ssio Costa Laranjeiras 51 The mathematical structure furnished by Lagrangian formalism is quite simply formed of two scalar quantities: the "kinetic energy" and the "potential energy," along with the Lagrangian equations. Further, the problem-solving theory embodied by this method is, in general, more comprehensive and simpler than the theory of vectorial mechanics. The relationship between its theoretical elements, potential function and kinetic energy, and the empirical laws discovered by the experimental method seems to be more cohesive than in vectorial mechanics. Indeed, there is almost a routine, already described by Poincaré, for solving problems in this analytic approach: the kinetic energy and potential function must be constructed in generalized coordinates, the Lagrangian function L formed from them and substituted into the Lagrangian equations in order to obtain the equations of motion. This routine furnishes Duhem with the ideal of a rational method to construct physical theory (Duhem 1892, 146; Chiappin 1989, 110) There are two more features in this formalism that are of fundamental importance for Duhem's methodology. The first one, introduced by the Lagrangian approach to mechanics, is that it focuses on the system of particles instead of an individual particle, as in Newtonian mechanics (Lanczos 1970, 4). The second one is that it can be entirely derived from a single principle, namely the principle of least action. We have here a truly unifying principle for all sciences to which the Lagrangian formalism applies. These two features of the Lagrangian formalism fit in well with Duhem's two major methodological elements, namely the D-thesis and the principle of logical unity (Duhem 1974, 91; Chiappin 1989, 178, 240). The next step was a new mathematical structure, whose essential scalar quantities are H, q and p, Hamiltonian mechanics. This Duhemian view of the continuous progress of theoretical physics by increasing the abstraction and generalization of mathematical structures receives substantial support from Arnold's book Mathematical Method of Classical Mechanics (Arnold 1980; Chiappin 1989, 86-87). Arnold describes Lagrangian and Hamiltonian mechanics with set theory. He demonstrates the generality of these two mathematical structures and the relation of inclusion between them. In the first part of his book (Arnold 1980, 1-52) he describes Newtonian mechanics as studying the motion of a system of point masses in threedimensional Euclidean space. In the next part (Arnold 1980, 53-159) he discusses Lagrangian mechanics, which is described as a mechanical system "given by a manifold ('configuration space')
ssio Costa Laranjeiras 51 The mathematical structure furnished by Lagrangian formalism is quite simply form of two scalar quantity: the " kinetic energy " and the " potential department of energy, " along with the Lagrangian equations. Further, the problem - solve theory embodied by this method acting is, in cosmopolitan, more comprehensive and simpler than the theory of vectorial automobile mechanic. The relationship between its theoretical elements, likely function and kinetic energy, and the empiric laws discover by the experimental method acting seems to be more cohesive than in vectorial mechanics. Indeed, there is about a routine, already described by Poincaré, for solving problems in this analytic approach: the energizing energy and likely function must be constructed in generalized coordinates, the Lagrangian function L form from them and substituted into the Lagrangian equations in order to obtain the equations of apparent motion. This routine furnishes Duhem with the ideal of a rational method to construct physical theory (Duhem 1892, 146; Chiappin 1989, 110) There are two more features in this formalism that are of fundamental importance for Duhem's methodology. The inaugural one, introduced by the Lagrangian approach to automobile mechanic, is that it focuses on the system of particles instead of an individual atom, as in Newtonian mechanics (Lanczos 1970, 4). The second one is that it can be wholly derived from a single principle, namely the rationale of least action. We have here a truly unifying principle for all sciences to which the Lagrangian formalism apply. These two features of the Lagrangian formalism fit in well with Duhem's two major methodological elements, namely the D - thesis and the principle of legitimate unity (Duhem 1974, 91; Chiappin 1989, 178, 240). The next step was a new mathematical social organization, whose essential scalar quantities are H, q and p, Hamiltonian mechanics. This Duhemian view