Ch. 7 Lagrangian and Hamiltonian dynamics Homework Problems 7-3, 7-7, 7-15, 7-16, 7-17, 7-18, 7-34, 7-37, where y'(x) dy dx Δ Δ Δ. f x.

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1 Ch. 7 Laranan an Hamltonan namcs Homewor Problems A. revew o calculus o varatons (Chapter 6. basc problem or J { ( '(; } where '( For e en ponts an ntereste n the path or whch J s mnme.. Euler's equaton 0 s a necessar conton o the path or whch J s an etreme. Can vsuale ths conton wth a varaton to ( that s locale aroun pont. The total chane to the entre nteral s J / / ( δ ( ( ( 0 Thereore Euler s equaton must hol or an on the path. 3. n orm o Euler s equaton I 0 then const. 4. Functons wth several nepenent varables { } n ; ( ( L n 0 L 5. Larane s unetermne multpler Tas: n mnmum o uncton F alon a path ene b G0. Trc: n the pont where two raent vectors are parallel G G F F // 0 G F λ ; 0 G F λ For eample: ma/mn stance rom - to orn λ ( 0 λ 0-0 λ - ½ /sqrt( What there are more than one constrants? 6. Euler s equatons (aular contons mpose { } ; { } const ;

2 λ( 0 λ( 0 B. Hamlton's prncple. O all possble paths between two e ponts the actual path taen s one that mnmes t t L { (t ẋ (t t} t tme nteral L { (t ẋ (t t} T {ẋ (t} - U { (t ẋ (t t} netc potental. varatonal problem L - L t 0 or each Larane eqn o moton ẋ orces actn on the sstem (not constrants must be ervable rom potental(s 3. eample: harmonc oscllator [snle partcle snle coor] T (ẋ / m ẋ U ( / L ( ẋ / m ẋ - / L - L t ẋ C. enerale coornates. q q q n partcles q ( ; t. can n prncple nvert these equatons (q q ; t (q q ; t 3. enerale veloctes & q & q eamples cm an relatve t t m ẍ & X & m & m m rotatn polar & r& cos(θωt - r θ & sn(θωt rω sn(θωt 4. netc ener T m (& & & eamples

3 cm an relatve T (m m ( X & Y & Z & rotatn polar M mm m m T (m r& mr θ & mr ωθ & mr ω μ ( & & & T a l l b T 0 l 5. enerale momentum L p eample: cm an relatve p X (m m X & mm p & etc. m m eample: rotatn polar p r m r& p θ mr θ & mr ω [n re polar ths woul be anular momentum] D. Larane s equatons (enerale coornates (a δ L [ q ( t ( t t] t 0 t t L [q (t (t t] T [q (t (t t] V [q (t t] L L - 0 or each q t & q q Arbtrar term o the orm ( q t can be ae to L wthout aectn eqn o moton t (b eample: cm an relatve m an m eert ravtatonal orce on each other; no eternal orces T (m m ( X & Y & Z & m m ( & & & m m V - Gm m L (m m ( X & Y & Z & μ ( & & & L ( X & (m m X & L X L L ( μ & & t & t L X & (m m X & 0 (m m X & 0 μ & & Gm m X & const. 3

4 L Gmm 3 / ( Gmm μ & & 0 E. sstems wth constrants. partcles m nepenent equatons o constrant s - m enerale coornates. Larane s equatons stll appl 3. eample: two partcles m an m connecte b r ro lenth l 6 Cartesan coornates equaton o constrant: ( ( ( l nee 5 enerale coornates: e.. X Y Z θ φ 4. Tetboo Eamples m Tr solvn t n Cartesan coornates: L ( & & & m wth constrant: tan α Euler equatons: m& λ 0 m& λ 0 m & m λ tan α 0 ot much help! Use enerale coornates r an θ ( r cotα Euler equatons ve us: r & θ const. an & r rθ& sn α snα cosα 0 a cosωt bsnθ a snωt b cosθ m L [ a ω b & θ b & θaω sn( θ ωt ] m( a sn ωt b cosθ & ω θ a cos( θ ωt snθ b b Frst solve t b ree-bo aram : no tanental orce no net torque. Wre cr slope: cr must equal to rato between perpencular mrω ω orces: cr c m m Solve b Laranan L ( r& 4c r r& r ω mcr & r ( 4c r 4c rr& r(c ω 0 For unorm crcular moton & r r& 0 ω c 4

