PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle

Transcription

1 1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple

2 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo f ) ( a) F p We can wte down, when we nclude dynamcs, Ths s the D Alembet s Pncple. f 0 0 Agan, snce the coodnates (and the vtual vaatons) ae not necessay ndependent. Ths does not mples,. F ( a) p We now need to look nto changng vaables to a set of ndependent genealzed coodnates so that we can wte and set? 0 the ndependent coeffcents n the sum to zeo. 0? q 0

3 3 Devaton of Lagange Equatons ( a) F p Beak nto two peces: 1. F ( a) (1) 0 Assume that we have a set of n=3n-k ndependent genealzed coodnates q and the coodnate tansfomaton, Fom chan ule, we have q, q,, q, t 1 2 n q q (note: t 0 snce t s a vtual dsp) t (Index conventon: goes ove # patcles and ove genealzed coods)

4 4 Devaton of Lagange Equatons Ths lnks the vaatons n to q, substtutng t nto expesson (1), we have, Defnng F F q F q ( a) ( a) ( a) q q Q F ( a) q as the genealzed foces We can then wte, F ( a) (Note: Q needs not have the dmensons of foce but must have dmensons of wok.) Q q (1') Q q

5 5 Devaton of Lagange Equatons Now, we look at the second pece nvolvng : p 2. p m m (2) q (don t foget the - sgn n the ognal Eq) (mass s assumed to be constant) m q (2 a) q q

6 6 Devaton of Lagange Equatons Let, go backwad a bt. Consde the followng tme devatve: d d m m m q q q Reaangng, the last tem (fom the pevous page) can be wtten as, d d d m m m (2 b v v ) q whee v q q Now, consde the blue and ed tems n detal,

7 7 Devaton of Lagange Equatons blue tem: q Snce we have, applyng chan ule, we have v q, q,, q, t 1 2 n d q q t k k k Takng the patal of above expesson wth espect to q, we have v q q (note: does not depend on q ) ed tem: d q d v q q (swtchng devatve ode) Is t ok? Check

8 8 explct check q, q,, q, t 1 2 n LHS: d qk q k q k q t q RHS: d q k q q k qk t q q q q t k k k q k k q k q tq Check! The two tems ae the same.

9 Devaton of Lagange Equatons Puttng these two tems back nto Eq. (2b): Wth ths, we fnally have the followng fo expesson (2): d m m m q q q v v v v m q q p (emnde: sums ove # patcles and sums ove genealzed coods) d m m d m q d q t q v v 9 (2 ) q q c d m m q v v v v

10 10 Devaton of Lagange Equatons We ae almost thee but not qute done yet. Consde takng the q devatve of the Knetc Enegy, T q q mv m 2 q v v 2 1 v v m v v 2 q q v mv q Smlaly, we can do the same manpulatons on T wt to, T q 1 2 m m q v 2 v v q q

11 Devaton of Lagange Equatons Substtutng these two expessons nto Eq. (2c), we have: Fnally, econstuctng the two tems n the D Alembet s Pncple, we have: T T q d m m q q q q d q v v p v v ( ) 0 a F p 0 d T T Q q q q 11

12 12 Devaton of Eule-Lagange Equatons q Now, snce all the ae assumed to be ndependent vaatons, the ndvdual backeted tems n the sum must vansh ndependently, d T T q q Q (3) Thee ae 3N-K of these dffeental equatons fo 3N-K q and the soluton of these equatons gves the equatons of moton n tems of the genealzed coods wthout the need to explctly knowng the constant foces. Also, note the advantage of ths equaton as a set of scala equatons (wth T) nstead of the ognal 2 nd law whch s a vecto equaton n tems of foces.

13 13 E-L Equaton fo Consevatve Foces Case 1: ( a) F devable fom a scala potental F U,,,, t ( a) 1 2 N (note: U not depend on veloctes) Q Q F U ( a) q q ˆ ˆ ˆ U xˆ ˆ ˆ k y z x y z q k U x U y U z x q y q z q U q

14 14 E-L Equaton fo Consevatve Foces Puttng ths expesson nto the RHS of Eq. (3), we have, d T T U Q q q q d T T U q q Notce that snce U does not depends on the genealzed velocty, we ae fee to subtact U fom T n the fst tem, 0 q d T U T U q q 0

15 15 E-L Equaton fo Consevatve Foces We now defne the Lagangan functon L = T U and the desed Eule-Lagange s Equaton s: d L L q q 0 Note: thee s no unque choce of L whch gves a patcula set of equatons of moton. If G(q, t) s a dffeentable functon of the genealzed coodnate, then L' qqt,, L qqt,, dg wll be a dffeent Lagangan gvng the same EOM.