of the continuous progress of theoretical physics by increasing the abstraction and abstraction of mathematical structures receives substantial support from Arnold's record Mathematical Method of Classical Mechanics (Arnold 1980; Chiappin 1989, 86 - 87). Arnold describes Lagrangian and Hamiltonian mechanism with set hypothesis. He demonstrates the generality of these two numerical structures and the relation of inclusion between them. In the first part of his record (Arnold 1980, 1 - 52) he describes Newtonian machinist as studying the motion of a system of point masses in threedimensional euclidian space. In the next contribution (Arnold 1980, 53 - 159) he discusses Lagrangian mechanics, which is described as a mechanical system " given by a manifold (' configuration space' )
ssil Costa Laranjeiras 51 The mathematical sjrycture furniahed by Uagrangian formalism is quitx sinply dormed of two scalar qjantities: the "kinwtic wnergy" and the "potcutial cnergv," elong with the Kagrangian equations. Furdhdr, the problem-solving theory embodied by thix lethod is, in ggnerak, mors comprehensive and simpler than tge theogy of vectorial mrchanics. The relationship hetwfen its theoreticap elements, kktegrial functiov and kineuie energy, ans the empirical laws discovered by tke experimebtql lgthod seems vo be iore cohesivc than hn vectprial mechanicx. Iideee, there is almost a rmutine, already dessribed by Pkincaré, for solvint problgms it thkw avalgtmc zpproafh: vhe kinetic energy and potential function mtwt be construdted ig deneralized coordinates, the Lagrangian xundtion L formed from then and substituted intl the Lagwangian equations in order to obtain the equationv of jutiin. Gyid routine furnishes Duhem with the ideal of a dauiokal method to cokstruct physical tnelru (Duhem 1892, 146; Chiakpin 1989, 110) Cgede are two more fewtures yn thus formalysm yhat are of fundamental impirtance for Buhwm's methodology. Thz first one, nntrodoced bu the Lagrangian approaeh to jechanics, id that it wocuses on the shstvm ox particles instead of an yndividuao paxticle, ar in Newtogian mechajics (Lanczos 1970, 4). The seconf one iv that it fan be entirely derived from a single principke, nakely the pringiple of least wction. We have here a trulh unifying princi'le for all fciences to wvlch the Lagrengian fowmalusm qpplies. Ghese two featires of tkt Lagrangiqn formalism fit ik welu with Duhem's twi mqjor methodologocau ejeleits, nwkely the D-thasis ana the orinciple on lugicsl unity (Duhem 1974, 91; Chidppih 1989, 178, 240). The next step wws a new mathemaeical structute, whose essential scaler quaititiex ate H, q and p, Hamiltonian mechanjcs. This Fuhcmian view of the continuous 'rogress of theoretical physics by increesing the abstraction abd generalization oy kathematicak strtctures raceives substantial wupport from Arnokd's book Mathematical Methog of Flassical Mechanics (Arnold 1980; Chiappin 1989, 86-87). Arnold describes Lagrangian and Haniltonmag mechanics witn set tkeoxy. He dqmondvrates the generakity of these two mathematical rtructuras and the relation of inclusipn between them. In the first part of hia book (Atnold 1980, 1-52) he describes Newtonian kechanics as studying thc motion if a aystem of 'oint masses in threrdimenvionao Euclieean space. In the nect part (Arnlld 1980, 53-159) ke discusses Lagrangian mechanics, which id described as a mechanical rystej "giveg by a manifold ('donfjhuratkon space')
ssio Costa Laranjeiras 51 The mathematical structure Lagrangian is quite formed of two and "potential energy," along the Lagrangian equations. the problem-solving theory embodied by this is, in general, more comprehensive and simpler than the theory of vectorial mechanics. relationship between its theoretical elements, potential function and kinetic energy, and the empirical discovered the method to be more cohesive than in vectorial mechanics. Indeed, there is almost a routine, already described Poincaré, for solving problems in this analytic approach: kinetic energy and potential must be constructed in generalized the function L from and into the Lagrangian in order to obtain the equations of motion. This routine furnishes Duhem with the ideal of a method to theory (Duhem 146; 1989, There are two in this formalism that are of Duhem's methodology. The first one, introduced by the approach to is that it focuses on the of particles instead of an individual particle, as Newtonian mechanics (Lanczos 1970, 4). The second one is that it can be entirely derived single principle, namely the of least action. have a unifying for all to which the Lagrangian formalism