6. Hamilton s Equations

Transcription

1 6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant of tme s comletely secfed by a set of n varables we'll call theq s. The n -dmensonal q sace s (naturally) called confguraton sace. It s lke a freeze frame, a snashot of the system at a gven nstant. Subsequent tme evoluton from that state s unquely determned f we're also gven the ntal veloctes. The set of q 's and 's together defne the state of the system, meanng both ts confguraton and how fast t s changng, therefore fully determnng ts future (and ast) as well as ts resent. The 2n - q, s the state sace. dmensonal sace sanned by The system s tme evoluton s along a ath n confguraton sace arameterzed by the tme t. That, of course, fxes the corresondng ath n state sace, snce dfferentatng the functons q ( t) along that ath determnes the ( t). 1-D Smle Harmonc Oscllator Path n Confguraton Sace Trval one-dmensonal examles of these saces are rovded by the one-dmensonal smle harmonc oscllator, where confguraton sace s just the x axs, say, the state sace s the ( xx, ) lane, the system s tme ath n the state sace s an ellse. For a stone fallng vertcally down, the confguraton sace s agan a lne, the ath n the ( xx, ) state sace s arabolc, x x. Path n State Sace Exercse: sketch the aths n state sace for motons of a endulum, meanng a mass at the end of a lght rod, the other end fxed, but free to rotate n one vertcal lane. Sketch the aths n ( θθ, ) coordnates. In rncle, the system s ath through confguraton sace can always be comuted usng Newton s laws of moton, but n ractce the math may be ntractable. As we ve shown above, the elegant alternatve created by Lagrange and Hamlton s to ntegrate the Lagrangan (,, ) = (, ) (, ) L q q t T q q V q t

2 along dfferent aths n confguraton sace from a gven ntal state to a gven fnal state n a gven tme: as Hamlton roved, the actual ath followed by the hyscal system between the two states n the gven tme s the one for whch ths ntegral, called the acton, s mnmzed. Ths mnmzaton, usng the standard calculus of varatons method, generates the Lagrange equatons of moton n q,, and so determnes the ath. Notce that secfyng both the ntal q s and the fnal q s fxes 2n varables. That s all the degrees of freedom there are, so the moton s comletely determned, just as t would be f we d secfed nstead the ntal q s and s. Phase Sace Newton wrote hs equaton of moton not as force equals mass tmes acceleraton, but as force equals rate of change of momentum. Momentum, mass tmes velocty, s the natural "quantty of moton" assocated wth a tme-varyng dynamcal arameter. It s some measure of how mortant that coordnate's moton s to the future dynamcal develoment of the system. Hamlton recast Lagrange's equatons of moton n these more natural varables ( q, ) momenta, nstead of ( q, ). The q 's and 's are called hase sace coordnates., ostons and So hase sace s the same dentcal underlyng sace as state sace, just wth a dfferent set of coordnates. Any artcular state of the system can be comletely secfed ether by gvng all the, q,. varables ( q ) or by gvng the values of all the Gong From State Sace to Phase Sace Now, the momenta are the dervatves of the Lagrangan wth resect to the veloctes, = L( q, ) /. So, how do we get from a functon L( q, ) of ostons and veloctes to a functon of ostons and the dervatves of that functon L wth resect to the veloctes? How It's Done n Thermodynamcs To see how, we'll brefly revew a very smlar stuaton n thermodynamcs: recall the exresson that naturally arses for ncremental energy, say for the gas n a heat engne, s de S, V = TdS PdV, where S s the entroy and T = E / S s the temerature. But S s not a handy varable n real lfe -- temeraturet s a lot easer to measure! We need an energy-lke functon whose ncremental change s some functon of dt, dv rather than ds, dv. The early thermodynamcsts solved ths roblem by ntroducng the concet of the free energy, F = E TS

