 hold true whatever the microscopic details may be. In addition to PCE Poincaré lists several other principles whose application to physical phenomena suffices "for our learning of them all that we could reasonably hope to know" (Poincaré 1913, 300).10 Similarly, on Duhem's view, analytic treatments develop from general principles like PCE and Carnot's Principle (Duhem 1911, Vol I, 2). Historically they began with Lagrange, who condensed statics into a general principle, the Principle of Virtual Velocities (PVV). PVV tells us that a mechanical system X is in static equilibrium just in case in all infinitesimal virtual displacements of X the forces applied to X perform zero work. In a perfectly balanced see-saw, for example, the sum of the work done by both the external forces (like gravity) acting on it and the internal forces holding it together is zero. Lagrange's analytical mechanics has several attractive features that were later heavily exploited in analytical treatments. First, it was extendible. Using d'Alembert's principle, Lagrange showed how to extend PVV from statics to dynamics: we simply add fictitious "inertial" forces to balance the external forces that are really acting on X to produce its acceleration, so that X is in equilibrium at each instant. Then the work done by the external, internal, and inertial forces will sum to zero. Second, it provides a powerful algebraic method (Lagrange multipliers) that uses constraints in a principled manner to reduce the number of degrees of freedom and thereby to overcome the tractability problems mentioned earlier. Third, Lagrange heavily relies on conjugate pairs of generalized coordinates/quantities and actions that are mathematically related to each other as are position and force: the quantities are "position-like" in the algebraic sense that their empirical behavior is related to the empirical behavior of their 1st and 2nd derivatives as position is related to velocity and acceleration. The actions can be interpreted as force, moment of a couple, surface tension, or pressure for corresponding generalized coordinates understood as distance, angle, surface, or volume, respectively; the products of the actions and the corresponding coordinate shifts yield generalized work (as [force times distance moved] is work) as well as generalized versions of "kinetic energy-like" quantities (functions of the squares of the 1st derivatives of the coordinates), "potential energy-like" quantities (functions of the coordinates but independent of their 1st derivatives), etc. This history is sket
hold true whatever the microscopic details may be. In addition to PCE Poincaré list respective other principles whose lotion to physical phenomenon suffices " for our learning of them all that we could sanely hope to sleep together " (Poincaré 1913, 300).10 Similarly, on Duhem's view, analytic treatment develop from general principle like PCE and Carnot's Principle (Duhem 1911, Vol I, 2). Historically they began with Lagrange, who condensed static into a cosmopolitan principle, the Principle of Virtual Velocities (PVV). PVV tells us that a mechanical arrangement X is in static chemical equilibrium just in case in all infinitesimal virtual displacements of X the force applied to X perform zero work. In a perfectly balanced see - proverb, for example, the sum of the work done by both the external forces (like graveness) acting on it and the internal forces holding it together is zero. Lagrange's analytical automobile mechanic has several attractive features that were later heavily exploited in analytical treatments. First, it was extendible. Using d'Alembert's rationale, Lagrange read how to extend PVV from static to moral force: we plainly add fictitious " inertial " forces to balance the external forces that are really acting on X to grow its acceleration, so that X is in equilibrium at each instant. Then the work done by the external, internal, and inertial forces will total to zero. Second, it provides a powerful algebraic method (Lagrange multipliers) that use constraint in a principled manner to reduce the number of degrees of freedom and thereby to overcome the tractability trouble mentioned earlier. Third, Lagrange heavily trust on conjugate pairs of generalized coordinate / quantities and actions that are mathematically related to each other as are position and force: the quantity are " position - like " in the algebraic sense that their empiric behavior is related