Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011
Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory, lws of thermodynmics Sttisticl thermodynmics Cnonicl ensemble, Boltzmnn sttistics, prtition functions, internl nd free energy, entropy Bsic electrosttics Clssicl mechnics Newtonin, Lgrngin, Hmiltonin mechnics Quntum mechnics Wve mechnics Wve function nd Born probbility interprettion Schrödinger eqution Simple systems for which there is n nlyticl solution Free prticle Prticle in box, prticle on ring Rigid rottor Hrmonic oscilltor Bsics Uncertinty reltion Opertors nd expecttion vlues Angulr momentum Hydrogen tom Energy vlues, tomic orbitls Electron spin Quntum mechnics of severl prticles (Puli principle) Mny electron toms Periodic system: structurl principle Molecules Two-tomic molecules (H2+,H2, X2) Mny-tomic molecules Chemistry... Informtics... Emilino Ippoliti 2 Wednesdy, October 12, 2011
Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory, lws of thermodynmics Sttisticl thermodynmics Cnonicl ensemble, Boltzmnn sttistics, prtition functions, internl nd free energy, entropy Bsic electrosttics Clssicl mechnics Newtonin, Lgrngin, Hmiltonin mechnics Quntum mechnics Wve mechnics Wve function nd Born probbility interprettion Schrödinger eqution Simple systems for which there is n nlyticl solution Free prticle Prticle in box, prticle on ring Rigid rottor Hrmonic oscilltor Bsics Uncertinty reltion Opertors nd expecttion vlues Angulr momentum Hydrogen tom Energy vlues, tomic orbitls Electron spin Quntum mechnics of severl prticles (Puli principle) Mny electron toms Periodic system: structurl principle Molecules Two-tomic molecules (H2+,H2, X2) Mny-tomic molecules Chemistry... Informtics... Emilino Ippoliti 3 Wednesdy, October 12, 2011
Clssicl Mechnics In Mechnics one exmines the lws tht govern the motion of bodies of mtter. Under motion one understnds chnge of plce s function of time. The motion hppens under the influence of forces, tht re ssumed to be known. Emilino Ippoliti 4 Wednesdy, October 12, 2011
Fundmentl elements Mechnics dels with entities (point-like msses) tht cn be completely described by: their position, i.e. 3 rel numbers t ech time: r t ( ) = x t ( ( ), y( t), z( t) ) mss m, i.e. rel number tht gives mesure of their inerti. In this context the time cn be seen s independent process nd therefore free prmeter. Emilino Ippoliti 5 Wednesdy, October 12, 2011
Some definitions The velocity is given by r ( t) = d r dt The ccelertion is defined by r ( t) = d 2 r dt 2 = d r dt Emilino Ippoliti 6 Wednesdy, October 12, 2011
Newtonin Mechnics Newton, Isc (1643-1727) Emilino Ippoliti 7 Wednesdy, October 12, 2011
Inertil frmes To write down the equtions of motion for certin problem, one first hs to choose frme of reference. The gol is then to find frme of reference in which the lws of mechnics tke their simplest form. It cn be derived tht one cn lwys find frme of reference in which spce is homogeneous nd isotropic nd time is homogeneous. This is then clled n inertil frme. In such frme free body which is t rest t some instnt of time lwys remins t rest. The sme is true if the body moves with velocity constnt in both mgnitude nd direction. This is the Newton s First Lw. Emilino Ippoliti 8 Wednesdy, October 12, 2011
Glileo s Reltivity Principle Experiment shows tht there is not one, but n infinity of inertil frmes moving, reltive to nother, uniformly in stright line. In ll these frmes the properties of spce nd time re the sme nd so re the lws of mechnics. This constitutes the Glileo s Reltivity Principle Emilino Ippoliti 9 Wednesdy, October 12, 2011
Forces The development in time of the position is given by the equtions of motions: m r t ( ) = F r t ( ( ), r ( t),, t) where F is given function tht describes the forces cting on the system (this cn lso be considered the definition of force). This is the Newton s Second Lw. The bsic chllenge of the Newtonin Mechnics is to find the solution to this eqution with the boundry conditions: r t0 ( ), r ( t ) 0 Emilino Ippoliti 10 Wednesdy, October 12, 2011