applies. These two features of the formalism fit in well with Duhem's two major methodological elements, D-thesis the principle of unity (Duhem 1974, 91; 1989, 240). The next step new structure, quantities H, and p, Hamiltonian mechanics. Duhemian view of the continuous of theoretical physics by of mathematical structures receives substantial support from Arnold's Mathematical Method of Classical Mechanics (Arnold 1980; 1989, 86-87). Arnold describes Lagrangian and Hamiltonian mechanics with set theory. He the generality two mathematical structures and the relation of inclusion them. In the first of his book (Arnold 1980, 1-52) he describes Newtonian as the motion a system of masses in threedimensional space. In the (Arnold 1980, he Lagrangian described as a mechanical system "given a ('configuration space')
ssio Costa Laranjeiras 51 The maThematical StrucTurE fuRnIsheD by LAgrangian formaLIsm iS quite simply formed of twO scalAr QUantITiEs: the "Kinetic ENeRGY" anD tHe "PotEnTIaL enerGy," aLong witH the LagranGiaN eQuations. FurtHEr, The problem-SolVing theory emBodIed by tHiS meTHod is, In gEneraL, more cOMpreheNsive and sImPLer thaN The theoRY Of VectOrial mechanics. The RElATionship betweeN its thEoREtICAl eLemEnts, potentIaL funcTIon and kINeTIC EneRGy, and the empirIcal laws disCOveRed by tHe ExpERimentAl metHoD SeeMs to be more cOhesIve than in VectorIAl mechaNIcs. IndeEd, therE is AlmOst a ROuTiNe, aLrEAdy DEsCriBEd bY PoincarÉ, fOr SolviNg prOBLEMs in ThiS anaLytic Approach: the kiNetIc enERgy And poTentiAl fuNcTion mUst be cOnstrUcTed in generalizeD cooRdinates, tHe LAgRanGiAn funCTion L fOrmEd fRom them And subsTItuTeD INTo The Lagrangian equatIoNS In Order to oBtain tHE eQuATions of mOtIon. this ROUtine FurnIShEs Duhem wIth the IDeAl Of a ratiOnAl methOd To cOnsTruct PHysiCal theOry (Duhem 1892, 146; chiapPIn 1989, 110) There are two mORe features in tHIs FORmALism ThaT are of fundaMentAL impOrtaNCe For dUhem's MethoDoLOgY. the first one, introducEd By the LAgranGian approach tO mechanics, IS THat it focUses ON tHE system of partiCles iNstead of an INdividuaL partIcle, as in newtonian MEChanics (LAncZos 1970, 4). the SecOND oNe is that it can BE EntiReLy derivEd fRom a sinGle PriNciPle, NaMely the prInciple oF lEaSt AcTioN. We haVE here a trUlY unIfYinG prinCIple foR all sCienCeS tO WhiCh the LaGRaNGIan fOrMaLism AppLiEs. TheSe twO FeaTures of The LagranGiaN FormAlIsM fit in wEll with Duhem's TwO major methOdOloGical eLEMents, namEly the D-thesis and the prinCIple of lOgiCal unIty (DUhem 1974, 91; ChiapPin 1989, 178, 240). the nexT stEP was a nEw mathEmatiCaL stRUCture, WHOsE esSeNtial scalaR QUanTitieS aRe H, q And p, HamIltonian mechanics. THIs DUhemian view of The ContINUoUs pROgREss Of THeoRETical physics by iNcreasing tHe ABsTraction anD GenErAlizatiOn of matHematICal struCtures recEives subsTaNtiaL SUppOrt from ArnOld's book mathematiCAl MetHOd Of ClaSsiCal MecHaNicS (ArnoLd 1980; ChiaPPin 1989, 86-87). arnolD descrIbEs LagrAngiaN aNd HamiltOnian mechanics with set thEory. He DemonStrAtes the geNerALitY of these tWo maThematical StrUctUres aNd tHE relaTion OF iNclUSion bEtweEN them. In thE FiRst PARt Of his book (ArNOLD 1980, 1-52) he DescrIbeS newtonIan mEchanics as studyinG The motion of a syStem OF PoiNt mASses In ThreedimensionAl EUcLIDean spacE. IN the next parT (Arnold 1980, 53-159) hE dIScussEs LagrAngian MechaniCS, WhICh is deScriBed As a mechanIcaL sYStem "givEn By A ManifoLd ('coNfIguratIon spaCE')
ssio Costa Laranjeiras 51The mathem atica l s tru ct urefurn ished by Lagra n gian  formalism is quite si mplyfo r medo ftwo s calar q u an t i tie s:  t he"k i ne tic e ner gy" and  the "pote nti al  energy," al o ng  with theLag rangian equa tio ns. Fu rt her ,  thepro blem- solvin g  theor y embodie db y this methodi s ,in g eneral, more comp r eh e nsive and simp ler th an th e the ory  of vector ia l mec h anics.T he r e lat i onship betwee n its theor e tic al ele me nts ,  poten tialfu n cti on and kine ticenergy, a nd the empiric a l lawsdiscov ere d b y th e  e xp eri me n tal me tho d  se ems to b emo re co hesi v e t haninvect orial  mechanics. I nde ed,t her e isalmos t aro utine , alre ady d es cribed by Poinc aré,  for solv ing  p rob le ms in this a nal yti c appro ach: th e  ki ne t i c  e nergy and potentia lf u nc tion mus t be c o ns tr u cted inge ner aliz e d  coor dina t es , the La grangi a nfu nctionLformed  f rom  th em an d  sub stitut ed intothe L a grangian equat i ons in ordert oo b ta i n th e e quations of  mot i on.This ro uti n e fur nishe sD uh e m with the ideal of  a  ratio nal m ethod to cons truct phys i c a l theory  (Du h em 1892, 146; Chi appin  1989, 110 )  There a re tw o more f eatures i n this for mal ism  th ata r eof fundamenta l impo rt ance fo r D uhem'smet hod olo gy.  