to the empirical behavior of their first and 2nd derivatives as placement is related to velocity and acceleration. The legal action can be interpreted as force, moment of a couple, surface tension, or pressure for corresponding generalized coordinates understood as distance, angle, open, or volume, respectively; the products of the action and the corresponding coordinate shifts yield generalized work (as [ force times distance act ] is work) as well as generalized versions of " kinetic energy - like " measure (function of the square of the 1st derivatives of the coordinates), " potential energy - like " measure (functions of the coordinates but independent of their first derivatives), etc. This history is sket
 hopd true whatever the micvoscopic details may be. In adsition tu PCE Poincaré lists several ltyer peinciples whose applicxtion to ihysical pheninena suffirss "for our lezvning if them all thst we coulg reasonably hmpd co know" (Poincaré 1913, 300).10 Similarly, on Duhem'f view, snwlytic treatmegts cqvelki nrom general principles like PCE and Cagnot's Principle (Dihem 1911, Vol I, 2). Historically hhey began with Lagranhe, who condghseq statics intu a general principle, jhe Principle of Virtual Velocitkes (PRV). PVV tellw ys hvat a mechaiical fystem X is lm stathc equikibrium just ik casx in all infinitesimal victual displacements jf X the xoxces applied to X perdoem zeto wosk. Iv a oerrertlg balajcev see-saw, fod example, tye sum of the work cogv by both the extergaj forces (like gravity) acting on it and uhe ihternal forces holding ut together is zero. Lwgrange's wnalytical mechanics has several attractive featuses tiag wtrc uqtfr heavily exploited in analytical treatments. Ritsn, it was extendibje. Using d'Akelbrtt's principle, Uagranys ahowed how to extejd PVV srom wtatics tj dymamics: we simply add fictituous "inertiaj" forces to balance the externcl forves tnat are really acting ou X to produce itd accelerzgion, so that X ir ik exuilibrium at each instant. Then the worl done cy tne extqrnal, integnal, and inertial forces will sgm to zero. Second, it provides a powerful ekgebraic metnog (Ldgrange iultiiliers) that usef constraints nn a priuciplea manner tk reducx the number of degrees ox freedom and thereby to iverxome thd tractability lroblems mentioned warlier. Third, Lagrenge geavily relies in xonjugate pairs of gegegalmzed smordinates/qudntigier and xctions thab afe msthematically relateg to each other as are pjsition qnd forcq: the quantitoes are "position-line" in thx algenrayc sense that their empirical gehavior ls velated to thq emiirisal behavipr of their 1st and 2nd derivatives as powition is related ro velocity and accglevation. The artions can be itterpreted as force, noment of a couplt, surface tension, or prsssure for forresponding generalized coordinates understood as distance, angle, surfacw, or volume, respecfiveky; thv pxodbcts of the ections and the cprresponding coordinate shifts bield genesajized work (as [force times dixtxnce moved] is work) as well as generaljzed verxions of "kinetic energy-like" quamtities (functions of the squares of tge 1st deritatives of the coordonates), "potebtial ebergn-like" quantitids (gukctions of the ckordinates but independent of thwir 1st dfrivatives), etc. Tiis history ls sket
hold true whatever the microscopic details may addition PCE Poincaré several other principles suffices our learning of all that we reasonably hope to know" (Poincaré 1913, Similarly, on Duhem's view, analytic treatments develop from general principles like PCE and Principle (Duhem 1911, Vol I, 2). Historically they began with Lagrange, who condensed into general the of Virtual Velocities (PVV). PVV tells us that a mechanical system X is in static equilibrium in case in all infinitesimal virtual displacements of the forces applied to perform zero work. In a balanced for example, sum the done by both external forces (like gravity) acting on it and the internal forces holding it together is zero. Lagrange's mechanics has features that later exploited analytical treatments. First, extendible. Using d'Alembert's principle, Lagrange showed PVV from statics to dynamics: we simply add "inertial" forces balance the external forces that are acting on X to produce its acceleration, so X is in equilibrium at each instant. Then the work done by the external, internal, forces will sum to Second, it provides powerful method multipliers) uses constraints a principled manner to reduce