Bhatnagar, P L

Kothari, D S

English

IACS Institutional Repository

3 2A  NOTE ON THE PRINCIPLE OF ADIABATIC INVARIANCE^By P. L. BHATNAGARANDD. S. KOTH ARI{ R e c e i v e d  f o r  p u b l i c a t i o n ,  May 2, tgjfs)The principle of adiabatic invariance' first formulated by Holtztnann, in connection with his studies on the Kinetic Tlieory of gases, acquired added importance when Ehrenfest incorporated it in the old quantum theory. It is usual “ in discussing the principle, to illustrate it with the help of some simple examples of classical mechanics. The illustration of the simple pendulum, due to Uinstein, has almost become historic. Andrade ’  has also considered the case of a conical-pendulum where for E /i’ to be adiabatically invariant, it is no longer necessary to confine to small angles, a restriction which has tube imposed for a simple pendulum. In the above relation represents the energy and v the pulsatauce of vibration, where the puhsatance is defined as aw times the frequency.Because of its importance and many apjjlications, it is of some interest to have other simple illustrations of the principle of adiabatic invariance (besides those referred to above).The examples of (i) a compound pendulum, (//) an oscillating magnet* I and (Hi) an electric oscillating circuit are given below’ . The case of simple pendulum is treated in h ,  that of compound pendulum in §2, an elementary general treatment in §3 and the magnetic and electric cases arc given in §4.1,1. Simple pendulum.—We consider a simple pendulum, its string passing Ihroiigh a ring which may be moved vertically up and down to change the point of suspension [Fig. 1 (a)]. If I be the length of the pendulum, v is given, by-and dv_ __ dl2 ITlie tension T  in the string is■ T=m /^®'+m gcosO .and the vibratioitai energy K is' E = 4mg/^0)where 0^ is-the amplitude of the vibrationsin W,* Communicated by the Indian Physical, Society, t Paritfn ,(*940) baa referred to a very interesting application of this case272 P. L. Bhatnagar and D. S. KothariThe reaction R of the string the ring tiffll be 2T sin and it will be inclined at an angle (71/2-^/2) to the vertical.When the ring IS displaced nl)v^ l^rds, slowly ami coiitiinioiisly, llirongli a vertical height ti/, the vertical coiMpoiiciil of R does wcih and the loss of uicigy of the system is equal toR sin =  =neglecting terms higher than 0 .^Therefore^ the average loss of energy dU becomesi m g  0  ^ d l —l tng 6  ^ dl.Hence dlior K 'constant. ^ If the clisplaoenient of the ring did not take place slowly and continuously, but say at the instant when the loss of energy would be Amg V  on the other hand, if the dis­placements take place when the energy loss would ht zero.An alteration in the length of the simple pendulum may be brought about in two ways : (/) by moving the ring as discussed above, and ( l i)  by suddenly clamping the string a little below the original point of suspension and then rtpeatirig this procedure of successively clamp­ing at lower and lower points (the displacement of thi* pendulum remaining thesfmie during theclfimping) It is interesting to note that in the second case il is not but H*', Hint is, an adiabatic invariant.Suppose that when the pendulum has a dis]>laremcnt 0, the string is suddenly clamped at a displacement d J  from the initial point of mispension, 'I‘he kinetic energy will remain the same during the process but the potential energy of vibration will l)c altered by«f {m gi(i-*cos (?)j =  iU ,and, therefore, the average change \yill be ►and hence R»'-constantNote on the Principle of Adiabatic Invariance 273N 2 toml>ouud-pcuduluimj^^oaf,ider a c^iijunnid pciiduluin. TJic pulsa- lance of small oscillation is given l>y-  V ‘j and dv _d!^ . 1 dh k i  ft ’where/v'is the radius Of gyration about the axis of suspension and h the distance between the centre of gravity and the axis. The pendulum carries a sliding mass, whose motion alters the position of the centi'e of gravity along the line OO[Fig. Kb)].The vibrational energy of the pendulum isAt the instant when the angular displacement of the pendulum is 0, let a motion of the sliding vveight lower the centre of gravity by a small amount d/t, the change in the angle 0 during this alteration of the position of c*g, being supi)OvSed to be negligible. As the angular momentuni of the pendulum will remain constant during the above process, we haveiuk'^O — cousiBinttand as the kinetic energy T  isI )^'“1 - 1 o ,2 .we have dT _  2dk T  “  T  ’or d ' i '= - 2 ‘ /^' ih n k ^ h  k=  ~ ~  (iwk^BSlHence we havedT _  _d/c K ' ■The potential energy of the pendulum in the position 9 h0m g h { i  — cos B) =  m g h  - ,2and the increase dh in It increases the potential energy byd V ^ iin g  B^dh,and therefore ord V ~ iv ig  B^  d h ^ h n gdV i.dh,E  2 . X 'B^  dh274 P. L . Bhatnagar and D. S. Kothari'rilcieforedl-: _ d /r  , I d / ( ^ d - '  Iv a / ;  ' 'o rK=  eon.slant.S A)i clniicnlaiy general treatincnl :—Consider any simple liannonk- uioliou defined bycfS.v . m— ,, =  — Ax dl^’ A mand >'= a /\ fThe energy of vibration 1^  is given bywhere a is the auiplitude- When the displacement is .v, let A change to A 4-ciA ; then the increase in the restoring force is xdA, and the energy of vibration is increased byand averaging we haveandor<rJ d K . x  d x = i d \  x'-* 0S E ™  IdAa^dE _  i dA_d\' H 2 A VJv1 =  constant. ^ 4 (a). Small magnet :—Consider a magnet of moment M oscillating in a field of strength H. When the deflection is let H change to H + (iH, then the increase in tlie restoring couple is and, therefore, the increase in the vibra-lional energy will bej MIdH =aud averaging we have As the energy of vibration isdIi =  iMi9§dH.Hj==iMH(9g, , _  /MHaud ' — ^^ / - j—■ jwhere I is the niomeut of inertia of the m agnet about the a x is  of vibration.W'c liavt'I'licreforcI rfH „,i.'K ■ 2 H “  v 'Note on the Principle oj Adiabatic Invariance 275li constant.(b) Klcci nc oscillating circuit :—We consider an electric circuit coiisisliiij* )1 a capacity c and a self-inductance I, thenV= V /  andV /r V 2 cLet q  be the charj:*e on the condenser at instant i ,  thenq — qo sin v/,1 a'['lie electrostatic enen^y at this instant is and the total energy of the circuit2 CK isE =  ’2 cWhen the charge on the condenser is q, let its capacity be suddenly changed from ( to c-\'dc  ^ then the potential difference of the condenser will be cliaiiged byrfV -  q di.and, therefore, the change in the energy will bedK =  - ’ q d \ ' -  I ‘I[■ A -Tlie average change of energy isand lienee ue havef>i.1 r-dV\l:2 C ''=  constant.Ijnxvivksitv or Duur.r e f e r e n c e s' Jeans, J . H ., D y n a m i c a l  T l i c o i y  of G a s e s ,  ^ to, ei s e q  (1925), * Born, M,* M e c h a n i c s  o f  t h e  A t o m ,  52 et s e g  { i g z y ) .3 Andrade, E . N. Da C,, Structure of the Atom, 19^ ( i93l) ^ DarwiHi C. H., Nature, 146, 956 (1940).

A Note on the Principle of Adiabatic Invariance

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A Note on the Principle of Adiabatic Invariance

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