1 PHYS 705: Classcal Mechancs Calculus of Varatons II
2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary value F F y ( x), y '( x); x 1,2,, N y x 1 ( ) N B I F y( x), y'( x); x dx We need to consder N varatons n N dmensons. y (, x) y (0, x) ( x) 1,2,, N A y x 1 ( ) N Agan, we have, as a parameter x C x 2 ( ) are -smooth varaton at ( x ) ( x ) 0 A A
3 Varatonal Calculus: Generalzaton (no constrant) Everythng proceeds as before, and we get: x B di F d F ( xdx ) 0 d y dxy' xa ( x) If all the varatons are ndependent, ths equaton requres that each coeffcents n the ntegrant to vansh ndependently. Then, we have, y (note: sum over comes from chan rule n takng the dervatve: ) df ( y, y '; x) d F d F 0 1,2,, N y dxy' Ths s the same result (Euler-Lagrange Eq) as we have before. However
4 Varatonal Calculus: Wth Constrant (2D) When we have constrants, the coordnates and hence the varatons wll NOT be ndependent! e.g., the system s constrant to move on a lower dmensonal surface (or ( x) curve) and allowed varatons are coupled n order to satsfy the constrant. To extend our analyss to systems wth constrants, let frst consder the 2D case wth ( y, y ) y ( x) as our dependent varables wth one constrant. To specfy the Holonomc constrant, we have g( y1, y2; x) 0 lnkng the two dependent varables.
5 Varatonal Calculus: Wth Constrant (2D) Wth two dep varables, our varatonal equaton can be wrtten as, x B di F d F F d F 1( x) 2( x) dx0 (*) d y1 dx y1' y2 dx y2' x A Smlarly wth, y1(, x) y1(0, x) 1( x) y2(, x) y2(0, x) 2( x)
6 Varatonal Calculus: Wth Constrant (2D) The constrant lnks so that these two varatons are NOT ndependent! ( x) to ( x) di d 0 Thus, we can t say that (Eq. *) requres the ndvdual coeffcents wthn the ntegrant to go to zero ndependently as before. x B di F d F F d F 1( x) 2( x) dx0 (*) d y1 dxy1 ' y2 dxy2' x A
7 Varatonal Calculus: Wth Constrant (2D) The constrant lnks so that these two varatons are NOT ndependent! ( x) to ( x) di d 0 Thus, we can t say that (Eq. *) requres the ndvdual coeffcents wthn the ntegrant to go to zero ndependently as before. To get around ths, we consder the total varaton of g (total dervatve) wrt : dg g y g y d y d y d 0 (**) NOTE: Ths expresson needs to be dentcally zero snce the NET varaton must vansh and be consstent wth the constrant (everythng must stay on the surface defned by g( y, y ; x) 0.
8 Varatonal Calculus: Wth Constrant (2D) From the defnton of the varatons, we have y1 y2 1( x) and 2( x) Puttng these nto Eq. (**), dg g g d y y 1 2 0 Rearrangng terms, ths then gves, g y 1 2 1 g y2 (Note: the varatons are lnked through the equaton of constrant g.)
9 Varatonal Calculus: Wth Constrant (2D) 2 Substtutng nto Eq. (*), we have, di F d F F d F d y dx y y dx y g y A xb g y 1 1dx 1 1' 2 2' x 2 0 1 Snce s an ndependent varaton, we can now argue that the coeffcents { } n front of t must vansh. F d F F d F g y y1 dxy1' y2 dxy2' g y2 1 0
10 Varatonal Calculus: Wth Constrant (2D) Rewrtng the expresson wth separately on the LHS and RHS, we have the followng equaton: y ( x)and y ( x) Note that these two terms are dentcal wth y y F d F 1 F d F 1 y1 dxy1' g y1 y2 dxy2' g y2 ( x) And, snce these two terms EQUAL to each other and depend on ndependently they can only be a functon of x ONLY! y1( x)and y2( x) And, we wll call ths as yet un-determned functon (multpler): ( x)
11 Varatonal Calculus: Wth Constrant (2D) Wth ths multpler, we can rewrte the LHS & RHS as two decoupled equatons: ( x) F d F g ( x) 0 1,2 (1) y dxy' y g( y, y ; x) 0 Together wth, these 3 equatons can be solved for the three unknowns: y ( x), y ( x), and ( x) ( x) s call the Lagrange undetermned multpler
12 Varatonal Calculus: Geometrc Vew Pont In our prevous 2D problem, the ntegral B I[ y Fy ( x), y '( x); xdx A s a functonal whch takes n two functons and gves back out a real number and the frst two terms on the LHS of EL equatons I F d F, 1,2 y y dxy' y ( x) and y ( x) s n fact a functonal dervatve of I [y wth respect to the functon y. Smlar to regular dervatves n vector calculus, t gves the rate of change of I wth respect to a change n the functon y. F d F g ( x) 0 y dx y' y
13 Varatonal Calculus: Geometrc Vew Pont So, one can thnk of the followng combned vector object as a functonal gradent smlar to a gradent vector n multvarable calculus. Ths gradent wll pont along the drecton of the largest ncrease of I n the space of functons correspondng to y ( x) and y ( x) and normal to the level curves of I. Smlarly, we can make a same assocaton wth the functonal dervatve of g wth respect to I I I[ y, y1 y2 y1( x) and y2( x) g g g g gy [,, y y y y (Prevously, we ddn t make ths dstncton but g s n fact a functonal.)