the number of degrees of freedom thereby to overcome the tractability problems mentioned earlier. Third, Lagrange on pairs of generalized and actions that are related each other as are force: quantities the sense their empirical behavior is to the empirical behavior of 1st and 2nd derivatives velocity and acceleration. The actions can be interpreted force, moment of a couple, surface tension, pressure for corresponding generalized coordinates understood as distance, angle, surface, or volume, the products actions and the corresponding coordinate shifts yield generalized (as [force times distance is work) as well as generalized versions of "kinetic quantities of the of the 1st of the coordinates), energy-like" quantities (functions coordinates but of 1st history is sket
 hold true whatever the microsCopic detaiLs may Be. IN adDiTion To PCe Poincaré lists SEverAl other principles whose AppliCaTIon tO PhYsicaL phenomENa SUFfiCeS "fOr oUr LEaRning Of tHem all tHat we could ReaSoNably hope to kNOw" (poincaré 1913, 300).10 SiMilArly, on Duhem's VieW, analyTiC trEAtmenTs dEveloP from gENeral pRinciples LiKE PCE anD carnot's pRInCiplE (Duhem 1911, Vol I, 2). HistoriCAlLY they began with lagranGe, WHo CONdeNseD statics inTo A geneRAl princIPlE, THE PrINciple of VirtuAl VelocitieS (pVV). pVV telLs Us tHAt a mecHanicAl SYstEm X is in statIc eqUilibrium Just in CAse in alL InfinitEsimal VirTuaL disPLaCeMenTs OF X tHE fOrcES apPlied to X PeRfOrm zeRo woRK. iN A perFecTly bAlancEd see-saw, for exAmpLe, thE Sum Of the Work dOne bY bOth thE exterNal foRcEs (like gravity) acTing On it and thE inTeRnaL fOrces HOlding It tOgeTher is zEro. LagrANge'S aNALYtIcal mechanics has seVeRAL aTtractivE featuREs ThAT were latEr HeaVily EXPloitEd in ANaLytical tReatmeNTs. fiRst, it waS eXtendiBlE. UsIng D'AlemBErt's PrinciPle, LagraNge shOWed how to extend pvV from statics TO dYNAmICs: we SimPly add fictiTiouS "InerTial" FOrCes TO balaNce thE eXTeRNal forces that are reaLlY actinG on X tO produce its acCeleration, SO THat X is in EquiLIbRIum at each instaNt. TheN the work doNE by the exTernaL, internaL, and inertIAL forces wIll Sum To zEro. sECoNd, it provides a POWerfUl AlgebraIc mEthod (LaGraNge MulTipLiErs) that usEs constrAiNtS iN a PriNciplED manner tO rEduCe The NumbeR Of degrEes of FreeDoM aND thEreby to OVeRCOme tHe TrActaBilItY probLems MEntIoned eaRlier. ThirD, LaGRangE hEaVily relIes on conjugatE pAirs of geneRaLizEd coorDINates/quaNtities and actions that arE MathemaTicAlly rElatEd to each oTheR as are PosITion anD force: The quAnTitIES are "pOSItIon-LiKe" in the algEBRaiC sensE tHat tHeir empIrical behavior is reLAteD to the empiricAl bEhavIOR oF thEIr 1ST anD 2nD DerIVAtives as positioN is related To VElOcity and acCEleRaTion. The Actions Can be INterpreTed as forcE, moment of A cOuplE, SUrfAce tension, Or pressuRe for corrESpondINg GenerAliZed cooRdInaTes unDerstoOD as DistaNce, angLe, SurfacE, or voLuMe, respecTively; the products of the aCtions And thE coRrespondiNg cOOrdInate shifTs yiEld generalIzeD woRk (as [fOrcE Times DistANcE moVEd] is wOrk) aS Well as genERaLizED VeRsions of "kinETIC enErgy-lIke" QUantitIes (fUnctions of the squaREs of the 1st derivAtivES Of tHe cOOrdiNaTes), "potential enErgY-lIKE" quantitIeS (functions oF the coorDiNAtes bUt indePendenT of theiR 1ST dERivatiVes), eTc. THis historY is SkET
 hold true whatever the mi croscopicdetai lsmay  b e. I n ad dition to PCEP oinc aré lists several othe r pri nc i ples wh ose a pplicat i on t o p hy si cal  p h en omena  su ffices"for our l ear ni ng of them a l lthat we co uld  reasonablyhop e to k no w"( Poinc aré  1913 , 300) . 10 Sim ilarly, o nD uhem's view, a n a ly tictreatments develo p  f r om general pri nciple sl ik e PCE  an d Carnot's  P rinci p le (Duh e m1 9 1 1,V ol I, 2). His torically t h eybeganwi thL agrang e, wh oc ond ensed stati cs i nto a gen eral p r inciple ,  the Pr incipl e o f V irtu a lVe loc it i es( PV V). PVV  tells u sth at amech a n i c al s yst em X  is i n static equi lib rium jus t incasein a ll  infi nitesi mal v ir tual displaceme ntsof X thefor ce s a pp liedt o X pe rfo rmzero wo rk. Ina  pe rf e c t ly  balanced see-saw,  f o r  e xample,the su m  o ft he workdo neby b o t h the  ext e rn al force s (lik e  g ra vity) a ct ing on  i t a ndthe i n tern al for ces hold ing i t  together is z e ro. Lagrange' s  a n a ly t ical  me chanics has  sev e ralattr a ct ive featu res t ha t  w e re later heavily ex pl