Phys 331: Ch 7,.2 Unconstrained Lagrange s Equations 1

Transcription

1 Phys 33: Ch 7 Unconstrane agrange s Equatons Fr0/9 Mon / We /3 hurs /4 7-3 agrange s wth Constrane 74-5 Proof an Eaples 76-8 Generalze Varables & Classcal Haltonan (ecoen 79 f you ve ha Phys 33) HW7 ast te we learne that the ntegral of the agntue of oentu over the path that a syste really takes s nal that s the prncple of least acton hols S f ps hrough a lttle bt of ath we foun that ths stateent s equvalent to sayng that te ntegral of the fference between knetc an potental energy e the agrangan s nze U egarless of what coornates we epress t n ters of ( t q ( t) q ( t) q ( t) q ( t)) U ( t q ( t) q ( t) q ( t) q N N N N ( t)) ( t q ( t) q ( t) qn ( t) q N ( t)) Of course that s equvalent to sayng that the agrangan satsfes the fferental equatons q q 0 for all the nvual coornates We showe that at least for Cartesan coornates that n turn s equvalent to Newton s n law As was our practce n Chapter 6 we focuse uch ore on ths fferental equaton whch the ntegran satsfes than on the ntegral We worke through a few falar an nterestng probles hey coul all be characterze as unconstrane n that there weren t any restranng forces that say force the syste to only ove on a surface or along a curve hs te we wll conser constrane systes Constrane Moton: Often t oes not take 3N paraeters to escrbe the oton of N partcles because they are subject to constrants For eaple t only takes paraeters to escrbe the oton of a partcle slng on a plane

2 Suppose you are tryng to escrbe the oton of N partcles he paraeters q q q n ( n 3N) are a set of generalze coornates f the poston of each partcle can be epresse n ters of q q q n an possbly the te t: r r q q q n t N It wll also be possble to epress each of the generalze coornates n ters of the postons of the partcles an possbly the te t: q q r r r N t n he nuber of egrees of freeo for a syste s equal to 3N nus the nuber of constrants (see Eaple # below) If the nu nuber of generalze coornates requre to copletely escrbe a syste s equal the nuber of egrees of freeo the syste s holonoc hat s the easer type to hanle so we won t conser nonholonoc systes An eaple of a nonholonoc syste s a ball rollng wthout slppng on a flat surface he ball has two egrees of freeo but t takes ore than two coornates to specfy the orentaton of the ball he ball can take fferent paths between two ponts so t can en up wth fferent ponts at the top More coornates are neee to specfy whch pont s at the top of the ball he agrangan can be wrtten as a functon of the generalze coornates ther te ervatves an te: q q qn q q qn t an there wll be a agrange equaton assocate wth each generalze coornate: q q n he equatons of oton foun ug the agrangan approach are equvalent to the ones foun ug the Newtonan approach However the agrangan etho s uch spler for any probles It also allows the use of whatever coornates are convenent for escrbng a syste Eaple #: How any generalze coornates are neee to escrbe the syste pcture below? (How any egrees of freeo oes the syste have?) k Just one he stance fro the wall of the block on the horzontal surface or the stretch of the sprng are goo choces he nuber of egrees of freeo for the two partcle s: DOF=3() - (objects ove n plane) - (objects ove along lnes) - (lnke together) =

3 How about workng t through Eercse #: wo equal asses are constrane by the sprng-an-pulley syste shown below Assue that there s no frcton an that the pulley s assless et be the stance the sprng has stretche k (a) Wrte the agrangan for the syste n ters of he knetc energy of each block s the sae so the total KE s: he potental energy s: an the agrangan s: (b) Fn the equaton of oton of the syste U U spr U grav k g U k g k g k g 0 Fro the prevous chapter that woul have the soluton g ( t) A t where k k 3

4 Say Eercse #: A bea of ass sles wthout frcton along a wre bent nto a parabola y a where the y as ponts upwar y (a) Wrte the agrangan for the syste n ters of hey Do: he knetc energy s: but: so: he potental energy s: so the agrangan s: (b) Fn the equatons of oton of the syste I o: 4a y y y a y a a 4a U gy ga U 4a ga ga 4a 4a 4a 4a ga 0 8a I ll confess to not beng terrbly eager to fn an analytcal soluton for ths epresson However t s perfectly reasonable to use ths n a coputatonal sulaton 4