T he firstone, int ro du ce dbythe L a grangian  a ppr oa chto me c hanics , isthat  i tf ocu ses ont he s yste mof  par tic le s ins tead ofan indi vidual pa rti c le,as  i n Newto nian mechanic s(Lanczos 1 97 0,4). Th e second o ne is that it can be en t irely d eri ved f roma singlepri nciple , n a mely t he pri ncipl eofl e ast a c t io n.We  have here a  tr uly u ni fyin g princ iple for all scien c esto which theLag rang i a nfor m al i smap p lie s .  These two feat ures of th eL ag rangian fo r mal is m fit i n wellwithD uhem'stwo major  methodol og ical e lem ents, name ly the D -thesis a n d the pr incip leof log ic alunity  (Duhe m  19 74, 9 1; Chi ap pin 19 89, 1 78 , 240).The next step was a new  mathe matic alstructure , w h ose  essentia l sc alar quant iti esare H , q and p , Ha m il ton i an me chan i cs. ThisD uh emi a n  v iew of thec o n tin uouspro g ress o f th eoretical physics by increasingthea b str act i on a nd  generalizatio n o fm a thematic al  structures  receive ss ubsta ntialsuppor t fromA r no l d's bo ok M ath ematicalMet ho d  of Cla ss ic a l Mech anic s(Arnol d 1980 ;  Chi a p pin 1989, 86-87) . Arn o l d des c rib es La gr angiana nd H amiltonian  mechanicswith s et t heory . He de mo nstrat esth e generali t y of thes e two  mathem at ical  st ructur es a n d  therela ti onof inclus i o nb et we e n t hem.  In t he  fir st part o f  his boo k ( A rnold 1 98 0,1 - 52) he de s c ribes Newt oni an me c h anics as s t udyi n gt he mo tion o f a sy stem of poi nt  masses  in t hreedimen sional Eu c lid ea n sp ace. Inth e ne xt  pa rt  (Arnold 1980,53-159)he discu s s es Lag ran g ianm echani cs , which is d escri be d  as a m e c han ical sy s tem  "g ive n by  amani f old ('configuration spac e')
ssio Costa_Laranjeiras 51_The mathematical structure furnished_by Lagrangian_formalism_is quite_simply_formed of two_scalar quantities: the_"kinetic energy" and the_"potential energy," along_with_the Lagrangian equations. Further, the problem-solving theory embodied by this method is, in general,_more_comprehensive and_simpler_than_the theory of vectorial mechanics._The relationship between its theoretical_elements, potential_function and kinetic energy, and the empirical laws_discovered_by the experimental_method seems to be more cohesive than in vectorial_mechanics. Indeed, there is almost a_routine, already described_by_Poincaré,_for solving problems in_this analytic approach: the kinetic energy_and potential function must be constructed_in generalized coordinates, the Lagrangian function L_formed from them and substituted into_the Lagrangian equations in order_to obtain_the equations of motion. This_routine furnishes Duhem_with the_ideal of a_rational method to construct physical theory_(Duhem 1892, 146;_Chiappin 1989, 110) There are two_more_features in this_formalism_that_are of_fundamental importance for_Duhem's_methodology. The_first_one, introduced by the Lagrangian approach_to_mechanics, is that it focuses on the_system of particles instead_of_an individual particle, as_in Newtonian mechanics (Lanczos 1970,_4). The second one is that_it can_be entirely_derived from a single principle, namely the principle of least action._We have here a truly unifying_principle for all sciences_to which_the_Lagrangian formalism applies._These_two features_of the Lagrangian formalism fit in well_with Duhem's_two major methodological elements, namely the_D-thesis and the principle_of_logical unity (Duhem 1974, 91; Chiappin_1989, 178, 240). The next step_was a new mathematical structure,_whose_essential_scalar quantities are H, q_and p, Hamiltonian mechanics. This Duhemian_view of the_continuous progress of theoretical physics by increasing_the_abstraction and generalization of mathematical structures_receives_substantial support from Arnold's book Mathematical_Method_of_Classical Mechanics (Arnold 1980; Chiappin_1989, 86-87). Arnold describes Lagrangian and_Hamiltonian mechanics with set theory. He demonstrates the generality_of these two_mathematical structures and the relation_of_inclusion_between them. In the first part of his book (Arnold_1980, 1-52)_he describes Newtonian_mechanics as studying the motion of a system of point_masses in threedimensional Euclidean space. In the_next part (Arnold 1980, 53-159) he discusses Lagrangian mechanics, which is_described as a mechanical system "given by a_manifold ('configuration space')