Potentil Energy Especilly simple nd importnt kind of forces re of the form F r ( ) = U r ( ) where U ( r ) stnds for the potentil energy. From the eqution of motion, inserting this kind of force we get in the coordinte nottion: m r i ( ) ( t) = d U r i dr i Emilino Ippoliti 11 Wednesdy, October 12, 2011
Motion constnt: Totl Energy Tking the sclr product with r i ( t) this becomes m r i ( t) r i t ( ) = dr i Using the derivtive chin rule: dt d ( ) dr i V r i d dt ( mr ( t) 2 ) = 2mr r i i i d dt U r i ( ( t) ) = i du dr i dr i dt we cn rewrite our eqution s the totl differentil d dt 1 2 m r 2 t Kinetic Energy ( ) ( ) + U r ( t) = 0 Emilino Ippoliti 12 Wednesdy, October 12, 2011
Lgrngin Mechnics Lgrnge, Joseph (1736-1813) Emilino Ippoliti 13 Wednesdy, October 12, 2011
Principle of Lest Action Let the mechnicl (n prticles) system fulfill the boundry conditions r ( t ) 1 = r 1 r ( t ) 2 = r 2 Then, the condition on the system is tht it moves between these positions in such wy tht the integrl S = t 2 t 1 L( r, r,, 1 2 r1, r2,,t)dt is minimized. S is clled the ction nd L is the Lgrngin ssocited with the given system. Emilino Ippoliti 14 Wednesdy, October 12, 2011
Lgrnge s Equtions Using the vritionl clculus it cn be proved tht minimizing the ction is equivlent to solve the system of 3n second-order differentil equtions: d dt L r L r = 0 = 1,,n These re the Lgrnge s Equtions nd re the equtions of motion of the system (they give the reltions between ccelertion, velocities nd coordintes). The generl solution contins 6n constnts, which cn be determined from the initil conditions of the system (e.g. initil vlues of coordintes nd velocities). Emilino Ippoliti 15 Wednesdy, October 12, 2011
Uniqueness If one considers two Lgrngin functions differing only by totl derivtive with respect to time of some function f(r,t): one finds tht the ction t 2 L ( r, r,t ) = L( r, r,t ) + d dt f ( r,t) S = L ( r, r,t )dt = L( r, r,t )dt + f ( r,t)dt = S + f r 2,t 2 t 1 t 2 t 1 t 2 t 1 ( ) f ( r 1,t ) 1 differs only by quntity which gives zero on vrition. This mens the conditions δs = 0 nd δs = 0 re equivlent nd the form of the equtions of motion is unchnged. Therefore, the Lgrngin is only defined up to n dditive totl time derivtive of ny function of coordintes nd time. Emilino Ippoliti 16 Wednesdy, October 12, 2011
Lgrngin of free prticle The Lgrngin ssocited to system hs to hve the sme symmetry properties of the system itself: - In free prticle system the time is homogeneous dl = 0 L = L( r, r ) dt - In free prticle system the spce is homogeneous dl dr = 0 L = L r - In free prticle system the spce is isotropic L cnnot depend explicitly on the time t L cnnot depend explicitly on the vector position r ( ) L = L( r 2 ) = L( v 2 ) = L( v 2 ) L cnnot depend on the direction of the vector velocity r Emilino Ippoliti 17 Wednesdy, October 12, 2011
L L v 2 Emilino Ippoliti 18 Lgrngin of free prticle If two frmes K nd K move with n infinitesiml velocity to ech other, then v = v + ε But since the equtions of motion must hve the sme form in every frme, the two Lgrngins L(v 2 ) nd L (v 2 ) cn only differ by totl time derivtive of function of r nd t: ( ) = L( v 2 ) = L( v 2 + 2v ε + ε 2 ) L( v 2 ) + L (v 2 ) 2 v ε The lst term cn only be totl time derivtive if it is liner function of the velocity. Therefore, ε is independent of the velocity, which mens tht the Lgrngin cn be written s L = 1 2 mv2 r Wednesdy, October 12, 2011
Additivity If mechnicl system consists of two seprte prts A nd B, ech prt would hve seprte Lgrngin LA nd LB respectively. If these prts do not interct, e.g. in the limit where the distnce between the prts become so lrge tht the interction cn be neglected, the Lgrngin of the system is given by L = L A + L B This dditivity sttes tht the equtions of motion of prt A cnnot be dependent on quntities pertining to prt B nd vice vers. Emilino Ippoliti 19 Wednesdy, October 12, 2011
