Energy in Closed Systems

Transcription

1 Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and pont out the beakdown. The beakdown s fst demonstated wth a two body system and fnally wth a many body system. Intoducton. We consde consevaton of enegy [] wth a many body system n elaton to the nvese squae law and pont out the beakdown. The beakdown s fst demonstated wth a two body system and fnally wth a many body system. Infntesmal wok dw s gven by, Basc calculatons dv dw = dw = F d = m d () dw:total nfntesmal wok done on the system by extenal+ntenal foces, coveng all the patcles dv dw = m d = m v dv = m v dv = m d(v v ) dw = m dv dw = d ( m v ) () Let s now consde a patcle n a consevatve feld: E = U dw = E. d = U. d = du f the potental functon s ndependent of tme

2 If U depends on tme then du = U t We have fo tme ndependent potentals Agan U + U. d U. d = du ; U. d du t dw = du dw = KE f KE The last equaton s the wok enegy theoem whch s unvesally vald KE f KE = du KE f KE = U U f KE + U = KE f + U f (4) The above s vald fo a tme ndependent consevatve feld. If the mass[magntude of souce] of one body s much lage than the othe bodes nvolved then the above two conons ae satsfed to a good/hgh degee of appoxmaton. The potentals of the smalle bodes n moton ae gnoable. Thus tme ndependence s acheved. Invese squae Law, Gavtaton Potental enegy n a thee body gavtatonal system: dw = F d + F d + F d = (F + F )d + (F + F )d + (F + F )d whee F j s the foce on the th patcle fom the j th one due to gavty. dw = (F + F )d + (F + F )d + (F + F )d = (F d + F d ) + (F d + F d ) + (F d + F d ) = (F d F d ) + (F d F d ) + (F d F d ) = F (d d ) + F (d d ) + F (d d ) = F d( ) + F d( ) + F d( ) = F d + F d + F d ; j = j dw = Gm m d + Gm m d + Gm m d (5)

3 = Gm m d(. ). Gm m d(. ) Gm m d(. ) dw = Gm m d. Gm m d Gm m d w = Gm m + Gm m + Gm m + constant (6) w excludes extenal wok[t s wok done by the system].it aplles to a closed thee body gavtatng system. Compang () and (5) we have fo a closed system whee extenal wok s zeo, d ( m v ) = Gm m d + Gm m d + Gm m d w = m v + C = Gm m d + Gm m d + Gm m d + C (7) If m v nceases Gm m d + Gm m d + Gm m d has to ncease (Gm m d + Gm m d + Gm m d ) deceases Fo a closed system,we have We ecall m v (Gm m + Gm m + Gm m ) = constant (7)(8) dw = Gm m d( ). Gm m d( ) Gm m d( ) w = +Gm m + C (, ) + Gm m +C (, ) + Gm m + C (, ) w = 0 = Gm m + C (, ) + Gm m +C (, ) + Gm m + C (, ) = 0 Dffeentatng the above wth espect to tme

4 4 d Gm m + Gm d m + Gm d m = (C (, )+C (, ) + C (, )) d Gm m + Gm d m + Gm d m = d(c (, )+C (, ) + C (, )) 0 d Gm m + Gm d m + Gm d m = d(c (, )+C (, ) + C (, )) 0 () The functons C, C and C ae abtay: we can fx them up accodng to ou choce The above s not tue when dw=0. Fom(5) we expect the ght sde to be zeo when dw=0. Theefoe C (, )+C (, ) + C (, ) should be ndependent of tme. Theefoe the constant n(7) s tme ndependent.,j; j Gm m j = constant,ndependent of tme. Else the ght sde (8) wll be non zeo when we j eque t to be zeo[fo w=0]. Fom the Lagangan fomulaton If the potental functon s ndependent of the genealzed veloctes then we have the elaton [][] T + V =constant; T: total knetc enegy; Vpotental functon of the system of patcles; we consde the potental functon to be velocty ndependent.: ;patcle ndex j:component ndex[j = x, y, z] satsfes V x j = F j Gm m j = V (9) j,j; j

