XII AGRANGIAN & HAMITONIAN DYNAMICS Iouco Hamlo aaoal Pcple Geealze Cooaes Geealze Foces agaga s Euao Geealze Momea Foces of Cosa, agage Mulples Hamloa Fucos, Cosevao aws Hamloa Dyamcs: Hamlo s Euaos agaga & Hamloa Mechacs: Aohe way of loog a mechacs, ohe ha fom he Newoa pespecve Usg Newo s aw Euao of moo havg al coos Rescos: * All foces shoul be ow Poblems such as: escbes moo of he sysem * Usg coaes : ecagula pola, cylacal o sphecal * A mass cosae o move o a sphecal suface * A bea ha sles o a we No smple o eal wh. Dffcules: he uow fom of he foces of cosa. Usg cooaes ecagula, pola, may mae mpossble o eal wh.
Two ffee mehos: agage s Euaos Hamlo s Euaos These Mehos: * Ae o he esul of ew heoes * Devable fom Newo s law * Offe ease halg ffcul poblems The ffeece bewee hese mehos: I agage s fomalsm: The geealze cooaes use ae poso & velocy. Resulg: seco oe lea ffeeal euao. I Hamlo s fomalsm: The geealze cooaes use ae poso & momeum. Resulg: fs oe lea ffeeal euao. These Techues use: * Geealze cooaes sea of ecagula cuvlea cooaes such as: velocy, lea o agula momeum, o legh. * A eegy appoach ealg wh scala uay ahe ha veco uay: foce & acceleao These mehos ae coae Hamlo aaoal Pcple. Fs: Posulag Hamlo Pcple : 3 4
Exema Pcple mmal pcple: O mahemacally: Objecs follow pahs hough space a me base o exema pcples. J T Examples: * lgh ay follows a pah ha mmzes s as me. * movg boes ae shoes sace bewee wo pos o a gve geomecal suface. Hamlo s aaoal Pcple: fs aouce 834 Of all he possble pahs alog whch a yamc sysem may move fom oe po o aohe wh a specfe me eval, he acual pah followe s ha whch mmze he me egal of he ffeece bewee he ec a poeal eeges. Whee: a: T agaga of he sysem T Kec Eegy Poeal Eegy A vaao of ay pacula sysem paamee by a fesmal amou. T T x x x, x x, x So Hamlo s vaao pcple saes ha he egal J ae alog a pah of he possble moo of a physcal sysem s a exemum whe evaluae alog he pah of moo ha s he oe acually ae. 5 6
Geealze Cooaes: Cooaes: use o efe he poso of a sysem of pacles space. Thee cooaes epee : Recagula x, y, z Cylcal x, y, z Sphecal,, Sysem of N pacles: To specfy he sae of such a sysem a a gve me, s ecessay o use vecos; hee cooaes escbe each veco. To escbe he posos of all he pacles: 3N cooaes No cosas 3 N m cooaes m cosas Havg cosas o he moo of pacle ess ha hee cooaes Cosas: o Move plae wo cooaes o Move sagh le Oe cooae Cofguao sae of a sysem: Smulaeous poso of all pacles. The umbe of egee of feeom. I geeal we ca selec ay se of cooaes o escbe he moo of a physcal sysem. cooaes: No ee o be ecagula o cuvlea. ca be ay paamees: legh, legh, agle, eegy, mesoless uay as hey escbe he cofguao of sysem. 7 8
Geealze Cooaes: ae ay se of epee cooaes, o coece by ay euaos of cosa ha escbes uuely he cofguao sae of a sysem of pacles. These cooaes ae show by:,, 3,...,, o whee,, 3,...,, a o esce by ay cosas. Holoomc Sysem cosas: Each cooae vay epeely of he ohes; he fom: f x, y, z, j,,3,..., N j,,3,..., m Example: The bob of a sphecal peulum of legh l s esce o move o a sphee escbe by x y z l. The euao may be we he fom: f x, y, z x y z l Holoomc Cosa. The moo of a gas a cubc coae of se l s esce by he oholoomc cosas: # of cooaes = # of egee of feeom oholoomc cosa x l y l z l NoHoloomc Sysem: Cooaes oes o vay epeely; Cosas ca o be expesse as euaos of eualy. # of egee of feeom < # of cooaes 9