14 Varatonal Calculus: Geometrc Vew Pont In the space of, g y, y 0 y ( x), y ( x) [ I y gy [, y 0 - defnes a curve n the space y ( x), y ( x) y 2 [ g y g[ y ponts n the drecton normal to the constrant curve defned by gy [ 1, y2 0 y 1
15 Varatonal Calculus: Geometrc Vew Pont In the space of, y ( x), y ( x) [ I y 1, 2 I y y c defne sets of level curves n the space of y ( x), y ( x) y 2 [ I y ponts n the drecton normal to the level curves defned by 1, 2 I y y c y 1
16 Varatonal Calculus: Geometrc Vew Pont Then, the Euler-Lagrange Equaton has a specfc geometrc meanng: g y, y 0 * I[ y F d F g ( x) y dxy' y y 2 * g[ y I[ y ( x) g[ y The soluton at are chosen such that y, y * * y 1 * I[ y * & g[ y are collnear!
17 Varatonal Calculus: Geometrc Vew Pont g y, y 0 * I[ y * g[ y Recall that y, * * - The soluton at to the E-L s also a statonary pont to. I y y 1, 2 - AND, now we are also requrng that y mn, max, or nflecton pt I y, y y 2 the soluton to be consstent wth the gy [, y 0 constrant. y 1 It s a problem n constraned optmzaton.
18 Varatonal Calculus: Geometrc Vew Pont y 2 g y, y 0 y 1 * I[ y * g[ y As we vary along the constrant gy [, y 0 curve (red), we wll typcally traverse dfferent level curves of I (blue curves). Only at the locaton where the two curves touch tangentally, the I value wll not change as we move along the constraned soluton. Ths geometrc condton can be compactly wrtten as: I[ y g[ y y * *, y
19 Varatonal Calculus: Generalzaton If the system has more than one constrant, let say M: g ( y; x) 0 1,2,, N k 1,2, M k (# dep vars) (# constrants) Each of these M equatons descrbes a (N-1) dmensonal surface and the ntersecton of them wll be a (N-M) dmensonal surface on whch the dynamcs must lve. There wll also be M normal (vectors) to each of these M (N-1)-dmensonal surfaces gven by: g [ y k 1,2,, M k
20 Varatonal Calculus: Generalzaton g [ y Each of these gradents wll pont n a drecton AWAY from k (normal to) ts prospectve constrant surface. g k ( y ; x) 0 0 g y k g [ y k I[ y In a smlar geometrc fashon, ponts n the drecton away from (normal I[ y to) the level set of and n the drecton of ts greatest rate of change. I[ y I[ y c c 1 2 level set of I[ y [ I y
21 Varatonal Calculus: Generalzaton I[ y The geometrc condton for to have a statonary value wth respect to the constrants wll be smlar to our 2D prevous case,.e., the optmal set of s the one wth, Or, equvalently y,, 1 yn I[ y span g [ y I[ y k Ths means that, can be wrtten as a lnear combnaton of the M normal vectors from the M constrant equatons. I[ y g [ y 0 k k (ponts normal to ALL the constran surfaces.) M I[ y g [ y k k k 1 Instead of one, we have M ndependent Lagrange Multplers
22 Varatonal Calculus: Generalzaton Puttng ths back nto the orgnal dfferental equaton format, we can wrte ths condton as, M F d F gk k ( x) 0 1, 2,, N y dxy' k 1 y g ( ; ) 0 k 1, 2,, M k y x (for statonarty) (to be on constrants) y ( x ) k ( x) Here, we have N+M unknowns: and And, we have N+M equatons: top (N) and bottom (M) Ths problem n general can be solved!
23 Hamlton s Prncple Now, back to mechancs and we wll defne the Acton as, where L = T- U s the Lagrangan of the system I 2 1 L q, q, t dt Confguraton Space: The space of the q s. Each pont gves the full confguraton of the system. The moton of the system s a path through ths confguraton space. (Ths s not necessary the real space.) (Note: Here we don t requre all the generalzed coordnates to be necessarly q ' s ndependent. The can be lnked through constrants.) q Hamlton s Prncple: The moton of a system from t 1 to t 2 the acton evaluated along the actual path s statonary. s such that
24 Lagrange Equaton of Moton So, we apply our varatonal calculus results to the acton ntegral. We also further assume that the system s monogenc,.e., all forces except forces of constrant are dervable from a potental functon whch can be a functon of q, q,and t The resultant equaton s the Lagrange Equaton of Moton wth N generalzed coordnates (not necessary proper) and M Holonomc constrants: U q, q, t M L d L gk k () t 0 1, 2,, N q dtq k 1 q k 1, 2,, M gk( q; t) 0
25 Forces of Constrant Comments: 1. The Lagrange EOM can be formally wrtten as: M L d L gk k() t Q 1,2,, N q dtq k 1 q where the Q are the generalzed forces whch gve the magntudes of the forces needed to produce the ndvdual constrants. q - the generalzed coordnates are NOT necessary ndependent and they are lnked through constrants. - snce the choce of the sgn for k s arbtrary, the drecton of the forces of constrant forces cannot be determned.
26 Proper Generalzed Coordnates Comments: 2. If one chooses a set of proper (ndependent) generalzed coordnates n whch the (N-M) are no longer lnked through the constrants, then the Lagrange EOM reduces to: q j ' s L d L 0 j 1,2,, N qj dt q j M - In practce, one typcally wll explctly use the constrant equatons to reduce the number of varables to the (N-M) proper set of generalzed coordnates. - However, one CAN T solve for the forces of constrant here