On the application of the principle of least constraint to electrodynamics

Transcription

1 Ueber die Anwendung des Princips des kleinsten Zwanges auf die Elektrodynamik Sitz. Kön. Bay. Akad. Wiss. 4 (894) On the application of the principle of least constraint to electrodynamics By A. Wassmuth in Graz (Received 5 May) Translated by D. H. Delphenich A point m in a system of particles might move from a to b in time τ if it were free while its actual motion is represented by the line segment ac. It is known that the principle of least constraint that Gauss expressed will then say that m (bc) must be a minimum or that one must always have m (bc) < m (bd) when ad is a virtual motion (i.e. one that is consistent with the conditions on the system). If x y z are the coordinates of the point m on which the forces m X m Y m Z should act then any coordinate x will go to: dx d x x τ τ in a very small time τ under the actual motion and: x dx τ X τ for the free motion such that the square of the deviation bc 4 τ d x or the square of the coordinate differences will be equal to X. One then has an 4 expression Z that shall be called the constraint (from the German Zwang) and that takes the form: d x d y d z Z m X Y Z in which the summation extends over all particles and this is the function that must be d x d y d z minimized in regard to the various accelerations which shall be written briefly as xɺɺ yɺɺ zɺɺ If one differentiates the condition equations for the system ϕ 0 ϕ 0 twice with respect to time then as would emerge from the

2 Wassmuth On the application of the principle of least constraint to electrodynamics derivation above ( ) one must currently regard the coordinates x and their first d ϕ ϕ differential quotients as given. The equation 0 expresses only that the x t ɺɺ must possess unvarying values. For given values of x and dx / the xɺɺ shall then be determined in such a way that Z will be a minimum. One will then get the known equations: ϕ ϕ m( ɺɺ x X ) λ λ 0 x x Now n mutually-independent variables p p p 3 p n will be introduced in place of the coordinates x y z such that the virtual work will be equal to P δp P δp P n δp n and the vis viva will be equal to T a p p ɺ ɺ in which the P κλ κ λ and the a κλ a λκ depend upon only the coordinates and the Greek symbols go from to n as they always will from now on. Then as Lipschitz showed (loc. cit. pp. 330) the constraint Z will be expressed by: A Z a ɺɺ p a ɺɺ p p p p p P ɺ ɺ ɺ ɺ κ λ ɺɺ ɺɺ ɺ ɺ ɺ ɺ a p a p p p p p P in which represents the determinant of the a κλ and A represents the adjoint: and one sets: κ λ A a a a a p p p κ λ κλ λ κ. The expression for Z will become clearer and more suited to (some) physical problems when the vis viva T is introduced. Namely if one sets: then ( ) one will also have: T d T T pɺ p ( ) Lipschitz Borch. Journ. Bd. 8 pp. 36 (Rausenberger Mechanik I pp. 66). Gibbs Supplement IV pp. 39. ( ) Cf. e.g. Rayleigh Sound pp. (in the German translation). Stäckel Borch. Jour. 07 pp. 3.

3 Wassmuth On the application of the principle of least constraint to electrodynamics 3 T a ɺɺ p a ɺɺ p p p p p ɺ ɺ ɺ ɺ and one will have: i.e.: Z Z A [T P ] [T P ] A ( T P ) A ( T P ) A ( T P )( T P ). A3 ( T P )( T3 P3 ) (I) Since this expression for the constraint Z would seem to be new it would not be irrelevant to show that one comes to the Lagrange equations when one addresses the minimum condition for Z. Therefore as a result of the remark above one must regard the quantities p and pɺ as given or fixed in the differentiation of Z with respect to pɺɺ and make use of the relation: a κ a κ. ɺɺ p κ One then gets: ɺɺ p a A a A ( T P ) ( T P ) or since those sums are equal to each other: or ( T P ) a A ɺɺ p a A ( T P ) ( T P )[ a A a A ]. Now from a property of the determinant that a A a A or zero according to whether or > so it would follow from 0 that one must also ɺɺ p have T P 0; i.e. Lagrange s equation: d pɺ p P.

