)δt2t1dtδt2t1W dt0or else:(2.2.8)δt2t1∑i∂∂˙qi˙qidtδBA∑i∂∂˙qidqi0the last integral is intended for evaluation over all values ofqidefinitely contained be-tween statesAandBof the sort for which time does not enter; there is, therefore, nofurther place here in this new form to impose any time constraints. On the contrary, allvaried trajectories correspond to the same value of energy,W.1In the following we use classical canonical equations:pi∂∂˙qi. MAUPERTUIS’principle may be now be written:(2.2.9)δBA∑ipidqi0in classical dynamics whereEkinEpotis independent of ˙qiandEkinis a homoge-neous quadratic function. By virtue of EULER’s Theorem, the following holds:(2.2.10)∑ipidqi∑ipi˙qidt2EkinFor a material point body,Ekinmv22 and the principle of least action takes its oldestknown form:(2.2.11)δBAmvdl0wheredlis a differential element of a trajectory.1Footnote added to German tranlation: To make this proof rigorous, it is necessary, as it well known, toalso varyt1andt2; but, because of the time independance of the result, our argument is not false.