Emilino Ippoliti 20 Lgrngin of system of prticles To derive the Lgrngin for system of prticles which interct with ech other we cn strt by tking the Lgrngin for the non-intercting prticles: L = nd dding certin function of the coordintes, depending on the nture of the interction: L = 1 2 2 m v 1 2 2 m v U ( r 1, r 2, ) The first prt of the Lgrngin we cn recognize to be the kinetic energy (T), nd the second prt to be the potentil energy (U). Wednesdy, October 12, 2011
Lgrngin of system of prticles In fct, if we derive the equtions of motion for the system by replcing tht Lgrngin in the Lgrngin equtions: we obtin d dt L v m d v dt L r = 0 = U r = 1,,n = 1,,n i.e. the Newton s equtions, if we recognize the second term s the forces cting on the system: F = U r Emilino Ippoliti 21 Wednesdy, October 12, 2011
When Lgrngin Mechnics It is often necessry to del with mechnicl systems in which the interctions between different bodies tke the form of constrints. These constrints cn be very complicted nd hrd to incorporte, when working directly with the differentil equtions of motion. Here it is often esier to construct Lgrnge function out of the kinetic nd potentil energy of the system nd derive the correct equtions of motion from there. Emilino Ippoliti 22 Wednesdy, October 12, 2011
Hmiltonin Mechnics Hmilton, Willim Rown (1805 1865) Emilino Ippoliti 23 Wednesdy, October 12, 2011
Generlized coordintes The formultion of clssicl mechnics in terms of the Lgrngin is bsed on formulting the stte of mechnicl system in terms of coordintes ( r ) nd velocities ( ). But this r is not the only possibility! Sometimes it is more dvntgeous to choose nother set of quntities to describe mechnicl system, e.g. generlised coordintes q = q ( r, r ) nd moment p = L q ( r, r ) Emilino Ippoliti 24 Wednesdy, October 12, 2011
Legendre trnsformtion From the Lgrngin eqution: d L dt q L q = 0 L q = d p dt = p Then the totl differentil of the Lgrngin is given by dl = L q dq + L d q q = p dq + p d q Using the differentil chin rule pplied on the lst term dl = p dq + d p q ( ) q d p Emilino Ippoliti 25 Wednesdy, October 12, 2011
Hmiltonin Rerrnging this we end up with d p q L = p dq + Emilino Ippoliti 26 q d p Since the L = T - U nd we cn write (using Euler s theorem on homogeneous functions): the rgument of the totl differentil on the left side is the totl energy of the conservtive system (T+U), now only expressed in terms of coordintes nd moment. In this form we cll it the Hmiltonin or Hmilton s function of the system: H( q, p,t ) = p q L L q q = p q = 2T Wednesdy, October 12, 2011
Hmilton s equtions Then the totl differentil of the Hmiltonin is given by dh = p dq + q d p Using this we cn derive the required equtions of motion in terms of coordintes nd moment: Hmilton s equtions: set of 6n firstorder differentil equtions for the 6n unknown functions p nd (t) q (t). q = H p p = H q Emilino Ippoliti 27 Wednesdy, October 12, 2011
Lgrnge vs Hmilton The difference between the Lgrngin nd Hmiltonin formlism is only Legendre trnsformtion. For some problems it is esier to write L down, for others H. One is given by system of 3n second-order differentil equtions, while the other one describes the sme physicl problem with 6n first-order differentil equtions. Emilino Ippoliti 28 Wednesdy, October 12, 2011
Summrizing Newtonin mechnics Lgrngin mechnics Hmiltonin mechnics System ( = 1,,n) Forces Lgrngin Hmiltonin F L = L r, = F r, r, r,...,t r,t ( ) ( ) ( ) H = H q, p,t Equtions of motion ( = 1,,n) F = m r d dt L r L r = 0 q = H p p = H q Type of mthemticl problem 3n second(or more)-order differentil equtions 3n second-order differentil equtions 6n first-order differentil equtions Emilino Ippoliti 29 Wednesdy, October 12, 2011
Venn digrm Under some ssumptions, we cn represent the sets of the systems tht cn be described by Newtonin (N), Lgrngin (L) nd Hmiltonin (H) mechnics this wy: N L H Emilino Ippoliti 30 Wednesdy, October 12, 2011
References 1. M. Gnz. Introduction to Lgrngin nd Hmiltonin Mechnics. http://imge.diku.dk/gnz/lectures/lgrnge.pdf 2. H. Goldstein. Clssicl mechnics. Addison-Wesley, Cmbridge, 1950. 3. L.D. Lndu nd E.M. Lifschitz. Mechnics. Butterworth, Oxford, 1981. Emilino Ippoliti 31 Wednesdy, October 12, 2011