4 Wassmuth On the application of the principle of least constraint to electrodynamics 4 One can find even more expressions for the constraint Z in a way that is entirely similar to the way that one exhibits auxiliary forms ( ) for Lagrange s fundamental equation. It is important for the applications to note that the constraint Z can be represented in such a way that one acceleration e.g. pɺɺ will appear in it detached from the remaining ones. One has: Z ( Lɺɺ p M ɺɺ p N ) in which L M N do not include pɺɺ and L and M can be easily found from the equation If the virtual work P δ p and the vis viva T a ɺɺ p 0. κλ κ λ κ λ pɺ pɺ are given for a physical problem then the constraint on the system can be determined by means of equation (I). The minimum property of Z expresses a law for the system which is certainly new in many cases and entirely distinct from the fact that other perhaps already known laws would follow from Z by actual differentiation. That is how e.g. Boltzmann has derived Maxwell s equations for electricity in a wonderfully simple way from Lagrange s equations in volume I of his lectures and thus supported it by mechanical processes. It illuminated the fact that one could also start from the principle of least constraint in the form of equation (I) above as the main theorem and once the virtual work and vis viva were given one would have to arrive at Lagrange s equations by appealing to the minimum condition along with Maxwell s equations in which one now proceeds entirely as Boltzmann did. If it might also become very hard to simplify Boltzmann s classical methods (especially in Part of his lectures) any more in that way nonetheless the condition Z minimum will after all express a newly-recognized truth. As an example one might consider the case of two cyclic coordinates pɺ l and pɺ l (the slowly-varying parameters k will be ignored temporarily). Here one has: T a pɺ a pɺ a pɺ pɺ A B l l C l l when one introduces Boltzmann s notation. It will then follow that: a A a B a C A C C B AB C A B A C A A T d ( Al B l ) T d ( Bl C l ). ( ) Weinstein Wied. Ann. 5. Budde Mechanik I pp. 397.

5 Wassmuth On the application of the principle of least constraint to electrodynamics 5 In addition when there is friction or viscosity the dissipation function (loc. cit. pp. 08 and 09) that Rayleigh exhibited might be denoted by: F ɺ b pɺ b pɺ b pɺ pɺ. ( κ x ) As is known one then adds F to the forces P L. It is also clear that b 0 on ɺ p intrinsic grounds. Ultimately the constraint Z will then be expressed by: Z (A B C ) d B ( Al C l ) L b l d d C ( Al C l ) L b l ( Al C l ) L b l d A ( Al C l ) L b l and Z must be a minimum in such a way that one will have 0 and l l 0. In this (Boltzmann I pps. 34 and 35) l and l represent the current strengths in the two conductors b and b are their resistances L and L are the electromotive forces in them A and B are the coefficients of self-induction and and C is the mutual induction. If condensers (loc. cit. I pp. 35) are also included then terms of the form d l and d l will also enter into the brackets. The condition Z minimum then expresses a basic electrodynamical law and implies the theory of self-induction and mutual induction for current fluctuations that are not too fast. If one would also like to include ponderomotive forces then one would have to introduce a slowly-varying parameter k along with l and l as a third variable exhibit the general expression for Z and construct the equation k ɺɺ 0 in which k and kɺ are assumed to be constant. Only afterwards does one take k ɺ 0 and k ɺɺ 0 and obtain as Boltzmann did the relation: K T k l A l B C l l k k k. Acoustics also serves as another field of applications for the principle above. Frequently only purely-quadratic terms with constant coefficients appear in the expression for the vis viva in that context which makes the equation for the constraint take an even simpler form;

6 Wassmuth On the application of the principle of least constraint to electrodynamics 6 If one introduces the abbreviation: T P Q then the constraint Z will be given by: Z D A ( T P )( T P ) Q Q A Q A Q Q A n Q Q n A Q A n Q Q n A Q. nn n The condition ɺɺ p ρ 0 can be replaced with Q ρ 0. Namely one has: or since: one will have: ɺɺ ɺɺ p ρ Q Q Q ɺɺ p Q ɺɺ p Q ɺɺ p ρ ρ ɺɺ p a ρ a ρ anρ p ρ ρ a ρ (ρ n). Q Q Q n n n ρ Since the determinant D a does not vanish it will generally follow from these n equations that: 0. Q If one actually differentiates the expression above then that will yield the n equations: Q A Q A Q A n Q n 0 ( n) and since the determinant A D n can never be zero that will in turn imply the Lagrange equations: Q 0 Q n 0. If the force P has a potential U such that P U / p and the conditions do not include time t explicitly then one will have on the one hand the vis viva in the form: or T pɺ pɺ pɺ pɺ