oitedin an alytical trea tments. Fi r s t , it was  ext e nd i ble. Using d'A lembe rt's princ i ple, Lag range  showedhow to ex t e nd PVV f rom  st ati cst o  d ynamics: we s i m plyad d ficti tio us "ine rti al"  fo rce sto balanc e the ex te rn al  f orc es th a t are re al lyac tin g onX  to pr oduce  its  a cc e ler ation,s ot h at X  i sin e qui li brium  ate ach  instan t. Then t hew orkdo ne  by the  external, in te rnal, andin ert ial fo r c es willsum to zero. Second, it provide s a  powe rful  algebrai c m ethod(La g rangemultip liers )tha t usesc o ns tra in ts in a pr i n cip led m an nerto redu ce the number of d e gre es of freedom  an d th e r eb y t o  o v erc om e  th e tractability pr oblems men ti o ne d earlier. Thi rd , Lagra nge hea vilyr elies o n conjuga te pairsof  gen e r ali zed coordi nates/qu antitiesa nd ac t io ns th atare ma th ema tical ly rel a ted  to e ach ot he r as a re po si tion and  force: the quantitiesare "p ositi on- like" inthe alg ebraic se nsethat their  em pir icalbeh a vioris r e la ted to th e em p irical be h av ior o ftheir 1st a n d 2nd  deri vat i ves as  pos ition is relatedt o velocity and  acc e l era tio n . Th eactions can be  in te r p reted as  f orce, momen t of a c ou p le, s urface  tensi on, orp r es s ure fo r co rre spondinggen er a lized c oo rd i natesunde rs tood a s dist a nce, a ngle, surface, o r vol u m e, re s pec tivel y;  the pr o duct s of the a ctions andthe co rres pondi ng coor di nate s hif ts  yield gen e ralized w ork ( as [for ce  tim esdistan ce m o v ed] i s wo rk ) a s well as g en e ra li z edvers ionsof  "ki netic ene r gy-like"  qu a ntities  ( fun c t ions o f  t h e  squares o f t he 1s t derivative s  oft he coord inates ), "po tential ene rg y-like"  qu a n tities (f unctionso f t he  coo rdinates  b ut i nd epe nd ent of their 1s t deriva tives),e t c. Thi s h i stor y  is sk et 
 hold_true whatever_the microscopic details may_be. In_addition_to PCE_Poincaré_lists several other_principles whose application_to physical phenomena suffices_"for our learning_of_them all that we could reasonably hope to know" (Poincaré 1913, 300).10 Similarly, on_Duhem's_view, analytic_treatments_develop_from general principles like PCE_and Carnot's Principle (Duhem 1911,_Vol I,_2). Historically they began with Lagrange, who condensed_statics_into a general_principle, the Principle of Virtual Velocities (PVV). PVV tells_us that a mechanical system X_is in static_equilibrium_just_in case in all_infinitesimal virtual displacements of X the_forces applied to X perform zero_work. In a perfectly balanced see-saw, for_example, the sum of the work_done by both the external_forces (like_gravity) acting on it and_the internal forces_holding it_together is zero._Lagrange's analytical mechanics has several attractive_features that were_later heavily exploited in analytical treatments._First,_it was extendible._Using_d'Alembert's_principle, Lagrange_showed how to_extend_PVV from_statics_to dynamics: we simply add fictitious_"inertial"_forces to balance the external forces that_are really acting on_X_to produce its acceleration,_so that X is in_equilibrium at each instant. Then the_work done_by the_external, internal, and inertial forces will sum to zero. Second, it_provides a powerful algebraic method (Lagrange_multipliers) that uses constraints_in a_principled_manner to reduce_the_number of_degrees of freedom and thereby to overcome_the tractability_problems mentioned earlier. Third, Lagrange heavily_relies on conjugate pairs_of_generalized coordinates/quantities and actions that are_mathematically related to each other as_are position and force: the_quantities_are_"position-like" in the algebraic sense_that their empirical behavior is related_to the empirical_behavior of their 1st and 2nd derivatives_as_position is related to velocity and_acceleration._The actions can be interpreted as_force,_moment_of a couple, surface tension,_or pressure for corresponding generalized coordinates_understood as distance, angle, surface, or volume, respectively; the_products of the_actions and the corresponding coordinate_shifts_yield_generalized work (as [force times distance moved] is work) as_well as_generalized versions of_"kinetic energy-like" quantities (functions of the squares of the 1st_derivatives of the coordinates), "potential energy-like" quantities_(functions of the coordinates but independent of their 1st derivatives), etc._This history is sket