LAGRANGIAN MECHANICS

Transcription

1 LAGRANGIAN MECHANICS

2 Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1, y 1, z 1, x 2, y 2, z 2,... KE T ṙ = T x 1, y 1, z 1, x 2, y 2, z 2,... T ṙ Calculatng U r and can often be smplfed Usng a dfferent set of coordnates (q n ) for the system Examples: Pendulum and Double Pendulum KE and PE easly expressed n terms of θ 1 (and θ 2 ) Example: marble sldng n hemsphercal bowl What are some possble generalzed coordnates? q n Express U and T n terms of and q n

3 Newton's 2 nd Law Another Vew Derve F net = ma from energy conservaton (de / dt = 0) Can get EOM from knowng only U and T (don't need forces!) de dt T de dt Cartesan Coordnates U r Chan Rule: T ṙ = [ U ṙ r r ] ṙ = m 2 ṙ = m ṙ ṙ [ U r m r ] ṙ = 0 de dt n T Generalzed Coordnates U q n T q n, q n [ U q n q n q n q n] = m 2 ṙ Chan Rule: =?? q n =?? de dt [ U r d dt ṙ ] ṙ Newton's 2 nd Law EOM = 0 de dt [a bunch of dervatves ] q n = 0 n Generalzed Newton's 2 nd Law EOM

4 Example: Mass-Sprng System R (to CM) r m k 1 m 2 x 1 x 2 Sprng has equlbrum length L Cartesan: (x 1, x 2 ) Generalzed: (r, R) Fnd transformaton equatons between coordnate systems In each coordnate system: Express U and T n terms of coordnates Calculate the equatons of moton nterpret the results

5 Constrants Systems often have forces of constrant e.g. normal force on marble n bowl Mathematcally descrbed by constrant equatons Cartesan constrants often cumbersome Example: Calculate T x, y, z, ẋ, ẏ, ż for marble n bowl Generalzed coordnates can nclude constrants Keepng T and U n smpler forms y s System wth N partcles and M constrant equatons x Would requre 3N M generalzed coordnates

6 Constrant Example: Pendulum Cartesan r = x, y y Generalzed q n = Constrant: x 2 y 2 = L 2 U = mgy T = 1 2 m ẋ 2 ẏ 2 x (Constrant already ncluded) x = L sn y = L cos U = mgl cos transformaton equatons In each coordnate system: T = 1 2 m L2 2 Calculate the EOM nterpret the results How many generalzed coordnates f pendulum moves n 3-D?

7 Coordnate Transformatons Goal: use chan rule to plug nto de / dt = 0 T And construct a generalzed way to get equatons of moton 1 2 m 2 ṙ q n m ṙ m ṙ ṙ ṙ q n From the transformaton equatons: Now must use chan rule some more to evaluate these dervatves r q 1, q 2,... ṙ q 1, q 2,..., q 1, q 2,... = r n n q q n (Chan rule) ṙ Vewng as a sum of terms, can take dervatve = r ṙ q n Cancellaton of dots Also: ṙ = p r p q q p p q p r q p = d dt r

8 Generalzed Newton's 2 nd Law Pluggng n: m ṙ ṙ (chan rule) m ṙ d dt r (from transformaton equatons) = d dt m ṙ r n q m r r (product rule) = d dt m ṙ ṙ n q U r r (from transformaton equatons) = d n dt q U (chan rule) Ths dfferental equaton s a factory for equatons of moton Once T and U are expressed n generalzed coordnates just plug n

9 The Lagrangan Functon Conservatve forces U s a functon of q n only Generalzed Newton's 2 nd Law can be re-wrtten as: U q n = 0 T U d T U dt q = 0 L q n, q n T U n Lagrangan d n dt q = 0 Euler-Lagrange equatons of moton (one for each n) Lagrangan named after Joseph Lagrange (1700's) Fundamental quantty n the feld of Lagrangan Mechancs Example: Show that ths holds for Cartesan coordnates

10 Examples Mass-sprng on table-top (top vew) Sprng has equlbrum length r 0 Calculate EOM n polar coordnates Is crcular moton possble? Is t stable? Fnd frequency of small oscllatons n r Double Pendulum Calculate the EOM for θ 1 and θ 2 Approxmate EOM's for small θ 1 and θ 2 Does moton have consstent frequences?

11 Symmetry and Conservaton Laws Euler-Lagrange equatons of moton: d n dt q = 0 Notce that f = 0 s a conserved quantty q n q n s called the generalzed momentum n the q n drecton Common Examples: Conservaton of lnear momentum: Conservaton of angular momentum: ẋ CM = m ẋ CM = Constant = m r 2 = Constant Conservaton of energy: = 0 (assumed prevously) t

12 Lagrangan/Hamltonan Revoluton Dynamcs of a physcal system Can be descrbed by energy functons T and U n state space Mathematcally system need not be dvsble nto partcles Ths opens possbltes for new models of matter Matter dstrbutons ρ(q n ) wth equatons of moton.e. generalzed Newton's 2 nd Laws Idea eventually led to the development of Quantum Mechancs Generalzed coordnates: good for descrbng felds Value of feld (at one pont) generalzed coordnate(s) Scalar/vector feld f(x,y,z) state vector n state space Ponts n physcal space (x,y,z) unt vectors n state space