16 16 E-L Equaton fo Velocty Dependent Potentals Case 2: U s velocty-dependent,.e., Uq (, q, t) In ths case, we edefne the genealzed foce as, Q d T T Q q q Q U d U q q Now, substtute ths nto, we then have, d T T U d U q q q q

17 17 E-L Equaton fo Velocty Dependent Potentals Combng tems usng L = T U, we agan have the same Lagange s Equaton, T U T U d q q d L L q q 0 0 Ths s the case that apples to EM foces on movng chages q wth velocty v, U q Av whee s the scala potental 1 2 And, L mv 2 q Av and A s the vecto potental

18 18 E-L Equaton fo Geneal Foces Case 3 (Geneal): Appled foces CANNOT be deved fom a potental One can stll wte down the Lagange s Equaton n geneal as, Hee, d L L q q Q - L contans the potental fom consevatve foces as befoe and - Q epesents the foces not asng fom a consevatve potental

19 19 E-L Equaton fo Dsspatve Foces Example (dsspatve fcton): F kv, kv, kv f x x y y z z Fo ths case, one can defne the Raylegh s dsspaton functon: 1 k v k v k v x x y y z z 2 Then, the fcton foce fo the th patcle can be wtten as, F,, f, v, vx vy vz

20 20 E-L Equaton fo Dsspatve Foces Pluggng ths nto the component of the genealzed foce fo the foce of fcton, we can get, Q F f, q q q To see ths, plug n ou eale elaton :, we have v q Q F f, v q v v, q q

21 21 E-L Equaton fo Dsspatve Foces Then, the Lagange Equaton fo the case wth dsspaton becomes, d L L q q q Q - Both scala functon L and must be specfed to get EOM. - L wll contans the potental devable fom all consevatve foces as pevously.

22 22 Smple Applcatons of the Lagangan Fomulaton A patcle movng unde an appled foce F n Catesan Coodnates : In 3D, = (x, y, z) and thee wll be thee dff eqs fo the EOMs. The Lagangan s gven by, L T mx 2 y 2 z 2 Then, the x-equaton s gven by, 1 2 Ths gves, d L L F x x d mx F 0 x x (Note: Q F F ) smlaly fo y & z x x x m F

23 23 Smple Applcatons of the Lagangan Fomulaton Let edo the calculaton n Cylndcal Coods wth the same appled foce F: The coodnates ae: = (,, z) 1 2 Tanfomaton: x, yz,,, z Fom befoe, we have T mx 2 y 2 z 2 xcos x sn cos y sn y cos sn z z z z z y x Expessng T n Cylndcal Coods: T m ( sn 2 sncos cos cos 2 cossn sn z 2 )

24 24 Smple Applcatons of the Lagangan Fomulaton Combnng and cancelng lke-colo tems, we have T m ( ) (*) z It s constuctve to consde the followng altenatve way to get to ths expesson, Let ty to expess the speed n T n cylndcal coodnates, z ˆ zz y Stat wth the poston vecto, ˆ zzˆ Takng the tme devatve, ˆ x d dˆ ˆ zˆ v z Note: the dectonal vectos ˆ change n tme as the patcle moves

25 25 Smple Applcatons of the Lagangan Fomulaton To examne on how these dectonal vectos changes, consde the followng nfntesmal change, ˆ dθ ˆ ' θ ˆθ y d ˆ ' ˆ ˆ d t' t (') t () t x Notce that, dˆ ˆ ˆ ˆ θ d dθ dθˆ dˆ d ˆ θ ˆ d dˆ v ˆzzˆ θˆ ˆzzˆ v z and m 2 m T v ( z ) 2 2 Back to the v vecto, So, 2

26 26 Smple Applcatons of the Lagangan Fomulaton Now, let calculate the genealzed foce n cylndcal coodnates, : Q F Snce, ˆ ˆ zz, we have ˆ So, Q F ˆ F : Q F ˆ θˆ ˆ θ (ecall pevous page) z: Q z F Fzˆ F z So, ˆ Q F θ F (ths looks lke toque) z

27 27 Smple Applcatons of the Lagangan Fomulaton The EOM s then gven by: Recall, : d T T 2 F m m F m T z ( ) : d T T d 2 F m F dl N (ths s ) 2 2 m m F m 2 F z : d T T d F mz F mz F z z z z z

28 28 Smple Applcatons of the Lagangan Fomulaton Puttng the components of F togethe, F F ˆ F θˆ F zˆ z 2 ˆ F m m 2 θˆ mz zˆ (*) F m Is that the same that we have gotten pevously n Catesan Coods? Check: v θˆ ˆ z zˆ dv d = θˆ ˆ z zˆ d ˆ ˆ dˆ θ θ ˆ zzˆ dθˆ ˆ ˆ dˆ ˆ zˆ θ θ z

29 29 Smple Applcatons of the Lagangan Fomulaton Usng the dectonal vectos elatons that we had eale, dθˆ dˆ ˆ, θˆ θˆ θˆ θˆ ˆ ˆ zzˆ Collectng all tems n the same decton, ˆ 2 θ zzˆ 2 ˆ So the EOM n (*) on the pevous page s ndeed,.e., ˆ 2 F m zˆ θ z m 2 ˆ F m