5 Lagrangiana e Hamiltoniana

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FORMALISMO MA1 N particulle diamond.ac.uk de N N of de ) : = (91, / = dt = N:= = DI LIBERTA' del SISTEMA L = L me il del de in istante ed Le able L in d PRINCIPLE MINIMA W = t2 dt i MINIMO = + Al tz JL JL = + = PER PARTI t2 of + - dt t, Se the do me can ESTREMI FISSI, d NULLO, e del ) - JL = 0 (N- of ) of MOTO DI of ORDER De LANDAN & LIFSHITZ (1960), del suff e L in or per d N : L = T do in V(9) : = - CONSERVATIVO, in NON doh del do : JL = JV = - JV EQ. DEL 01 NEWTON JL = e FORZA = - JV In i do tale DI GENERALIZZATO = JL Dollo def / = = of JL = Jr. JL = L in del TEMPO dt d L = JL + JL + JL = = JL + Jg: ) JL + = at of JL ) at JL - ) = - It JL If me to L = V, me dt ( 2 - T+V) + = dt (T + = - JL 2T ad IF L does NOT depend from T+V= + = ENERGA del E = JL ) - L = + loscie L in wel in => energeFORMALISRO MA2 Le LEGENDRE L e diamond.ac.uk 9, 9 in d HAMILTONIANA del H = - per L = T Dolle H JH = DEL DI 2N - set del dd JH = - JL = A JL = - dp: = - p; dt DI LIOUVILLE : d M.Weiss (DISPENSE) TRASFORMAZIONI legg Q: = PN) = N P. = in, PN) Eye date se delle = done = = dt e d POIN es: 11 PHASE SPACE VOLUME N SS 2 SS of PHASE SPACE AREAS the let- 162 - A SHORT DEMONSTRATION OF LIOUVILLE'S THEOREM* M. Weiss CERN, Geneva, Switzerland ABSTRACT A brief demonstration of Liouville's Theorem is given by applying the Hamiltonian. An ensemble of particles evolving in a system of external forces (space and velocity dependent) and self forces (space charge) is described by two families of canonically con- jugated variables (coordinates) q and p. The equations of motion form a system of first- order differential equations of the coordinates q and , where the dot indicates derivatives with respect to time. If the system is non-dissipative, one can obtain the equations of motion from a function called Hamiltonian: q = p = . The Hamiltonian is in general also a function of time: H(q,p,t) . An ensemble of particles, at a given moment t, occupies a volume V(t) in the (q,p) space called the phase space. P V(t+At) V(t) ... vector of surface element w(t) df w(t) ... phase space velocity of surface element: = p q * Derivation shown at a discussion session- 163 - At the time t + the particles occupy another volume It can easily be shown that these volumes are the same: dV(t) dt = + = + = + др p) dv = 0 surface + volume integral integral ( - = 0 (Gauss Theorem) (Hamilton) The volume V(t) remains constant, if the motion can be represented by a Hamiltonian. This is true also when H is an explicit function of time. We conclude: In non-dissipative systems, the particles move like an incompressible fluid in phase space. This is Liouville's Theorem.EQUAZIONI HILL 30 do FORMALISMO SIF PROVAN E dt = + is 120 ANNI 1897-2017 DI MAXWELL wel ) E = - - Jt JA = x LAGRANG(ANA D' PARTICELLA Dd della Rel. Sp. he Principio determine in debbe esthe INVARIANTE DI t.c. = 9 Allone il dene W = = = INV. = dt 8 TEMPO un = INV. ph 1) 2) di d psin depende do rL - DIMOSTRA che del a Lp =- F ( of = 0 ; of dt - = = 2 = + = is + at dr of atLAGRANGIANA DI SINGOLA PARTICELLA 3) Ip ph = =0 LINEARE CARICA 5) LINEARE in L = - r - + 6) are LINEARE in N.B.: queste d. L = i) H esseciate ii) le d MOMENTO P. = JL = + e = + = CINETICO H " = in dolla def. ) is = = eA) = J = + = rmo + r + e - = 1 - + ] + e = + c + = C + E. PARTICELLA LIBERA : he = - = + r = + = + = + = + =FORMALISMO HAMILTONIAND : EQ. HILL : 31 LINEARI, TRASVERSI Ax = Ay = 0 A2 = 2) ADIABATICA = di dol SR CARTESIANO e - SERRAT) P 0 X Dolla del P C = - e Yc = tc = GENERATRICE f che d. = MOMENT. CONIUGATI Px, Ps = - - - che essa be : Xc = = tc) Spe e Px = Pxc + P = - JF = = = Ps = Pxc + (1+ P Ptc come: Ps =+(1+ P - Pxc ] Pz le component del A = = Atc fn Are, = (Ax, Ar,HAMILTOMANA 32 H = - + + = (Ps = C - 2 + + + dolla e H - = - Ps = - (1+ + P X ) [ H - Px2 - - - As = = - (1+ + P + = - + + eAs che descrive e As = Ps e [ P k) 2 + = By = = " + - e = (1+ P - strong Bx = = - e = - SY = JX = dx dt = Jpx 1 dpx = = = ps Ps Ps Jx J the - ) + e [ Pt Ps + X ( 1 p2 k) + k 2 ] = + P 2 ix - - k) = per d ds = Ps 1 ds = - = - psky = - ky " is + ky =0MATRICE SIMPLETTICKE e INVARIANII able Hill e he di = 1. cho il d tole one he ME conditions NECESSARIA e SUFFICIENTE, l 9 = - up 9 1 = = on -1 0 Can J = -1 9, JH P. J= -1 ) J9, 92 01 PH = JH P2 -1 0 Jp, JA V2 = Mv, 2> V2 = = = DH L per M=1 = + + = = MJDH MJM+ = V2 = JDH ; = J at det J = = deh - def J. 1 N.B. : se M le condizione d 1) VINCOLI di M INDIPENDENTI. Ex., M=1 TEO d = = on = = & Suptes il mello delle for do d (= in d use di in me se = = ! 2 : se = Lawlle ! (wed 3 : rl i in INSIEME INVARLANTI - CARTAN): = + + = = + + dydz = const. SE ; - gli (Ex, FLAT BEAM (= Ex + = Ey => Ex = k = CAUPLINGROUND BEAM (LINAC) k in coordinate rel 2 move - so COSTANTE MATRICE del m=2 = (x'> 6x ) : " che R(s) = R(s) R SIMPLETTICA Y , M2 y' R-1) Oxy = TX = = -1 e D(s) = 9 : 11 0 Gxy = = (R/R = Eeig = e E2 = 1 EIGEN- EMITTANCES In > E2 E+, E2:= (Ex + Ey = = (E+ + = + = = E+ E. = e =det 6xy = = det D = Eeig H L = X-Y H L Epg = 2-D Eig 0 E+ (x 2 = E+ E+E- 2 2 + = E. + L² = = + L + = 2 ( 2 + L²) (E+ + 4 E+