ERMAT We shall then propose a synthesis of these two which perhaps can be

Ermat we shall then propose a synthesis of these two

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ERMAT . We shall then propose a synthesis of these two, 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 dynamics In classical dynamics, the principle of least action is introduced as follows: The equations of dynamics can be deduced from the fact that the integral t 2 t 1 dt, between fixed time limits, t 1 and t 2 and specified by parameters q i which give the state of the system, has a stationalry value. By definition, , known as Lagrange’s function, or Lagrgian, depends on q i and ˙ q i dq i dt Thus, one has: (2.2.1) δ t 2 t 1 dt 0 From this one deduces the equations of motion using the calculus of variations given by L AGRANGE : (2.2.2) d dt ˙ q i q i where there are as many equations as there are q i . It remains now only to define . Classical dynamics calls for: (2.2.3) E kin E pot i.e., the difference in kinetic and potential energy. We shall see below that relativistic dynamics uses a different form for . Let us now proceed to the principle of least action of M AUPERTUIS . To begin, we note that L AGRANGE ’s equations in the general form given above, admit a first integral called the “system energy” which equals: (2.2.4) W i ˙ q i ˙ q i under the condition that the function does not depend explicitely on time, which we shall take to be the case below. dW dt i q i ˙ q i ˙ q i ¨ q i ˙ q i ¨ q i d dt ˙ q i ˙ q i i ˙ q i d dt ˙ q i ˙ q i (2.2.5)
2.2. TWO PRINCIPLES OF LEAST ACTION IN CLASSICAL DYNAMICS 17 which according to L AGRANGE , is null. Therefore: (2.2.6) W const We now apply H AMILTON ’s principle to all “variable” trajectories constrained to initial position a and final position b for which energy is a constant. One may write, as W , t 1 and t 2 are all constant: (2.2.7) δ t 2 t 1 dt δ t 2 t 1 W dt 0 or else: (2.2.8) δ t 2 t 1 i ˙ q i ˙ q i dt δ B A i ˙ q i dq i 0 the last integral is intended for evaluation over all values of q i definitely contained be- tween states A and B of the sort for which time does not enter; there is, therefore, no further place here in this new form to impose any time constraints. On the contrary, all varied trajectories correspond to the same value of energy, W . 1 In the following we use classical canonical equations: p i ˙ q i . M AUPERTUIS principle may be now be written: (2.2.9) δ B A i p i dq i 0 in classical dynamics where E kin E pot is independent of ˙ q i and E kin is a homoge- neous quadratic function. By virtue of E ULER ’s Theorem, the following holds: (2.2.10) i p i dq i i p i ˙ q i dt 2 E kin For a material point body, E kin mv 2 2 and the principle of least action takes its oldest known form: (2.2.11) δ B A mvdl 0 where dl is a differential element of a trajectory. 1 Footnote added to German tranlation : To make this proof rigorous, it is necessary, as it well known, to also vary t 1 and t 2 ; but, because of the time independance of the result, our argument is not false.
18 2. THE PRINCIPLES OF MAUPERTUIS AND FERMAT