Geealze cooaes evew: A suable se of geealze cooaes of a sysem s hose ha esuls euao of moo leag o a easy epeao of moo. These geealze cooaes fom a cofguao sae space, wh each meso epesee by a cooae. The pah of he sysem s epesee by a cuve hs cofguao space, whch oes o le self o he same epeao as a pah oay hee-mesoal space. Devaves of,, 3,, ae efe as geealze velocy o Geeal ule o choose mos suable se of geealze cooaes: NONE Cose a sgle pacle Caesa cooaes x, y, z as a fuco of geealze cooaes,, 3 : Fo moo space, hee egee of feeom: x x,, x y y z z 3,, 3 y,, 3 z Fo moo o a suface, wo egee of feeom: x x, x y y, y Fo moo o a le o cuve, oe egee of feeom: x x x Suppose he sysem chages fom al cofguao,, 3 o,, 3 3. The coespog chages Caesa cooaes: x y z 3 3 3 x y z
3 So fo a sysem of egee of feeom: x x y y z z vual splaceme acual splaceme Dsplaceme of a mass, m : Is calle a acual splaceme f he pacle moves fom o ug, ue he fluece of he apple foces a cosse wh he euaos of moo a cosse wh he cosas. Is calle a vual splaceme f s cosse wh oly he cosas. A vual splaceme oes o volve he euaos of moo o me. 4 agage s Euao of Moo fo Cosevave Sysem: Hamlo s pcple saes ha he egal: J always assumes a exeme value. So: T J Assume ha he agaga T s ow ems of geealze cooaes a geealze velocy:,, T x x T, Usg [ U U U ], he las em he ega: a ] [ a ] [
[ ] Poceue o solve poblems wh agage Euaos: Selec a suable se of geealze cooaes. agage Euao,, 3,..., agage euaos escbe he moo of a pacle a cosevave foce fel. Thee ae of hese euaos, a ogehe wh he m euaos of cosa a he al coos, hey compleely escbe he moo of he sysem. So: F he euaos of asfomao elag he Caesa cooaes o geealze cooaes F he ec eegy coespog o geealze cooaes a veloces. F he poeal eegy as a fuco of geealze cooaes. We agage s euaos of moo. To solve hese euaos: F T, he subsue o: agaga fuco,, mus be ow he appopae geealze cooaes. Noe: Eegy s SCAAR s a SCAAR fuco s vaa wh espec o cooae asfomao. gves he same escpo of he sysem ue gve coos, o mae whch geealze cooaes ae use. 5 6
Igoable Cooaes, Geealze Momea: Fo a sysem of egee of feeom, geealze cooaes ae eee. The agaga,, s escbe ems of geealze cooaes, a geealze velocy.,, [,,...,,,,...,, ] Defe p p x mx x p geealze momea coespog o as: Fo a cosevave sysem, f he agaga oes o explcly coa he cooae : Wa o show: Momeum s coseve alog he eco of a geealze cooae o explcly coae he agaga of he sysem. p p p Cose: A fee pacle movg wh velocy x a sagh le, alog x cooae. poeal eegy T m x ec eegy T agaga p mx T x If p p cos. Ths cooae s goable. 7 8
Foces of cosa: agage Mulples Assume a cosa f,, coecg wo geealze cooaes &. I geeal fo geealze cooaes a m euaos of cosas: f j,,3,..., j j j,, 3,..., m If we wa o cose he euao of cosa, he he agage s euao chages o: Noe: hee ae m uows: he a he m j. f [,,3, ] f j, fuco of me uow fuco O f,, 3,... Geealze foce of cosa {agage mulple em} f j f j,,3,..., j,,3,..., m I may saces, he cosas ca be eeme by o explc me evave: foce oue f f spaal cooae agula cooae f j,,3,..., j,,3,..., m These wo euaos ae euvale; seco oe s he oal ffeeal of he fs. 9
Geealze Foce: Cose foce F acg o a pacle, wh hee ecagula compoes: F x, F y, F. x Fx y Fy z Fz geealze foce assocae wh he geealze cooaes. Dmeso of epes o meso of. Fo a sysem of N pacles wh foces F,,..., N : z Fo cosevave foce: whee: x, y, z a F x A he agage euao: x, F y y,,,3,..., F z z N x Fy F z F x y z Fo ocosevave foce: The agage euao s,,3,...,,,3,...,