§ 6. Reciprocity for the forward and reverse motions

I call the motion of a system reversible when the sequence of positions that it passed through during its forward motion can also be traversed by the reverse motion without the action of other forces, and with the same intermediate times for each pair of equal positions. The reverse motion will be possible when the values of the kinetic potential is not changed when one changes the signs of all qa . However, if products and powers of the qa of odd degree occur, which happen, e.g., for the interference of hidden motions (§ 1), then the motion will be reversible only when it is mechanically possible to also make some of the constants (viz., the velocities of the hidden motions) negative in such a way that the quantity H does not change in value under a simultaneous setting to negative values of these constants and all qa . This is easily obtained from a consideration of the equations of motion (1c ) when one considers that dt must also assume the opposite sign under reversal. Law of reciprocity. In my acoustic investigations (**), I proved a law of reciprocity that I easily extended to small oscillations around a stable equilibrium position of an arbitrary, oscillating, mechanical system in my lectures. However, it is more general, and true for any moving system that is subject to the law of least action and has a reversible motion. The original motion A will be unchanged when one keeps all initial positions at time t0 unchanged, but increases one of the moments of motion s1 by ds1. In that way, the coordinate p2 must increase by dp2 at time t. If one then changes the moment of motion s2 in the reversed motion when it goes through the value pa of the coordinates by the same amount that one changed s1 then the coordinate p1 will be changed by just as much as p2 after the time interval t = t1 – t0 . Since all dt and dpa must be zero, we will have: (11) dsa

s dp p   ∂ ⋅   ∂  ∑ a b b b .

(* ) Jacobi, loc. cit., Lecture XX. (**) “Theorie der Luftschwingungen in Röhren mit offenen Enden,” this Journal, Bd. 57, pp. 27-30. Helmholtz – On the physical meaning of the principle of least action. 33 Of these, only ds1 should be non-zero. For the sake of brevity in our notation, we would like to write:

From (10b ), the quantities σa, b are the same as the σb, a . We denote the determinant of the quantities σa, b by D(σ) . If these are not identically zero then, from equations (11), with the restriction that was made, we will have: (11b) dp2 = ( ) 1,2 log D σ σ ∂ ∂ ⋅ ds1 . By contrast, if we demand that all dpa = 0, and likewise all dsa = 0, with the exception of ds2 , then we will get, with consideration given to (11a ), the corresponding equation: dp1 = ( ) 1,2 log ( ) D σ σ ∂ − ∂ − ⋅ ds2 for the forward motion. For the reverse motion, the signs of the moments of motion are inverted, and thus, those of the σa, b , as well; thus, one will have:

for them.

It follows from the combination of equations (11b ) and (11c) that:

dp2 : ds1 = dp1 : ds2 ,

with which, the theorem that was expressed above will be proved.

As far as the exceptional case is concerned, in which the determinant D(σ) is equal to zero identically, in that case, the dpb would not necessarily be equal to zero when all dsa are also equal to zero, without exception. Now, since the motion of the system must be determined completely, and therefore, the values of the pa would not be double-valued during the course of time t if the initial positions pa and the initial velocities were given at the start of the time interval t, this exceptional case could occur only if the qa were not determined completely by the values of the sa , which we excluded in the concluding remarks of § 1. It is therefore unnecessary to pay any attention to that exceptional case. The sudden changes in the values of the sa and the sa here, under which, the coordinates themselves should suffer no changes in their values, would come about mechanically in such a way that one lets forces Pa act during a very small time interval, Helmholtz – On the physical meaning of the principle of least action. 34 but with corresponding intensity. In that way, the various rising degrees of velocities can be traversed without also having to change the position of the time at which the greatest velocity is attained notably. For such an assumption, it follows from equation (1) that: − ∫ Pa ⋅ dt = s1 – s0 .

Since Pa denoted the force that was exerted upon the moving system from the outside, in the notation that was used there, (− Pa ) will be the opposite external force that is required in order to bring about the desired change in motion. Following Sir W. Thomson, we would like to refer to such a force effect as a push in the direction of the coordinate pa . In that regard, one must remark that, in general, the forces Pa are aggregates of components that act upon different parts of the system, and are distributed in such a way that the aggregate of forces Pa performs no work under any variation of the remaining coordinates, except for pa . Moreover, since we have to distinguish between forward and reverse motion, it would be preferable to consider values of the dsa that increase the forward moment sa , as well as displacements dpa that increase the distance (pa – pa ), to be positive for the forward motion, while for the reverse motion, values of the dsa that increase the reverse moment (− sa ) and displacements (− dpa ) that increase the distance (pa – pa ) are considered to be negative, and to treat them as equivalent to the positive changes in dsa and dpa for the forward motion. The reciprocity theorem can then be expressed as: If a push that increases only the moment sa by dsa at the initial position of the forward motion has increased the coordinate of the final position pb by dpb after the time interval t then an equivalent reverse push that increases the reverse moment (− sb ) by the same amount at the earlier final position would provoke the equivalent reverse change in the coordinate pa after time t.