Hamilton s principle for non-holonomic systems

Transcription

1 Das Hamltosche Przp be chtholoome Systeme, Math. A. (935), pp Hamlto s prcple for o-holoomc systems by Georg Hamel Berl Traslate by: D. H. Delphech I the paper Le prcpe e Hamlto et l holoomsme, Prace mt.-fz. 38, Kerer has prove the theorem: Holoomty s the ecessary a suffcet coto for Hamlto s prcple to be true, or, more precsely: Holoomty s the ecessary a suffcet coto for the correct equatos of mechacs to be equvalet to the equatos that are obtae from the Lagraga metho of varatoal calculus wth supplemetary cotos wth the help of parameters. It s goo to a that: The former equatos are completely equvalet to the latter oe a ot, say, obtae by specalzg the atoal parameter. Otherwse, the theorem ca be false. Geometrcally, the followg fact s fuametal: As s well-kow, o-holoomc cotos o ot reuce the meso of the space of moto. O the cotrary, ths s ot true for the recto of moto. Thu the eghborg paths that oe must coser the calculus of varatos are a space of hgher meso tha the oes that oe arrves at by supplemetary splacemets. Wth that, t ca happe that these splacemets volve a choce of paths the calculus of varato whch lkewse, alog wth the startg path are themselves the correct paths of mechac whle the theorem of Kerer extes the etty a thus the lowerg of the meso cout. I the followg, t wll be show:. The Hamlto prcple s always correct for a correct formulato, whch s a ol, but less metoe theorem. 2. The theorem of Kerer may be prove by my metho usg bref, effortless calculatos (see: De Lagrage-Eulersche Glechuge er Mechak, Zetschr. f.

2 G. Hamel, Hamlto s prcple for o-holoomc systems 2 Math. u. Phys. 50, Über e vrtuelle Verschebuge er Mechak, Math. Aale 70, a Über chtholoome Systeme, Math. Aale 92). 3. It ca be false, whe oe oes ot observe the atoal remark above. Hamlto s prcple From Alembert s prcple, Lagraga form: S m wδ r = S k δ r (S meas oe sums over the system, r s the posto vector, m, the mass elemet, w, the accelerato vector a k s the vector of apple force), what follow wth the always supplemetary assumpto that: s the cetral Lagraga equato: δ r δ r = 0, S mv δ r δ(e U) = 0, where v refers to the velocty, E, to the ketc eergy, a U, to the potetal eergy that are assume to be preset. It yel by tegratg over the tme terval from t to t 2, at whose es the vrtual splacemets shall be zero: t2 δ L = 0 t (L E U), hece, Hamlto s prcple. The varatos must therefore be regare as supplemetary splacemets here, from whch, the eghborg paths that oe arrves at wll ot be supplemetary to the rule. For the sake of smplcty, we take the system to be scleroomc. Let the Lagraga coorates be q, q 2,, q. I place of the qɺ, we thk of there beg learly epeet couplgs trouce betwee them: ϑ ω = b, sq ɺ or, whe solve: s s s= qɺ = c ω, = whch we ca o such a way that the o-holoomc coto equatos become: ω k+ = 0, ω k+2 = 0,, ω = 0 k <.

3 G. Hamel, Hamlto s prcple for o-holoomc systems 3 The commutato equatos may be: δϑ σ δ ϑ σ = β, σδϑ ϑs. They, together wth Hamlto s prcple, mmeately gve the correct equatos of moto for k < (sce δϑ σ = 0 for σ > k) p + β ω p (I), m s m m s qs, s c = 0, =, 2,, k. The p m = / ω are the mpulses. O the cotrary, the problem of the calculus of varatos: δ L = 0, wth the supplemetary cotos ω k+ = 0, ω k+2 = 0,, ω = 0, gves: or: + k + δ ( L λν ων ) = 0 ( pm + λm) δωm + δ qν = 0, m ν qν where λ = 0, λ 2 = 0,, λ k = 0, a the other λ are the Lagraga parameters. As before, t ow follows that the δϑ are all to be treate as arbtrary: (II) ( p + λ ) + β,, ω ( p + λ ) c, q whch s ow true for all. = 0, s m s m m s m s s 2. Proof of Kerer s theorem Shoul (I) a (II) both be correct, t the follows from a comparso of the frst k equatos that: =, 2,, k, β, mωsλm = 0 for s =, 2,, k, m or, sce the values of ω s ca be gve freely at ay locato:

4 G. Hamel, Hamlto s prcple for o-holoomc systems 4 (III) β, mλm = 0 s a =, 2, 3,, k. m= k +, k+ 2,, Thu the λ gve the fferetal equatos: ( p + λ ) + β ω ( p + λ ) c = 0, = k +, k + 2,,., m s m m m s qs Now, shoul a complete etty betwee both systems of equatos exst the sese escrbe the troucto, the the λ, whch ca be chose freely at ay locato, are subject to o fte restrctos lke (III), such that oe must have: β,m = 0 for s a =, 2,, k a m = k+, k + 2,,. However, the commutato equatos for the last ϑ the take the form: δϑ σ δ ϑ σ = β, δϑ ϑs, s = k +, k + 2,,., s= k +, k+ 2,, σ σ From a theorem of Frobeus (Crelle 82, pp. 267, see also the Ezyklopäe er math. W. II, A.5, 5, pp. 39, rem. 90), ths however, suffcet for us to coclue the tegrablty of our equatos of coto. Kerer s theorem s thus prove. 3. A couter-example The followg example shall show that the theorem s false whe oe oes ot requre the complete etty. Let: L = ( qɺ + qɺ + ω ), 2 2 wth the partcular o-holoomc coto: ω qɺ 3 + qqɺ 3 = 0. Because, from a well-kow theorem (see my frst paper, pp. 25) oe ca set ω = 0 from the outset, the correct equatos of moto rea: qɺɺ = 0, qɺɺ 2 = 0, ω = 0. O the cotrary, the equatos of the varatoal problem rea:

5 G. Hamel, Hamlto s prcple for o-holoomc systems 5 qɺɺ λqɺ 2 = 0, ( q ɺ 2 + λ q ) = 0, λ = 0, ω = 0, such that λ = cost. If oe ow as the restrcto that λ = 0 the oe obtas the correct equatos of moto. (Receve o 23, 0, 934)