5 m m T & U m m & m & & & m ( l l 3 ( ( ( l 3 & m (&& && m (&& 3 & ( m m m3 m (& && m (&& 3 & ( m m3 m F. ewton s n law Larane s equatons. enerale orces pp. 360 (a enerale coornates ( q t δ δq q & & q t q (b vrtual wor δw F δ F δq Q δq q Q F q (c conservatve orces potental ener V( V F - V V Q (- - q q. unconstrane sstem enerale coornates p. 364 L & (a {p } m & m & q p { L } ( t t m & & m & q t q & F m & q q T p Q compare to Cartesan q t L T V ( - Q - t q q (T V (T V ( - 0 or conservatve orces t & q q 3. Larane s equaton wth constrant (q ; t 0 F ( ewton s n : m & & F L L - q t & q λ ( t 0 q 5

6 Generale orce o constrant Q λ ( t q 4. Eample 7-0 ( r θ r a 0 L L L L - λ 0 - r t & r r θ t & λ 0 θ θ mr & θ m cosθ m& r λ 0 mr snθ mr & θ mrr& & θ 0 Set r a & r r& 0 & θ cosθ a a λ m ( 3cosθ G. Hamlton s equatons. chane varables Larane s eqns: q Hamltonan s eqns: q p T L p & q & q p (q j j t (q j p j t [potental oes not epen on Cartes veloc; thereore oes not epen on ]. useul equatons rearn L L L (a - q t & q 0 L - p 0 q t L q note: L (q p const (cclc L L L (b L ( q t & q t ( p q q L t 3. Hamltonan H p - L(q j j t H(q j p j t H q p p - L & ( ( p p - ( p L ( p q - t t H H H H ( q p t q p t 4. Hamlton s equatons o moton t q - L t t 6

7 H H - p q H L also - t t 5. Hamltonan an ener T H - L(q j j t (q q ; t & q T m & a l l l t b c assume t-nepenent enerale coornates: (q q ; not t etc. T T a l l A l l l & q l T Al l T l H T L T (T V T V 6. eamples (a smple harmonc oscllator L m & - H m& & - ( m & - m & H p H - - q (b partcle mass m constrane on surace o clner R subject to orce F - r enerale coornates φ T m (R φ & & & m p V r (R L m (R φ & & - (R p φ m R φ & p m & H m R φ & φ & m & & - L m (R φ & & (R p [equals E] m mr p φ p (R m 7

8 H φ & p φ p mr p & m H - q φ 0 - m R φ & const m& & - ote that φ s a cclc coornate an ma be remove rom the Hamltonan. (c Eample 7- T mb & θ mb sn θ & φ U mb cosθ H & θ p & θ φpφ T U (φ s cclc & p θ θ pφ cosθ θ θ mbsn 3 mb mb sn θ G. orces o constrant an Larane multplers 4. applcaton to mechancs can eep more than mn number o enl coors alon wth an eqn o constrant ma/mn L{q ; t} t subject to constrant (q ; t 0 L L - λ q t & q q 0 use these an (q ; t 0 to solve or q (t an λ(t λ q s orce o constrant assocate wth q 5. eample Problem 7-3 eqn o constrant: ρ (θ φ Rφ (R - ρ φ - ρθ 0 (φ θ wrte L n terms o both θ an φ T m ( R ρ θ Iφ U R ( R ρ cos θ m L L 0 - λ φ t φ& φ -m(r - ρ snφ - m(r - ρ φ λ(r - ρ 0 - mρ & θ - λρ 5 (R - ρ & φ ρθ & 7 0 -m(r - ρ snφ - m(r - ρ & φ 5 λ - mρ & θ - m(r - ρ & φ 5 5 & φ 5 7 R ρ snφ 0 & φ 5 7 R ρ φ 0 ω 5 7 R ρ 8