5 5 V x j = F j m v Gm m j j = constant (0),j Closed Systems Fo a closed system whee extenal wok s zeo we have m v + C = Gm m d + Gm m d + Gm m d + C [The ght sde nvolves only ntenal foces: gavtaton. Hence the wok s ntenal wok. Fo an n body system m v We apply ths dea two body system Gm m j j = constant (0),j m v + m v Gm m = C () Two Body moton Two gavtatng masses m A and m B aqe beng consdeed d B d AB d A d B d A = Gm B = Gm A AB AB AB AB = G(m A + m B ) AB AB = G(m A + m B ) AB AB () j = j d AB. d AB = G(m A + m B ) AB. d AB AB

6 6 d (d AB ) d (d AB ) = G(m A + m B ) d( AB. AB ) AB = G(m A + m B ) d( AB ) AB d (d AB ) = G(m A + m B ) AB d AB AB d (d AB ) = G(m A + m B ) d AB AB d ( d AB ) = G(m A + m B ) d AB AB ( d AB ) = G(m A + m B ) + E () AB E: constant of ntegaton ( d AB ) G(m A + m B ) = E[constant](4) AB Now, d AB = d( B A ) = d B d A = v B v A Theefoe, ( d AB ) = (v B v A ) (v B v B ) G(m A + m B ) AB = E = E v A + v B v A. v B G(m A + m B ) AB = E (5) Momentum of m A = p A ; Momentum of m B = p B Fo COM fame p A = p B = P

7 7 p A = p B = P m A v A = mb v A = P ; Equaton (4) stands as P m A + P m P. ( P ) B m A m B + K AB = E P [ m + A m + ] G(m A + m B ) = E B m A m B AB [ + ] G(m A + m B ) = E (6) m A m B AB P We obseve fom (5), m Av A + m Bv B Gm Am B AB = E (7) P P m A + P m B Gm Am B AB = E [ m A + m B ] Gm Am B AB = E (8) Dvng equaton by (6) we may solve fo AB whch becomes a constant Many Body Moton Smla beakdown may be obseved fo an n body system whee j = j By subtsacton, m d = Gm m + Gm m m d = Gm m + Gm m + Gm m Gm m Gm m n n n 4 Gm m n n n d d = Gm Gm + R

8 8 d = G m + m + R (9) Let R = G m + m G m + m ( ) + d (0) whee s a sutable vecto satsfyng the above equaton. Now we have two body moton d ( ) = G m + m ( ) () It s possble to locate an ogn fom whee the pa (m, m ) executes an equvalent two body moton We may wte : = = The equaton now stands as d = G m + m () d ( ) = G m + m ( ) In a geneal manne the above equaton and hence equaton () may be obtaned fom the followng two equatons[by subtacton] d = G m + f d = G m + f By applyng an acceleaton of f on the ogn we cancel out ths acceleaton. We nowdo have d = G m d = G m We now have ou nvese squae law and consequently a pa of equatons lke (6) and (8)!

9 9 Physcal Independence of foces Fst we wte the followng equatons [n an n body moton] fo the patal nteactons: m d = Gm m (.) m d = Gm m (.) m d 4 = Gm m 4 4 (.).. m d n = Gm m n n (. n) Each of the above equatons expesses an ndependent two body moton whch leads to equatons of the type (6) and (8)!Each holds fo evey nstant of tme as f the patal ntactons wee poceedng ndependently. We also have [total acceleaton= sum of patal acceleatons] d ( n ) = d = n + Ct + D We solve,, 4. n fom (.),(.)..(.n).then we apply = n + Ct + D to obtan the many body soluton subject to the dffculty mentoned[equatons [(6) and (8)] Fo valdaton(coss checkng) we may consde the followng j = j + j +. + jn + C j t + D j j = t C j + D j j j j

10 0 Fo Cente of Mass fame m j j = 0 j Concluson Thee s a clea ndcaton of laws gettng volated. Classcal physcs tself s a beacon to new Refeences Goldsten H., Poole C, Safko J, Classcal Mechancs, Thd eon, Addson Wesley,p6-6 Goldsten H., Poole C, Safko J, Classcal Mechancs, Thd eon, Addson Wesley,p6-6 Goldsten H., Poole C, Safko J, Classcal Mechancs, Thd eon, Addson Wesley,p