7 Wassmuth On the application of the principle of least constraint to electrodynamics 7 dt d pɺ ɺɺ p pɺ pɺ whereas on the other hand since T is also a function of p pɺ one will have: Since: dt pɺ ɺɺ p p pɺ d p ɺ p T it will follow upon subtraction that: to which one adds: du dt pɺ T pɺ T U p ɺ pɺ P pɺ P p and since T P Q one will ultimately arrive at the equation: d( T U ) pɺ Q pɺ Q pɺ Q R. n n One sees that for Q 0 Q n 0 one will also have R 0; i.e. T U must be equal to a constant or that the principle of the conservation of energy must also hold from the Lagrange equations that were found above. Both of them can be obtained simultaneously when one eliminates one of the quantities Q e.g. Q with the help of the relation R pɺ Q and presents the minimum conditions: 0 0 afterwards. It is Q Qn preferable to use the determinant form for the constraint Z in that. For example for n 3 one has: 0 Q Q Q 0 R Q Q which will make: Z D 3 Q a a a 3 Q a a a 3 Q a a a pɺ 3 R b b b 3 Q b a a 3 Q b a a b ( a pɺ a pɺ a3 pɺ 3) p ɺ T pɺ

8 Wassmuth On the application of the principle of least constraint to electrodynamics 8 b b a pɺ a pɺ a3 pɺ 3 b 3 b 3 a3 pɺ a3 pɺ a33 pɺ 3 Applying the minimum conditions will yield: ɺ p ɺ p 3. A R Q [ pɺ A pɺ A ] Q [ pɺ A pɺ A ] A R Q [ pɺ A pɺ A ] Q [ pɺ A pɺ A ] A R Q [ pɺ A pɺ A ] Q [ pɺ A pɺ A ] or since the determinant of this system namely ɺ p A pɺ D never vanishes R 0 and Q 0 Q 3 0; i.e. one has the law of energy and Lagrange s equations. If one eliminates say Q from the general equation: by using the relation: A Q A Q A n Q n 0 R pɺ Q then it will likewise follow naturally that: R 0 Q 0 Q n 0. Addendum Concerning linear current branches. If p p k are in turn cyclic coordinates T a pɺ a pɺ a pɺ pɺ is the vis viva and F b pɺ b pɺ b p ɺ p ɺ is the dissipation function that Lord Rayleigh introduced then the force F (b pɺ b pɺ ) will be ɺ p added to any force P and that will yield the principle of least constraint namely that: Z D A Q Q A Q must be a minimum for any Q. Therefore one has: A Q A Q Q ()

9 Wassmuth On the application of the principle of least constraint to electrodynamics 9 d Q a p a p P b p b p ɺ ɺ ɺ ɺ () D a κλ A κλ D. a κλ If one goes over to electrodynamics and sets: pɺ J pɺ J as well as: d Q a J a J P b J b J (3) then the minimum property () will imply a property of a linear current branch. In it a a a 33 are the coefficients of the self-inductions of the first second loops a a 3 a 3 are the coefficients of mutual induction and P is the constant electromotive force. Furthermore b is the resistance of the entire first loop b is that of the entire second loop etc and b as is the resistance of that piece of the conductor that is common to loops and. b is positive when J and J have the same directions and negative when they have opposite directions. Applying the minimum condition will imply that Q 0 i.e.: P b J b J b n J n d [a J a J ] (4) Those are (in a somewhat generalized form) the equations that H. von Helmholtz presented in 85 (Abhandlungen I pp. 435) for the induction in linear current branches which one must think of as being decomposed into the smallest-possible number of simple loops. For the work done by the retarding forces one gets: F F dp dp ɺ p pɺ ɺ F [ b J b J b J J ] up to sign; i.e. the Joule heat and all of the summands in the bracket are positive. Now assume that a a 3 0; a a a 3 a which is a case is not too difficult to realize experimentally. One will then have: Q a d [J J ] P b J b J and one must have that Z a Q Q is a minimum when a is taken to be small enough. For lim a 0 the current strengths will be independent of time thus constant and it will follow from the condition Z Q Q minimum. Q b J P 0.

10 Wassmuth On the application of the principle of least constraint to electrodynamics 0 with Since the determinant of the b does not vanish the equation J 0. For constant currents one then has to minimize: Z for every J. [( b J ) P ] Q 0 can be replaced Remark: One can get an oft-mentioned minimum property of constant currents from the well-known equation: d( T U ) F or du dt F F () in which U P p represents the potential of the constant force and as above: T a pɺ a J F b pɺ b J If the current strengths J J which are initially zero attain their full strengths J J for t (since dt dj ) and also dt 0 then the F that is found on J the right-hand side of (I) which consists of nothing but positive terms will attain its greatest value. Therefore the negative left-hand side of (I); i.e.: F du b J (P J P J ) must represent a minimum for any J.