5 y 4a ga 4a * t * t a ry ths for a = 0 = g = 98 v o =0 Eaple #: Suppose a partcle s confne to ove on top of a hesphere of raus Pck a goo set of generalze coornates an epress the heght an spee of the partcle n ters of the he sphercal polar coornates an wth the z as upwar an the orgn at the center of the sphere s a natural choce he Cartesan coornates of the partcle are: y an z he z coponent s the heght he coponents of the partcle s velocty are (reeber that s a constant): he spee square of the partcle s: y v y z an z v he cross ters of the frst two squares cancel so: v v v v 5

6 Say ths takes place on Earth so there s a gravtatonal potental of U gz g( ) relatve to the top of the sphere An we ll say we start wth the object on the top of the sphere hen U v gz g hey o: Whch ust satsfy So 0 C g g Now f we start wth the thng on the top of the sphere then = 0 so = 0 so C = 0 Of course the pont of C s beng a constant s that t stays the sae value 0 as the object sles Naturally wll change an so wll so the only way for C to rean 0 s for 0 hat eans the thng just sles straght own Now the agrangan ust also satsfy g g But we just reasone that 0 so we re left wth the ol falar g As for an nverte penulu g I o: What f you on t start t at the top an you o gve t an ntal rotaton aroun then you have g An cot 6

7 Agan nothng I crave solvng analytcally but soethng that s qute reasonable to hanle coputatonally hen r You coe ths t t an t t Avantages of the agrangan approach: he agrangan (& energes) are scalars nstea of vectors lke force an acceleraton It s easer to use whatever coornates are natural wth ths etho It proves a soewhat systeatc approach to gettng equatons of oton Wrte own the energes n ters of the chosen coornates an take soe ervatves of U Drawbacks of the agrangan approach: Frcton/rag can t be nclue easly You on t get the sae physcal unerstanng that ealng wth forces an torques gves Eaple #3: (E 08 of F&C 5 th e) Fn the equatons of oton for the syste shown below he block sles wthout frcton along the as an the penulu swngs n the y plane on a assless ro of length r y M X r he Cartesan coornates of the penulu bob are: X r an y r he ervatves of these are: X r an y r he knetc energy of the syste s: 7

8 he potental energy s U he equatons of oton are: whch gve: an: Xr MX y MX X r r MX X r X r gr so the agrangan s: U MX X r Xr gr gr X X an 0 MX X r r Xr r Xr X r he frst equaton eans that the total oentu n the recton s conserve: P MX X r constant he secon equaton splfes to: X g 0 r r If M the upper ass wll barely ore ( X 0 ) an ths reuces to the equaton for a sple penulu hese two couple fferental equatons are ffcult to solve but they are easy to erve ug the agrangan approach Eercse #4: Suppose a strng of length s connects a puck of ass on a frctonless table an an object wth ass through a hole (see the fgure below) r s -r 8

9 (a) Wrte the agrangan for the syste n ters of the puck s polar coornates he knetc energy for the puck s: an for the other ass s: r r r r he efne the gravtatonal potental energy to be zero at the level of the table so: he agrangan s: U g s r U r r r g s r (b) Fn the equatons of oton for the syste r r an r g r r an 0 r r g r an r constant he secon equaton s equvalent to conservaton of angular oentu he frst equaton can be rewrtten as: r g 3 r 9

10 Eercse #5: Suppose a ouble penulu conssts of two rg assless ros connectng two asses he asses are not equal but the lengths of the ros are he penulu only swngs n one vertcal plane (a) Wrte the agrangan for the syste n ters of the angles shown above ake the top pvot pont as the orgn an choose postve to the rght an postve y upwar he coponents of the poston of are: so the coponents of ts velocty are he coponents of the poston of are: so the coponents of ts velocty are he knetc energy s: an y an y an y an y an y y y he potental energy s: U gy gy U g g so the agrangan s: = g g 0

11 (b) Fn the equatons of oton of the syste he equaton assocate wth s: g g he equaton assocate wth s: g g hese equatons are heously coplcate but t s not ffcult to get ug the agrangan approach f you are careful hey woul be treenously ffcult to get ug the Newtonan approach! Net two classes: Monay More Eaples of agrange s Equatons