3 so that df = SdT PdV. Ths change of functon (and varable) was mortant: the free energy turns out to be more ractcally relevant than the total energy, t's what's avalable to do work. So we've transformed from a functon E( S ) to a functon F( T) F( E / S) are assve observers here). Math Note: the Legendre Transform = (gnorng PV,, whch The change of varables descrbed above s a standard mathematcal routne known as the Legendre transform. Here s the essence of t, for a functon of one varable. Suose we have a functon f ( x ) that s convex, whch s math talk for t always curves uwards, meanng 2 2 d f x / dx s ostve. Therefore ts sloe, we ll call t y = df ( x) / dx, s a monotoncally ncreasng functon of x. For some hyscs (and math) roblems, ths sloe y, rather than the varable x, s the nterestng arameter. To shft the focus to y, Legendre ntroduced a new functon, g( y ), defned by g ( y) = xy f ( x). The functon g( y ) s called the Legendre transform of the functon f ( x ). To see how they relate, we take ncrements: dg ( y) = ydx + xdy df ( x) = ydx + xdy ydx = xdy, (Lookng at the dagram, an ncrement dx gves a related ncrement dy, as the sloe ncreases on movng u the curve.) From ths equaton, x = dg ( y) / dy. Comarng ths wth y = df ( x) / dx, t s clear that a second alcaton of the Legendre transformaton would get you back to the orgnal f ( x ). So no nformaton s lost n the Legendre transformaton -- g( y ) n a sense contans f ( x ), and vce versa.

4 Hamlton's Use of the Legendre Transform We have the Lagrangan L q,, and Hamlton's nsght that these are not the best varables, we need to relace the Lagrangan wth a closely related functon (lke gong from the energy to the free energy), that s a functon of the q (that's not gong to change) and, nstead of the 's, the 's, wth = L q, /. Ths s exactly a Legendre transform lke the one from f g dscussed above. The new functon s from whch n H( q, ) = q L( q, q ), = 1 dh, q = dq + d, analogous to df = SdT PdV Ths new functon s of course the Hamltonan. Checkng that We Can Elmnate the 's We should check that we can n fact wrte as a functon of just the varables (, ) The answer s yes. Recall the n H, q q L q, q = = 1 q, wth all trace of the s elmnated. Is ths always ossble? s only aear n the Lagrangan n the knetc energy term, whch has the general form where the coeffcents, j T = a q qq j k j aj deend n general on some of the the s. Therefore, from the defnton of the generalzed momenta, k and we can wrte ths as a vector-matrx equaton, n L = = a ( q ), j k j q j= 1 = A. q k s, but are ndeendent of the veloctes,

5 That s, functon of the s a lnear functon of the j s. Hence, the nverse matrx 1 A wll gve us as a lnear j 's, and then uttng ths exresson for the nto the Lagrangan gves the Hamltonan as a functon only of the q 's and the 's, that s, the hase sace varables. The matrx A s always nvertble because the knetc energy s ostve defnte (as s obvous from ts Cartesan reresentaton) and a symmetrc ostve defnte matrx has only ostve egenvalues, and therefore s nvertble. Hamlton s Equatons Havng fnally establshed that we can wrte, for an ncremental change along the dynamcal ath of the system n hase sace, (, ) dh q = dq + d we have mmedately the so-called canoncal form of Hamlton s equatons of moton: H H q = q, =. Evdently gong from state sace to hase sace has relaced the second order Euler-Lagrange equatons wth ths equvalent set of ars of frst order equatons. A Smle Examle 1 2 For a artcle movng n a otental n one dmenson, L( q ) = m V ( q) Hence Therefore L = = m, =. m, H = q L = q 2 mq + V q 2 = + V q 2m (Of course, ths s just the total energy, as we exect.) The Hamltonan equatons of moton are.

6 H = = m H = = V ( q). q So, as we ve sad, the second order Lagrangan equaton of moton s relaced by two frst order Hamltonan equatons. Of course, they amount to the same thng (as they must!): dfferentatng the frst equaton and substtutng n the second gves mmedately V ( q) = m, that s, F = ma, the orgnal Newtonan equaton (whch we derved earler from the Lagrange equatons).