Integrals and Invariants of

Transcription

1 Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore 1

2 Outlne of the lecture Frst ntegrals of Euler Lagrange equatons Noether s ntegral Parametrc form of E L equatons Invarance of E L equatons What we wll learn: How to smplfy the E L equatons to easy to solve dfferental equatons n some cases How to take advantage of parametrc forms and change of varables G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton

3 More than formulatng equatons So far, we have learnt how to get dfferental equatons and boundary condtons usng the technques of calculus of varatons. Indeed t s powerful. We have learnt varous generalzatons: Multple dervatves Multple functons Two and three ndependent varables Equalty and nequalty constrants Varable end condtons Broken extremals and corner condtons There are a few concepts that become useful when we also want to solve them usng analytcal (rather than numercal) technques. We wll stll not get a soluton rght away but we get a smpler or easly solvable form of dfferental equatons. In some cases, we get some nsght nto the problem. Ths s the am of the content of ths lecture. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 3

4 Consder the brachstochrone problem From Slde 14 n Lecture 11 A L 1 Mnmze T dx y(x) ( ) 0 g(h y) y H g y(x) ) L B And we have Drchlet (essental) boundary condtons at both the ends. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 4

5 Looks formdable to solve At frst sght, ths dfferental equaton looks to be too complcated to solve analytcally And we are far from showng that the soluton of ths s a cyclod. Frst ntegrals of Euler Lagrange equatons provdes a way out of ths. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 5

6 Frst ntegrals of specal forms Solvng dfferental dff equatons means that we are ntegratng them. Ths s what we do whether we do t analytcally or numercally. So, frst ntegrals mply that we are ntegratng the dfferental equaton to some extent. For Euler Lagrange g equatons, some specal forms, are amenable for wrtng the frst ntegrals and thereby reduce ther degree and hence ther complexty. J x 1 x 1 x 1 F(x, y)dx J F(x, y )dx J F(y, y )dx J F(x, ( y, y )dx f (x, y) 1 y dx x 1 x 1 G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 6

7 Integrand of the form F(x, y) Mn y(x) J F y 0 x x 1 F(x, y)dx Euler Lagrange equaton has only one term, n ths case. f (x, y) 0 It s smply an algebrac equaton; not a dfferental equaton. So, there s nothng to ntegrate here. Notce also that t does not have a boundary condton too. Recall that the smplest boundary condton term nvolves y. See Slde 13 n Lecture 11 for an example. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 7

8 Integrand of the form F(x, y') Mn J y(x) d dx x 1 F(x, (, y )dx F Euler Lagrange equaton has only one term, n ths y 0 F C constant y y f (x,c) Euler Lagrange equaton has only one term, n ths case too. We can express y n ths form and now t can be drectly ntegrated ether analytcally (when t s possble to do) or numercally. 8

9 Integrand of the form F(y, Mn y(x) J x 1 F(y, y )dx F y d F dx y 0 F y y F y y y F y F y d dx F yf y 0 y ) Euler Lagrange equaton has two terms. y dy F y y dy y 0 F y yy 0 Expanded. A smple contracton of the terms. F yf y C constant t An elegant frst ntegral. Multply by y through out. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 9

10 Brachstochrone problem has the form F(y, y ) L 1 Mnmze T dx y(x) 0 g(h y) y Now, nstead of that, we get ths. F yf y C constant G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 10

11 Smplfcaton of the Brachstochrone dfferental equaton A much smpler form to solve. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 11

12 An nsght wth the frst ntegral Consder the Hamltonan for the dynamcs of a sprng mass system: Sde 33 n Lecture 3 k m x F Mn x(t ) T 1 Mn H m dx 1 kx Fx dt x(t ) dt 0 T Mn H KE PEdt x(t ) Mn H x(t ) 0 T 0 L dt J t t 1 F(x, x)dt Ths s of the form and hence s amenable for the elegant frst ntegral. Hamlton s prncple for dynamcs. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 1

13 An nsght wth the frst ntegral: conservaton of energy T 1 Mn H m dx 1 kx Fx dt x(t ) dt 0 F y F y C constant F - xf x C 1 mx 1 kx Fx x mx C 1 mx 1 kx Fx c KE PE constant Thus, the frst ntegral gave rse to the prncple of conservaton of energy. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 13

14 An ntegrand of the form f (x, y) 1 y Mn y(x) J x x 1 f (x, y) 1 y dx F f (x, y) 1 y F y F y 0 f y f y 1 y 1 y f x fy 1 y y 0 1 y f y y 1 y f f y (1 y ) f x y f y y fy 1 y 0 y 1 y 0 3/ f y f x y f y y fy 0 Not ntegrated, but s a smpler form to deal wth. 1 y G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 14

15 Now, try to solve ths functonal Mn y(x) J x x 1 y y dx It s of the form: F(y, y ) Therefore, F yf y C constant Thus, y y y y y y y C y y yy C Let us try change of varables: y y x u cosv y usnv No sght of soluton yet! (despte usng the frst ntegral) G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 15

16 Change of varables Mn y(x) J Mn v(u) F(x, y, y )dx x x(u,v) dx x u x v du x 1 y y(u,v) dy y u y v dv J dx x u x v dv du F(x(u,v), y(u,v), y u y v v x u x v v )( x u x v v )du du x u x v vdu dv dy y u y v du du y u y v vdu x 1 Now, ths s a new functonal n u and v where we need fnd v(u). What would be the Euler Lagrange g equatons for ths? G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 16

17 New functonal satsfes the old equaton! Mn v(u) J F(x(u,v), y(u,v), y u y v v x x v )( x u x v v )du u v x 1 Mn v(u) J u u 1 F 1 (u,v, v )du F 1 v d F 1 s satsfed by v(u) F du v 0 just as y(x) satsfes y d F dx yv 0 So, we need to get the new functonal n the form shown above, when we change varables G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 17

18 An example Wth x u v x u x v y u y v y tan 1 (v / u) ) u u v v x u vv u x v v u v y uv v u y v v u v And notng that Mn v(u) u v v Mn J y(x) u v x 1 u u v J F(x(u,v), y(u,v), y y u vv x u x v v )( x u x v v )du x 1 Check the algebra by workng t out n detal. y y dx Mn v(u) ( ) becomes J u u 1 1 v du G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 18

19 Example (contd.) Mn J v(u) u u 1 v du F 1 du u 1 u 1 d F 1 F t Thus the soluton of du v 0 1 v constant c v C 1 v C 1u C Thus, Wth x u v y tan 1 (v / u) ) Thus, the soluton of the dfferental equaton n slde 15 s y y yy C y y u x cos y or v xsn y xsn y C 1 x cos y C G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 19

20 A note about change of varables Change of varables s a great way to solve an otherwse dffcult problem to solve. But nobody can tell us whch change of varables wll work for a gven problem. You just have to guess. But note that calculus of varatons lets you use change of varables. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 0

21 Parametrc form and Euler Lagrange equatons Mn y(x) ) J x x(t) x 1 Parametrc form y y(t) Where F(x, y, y )dx dx x dt y' x y y F x(t), y(t), x ( x y x d xx dt x Then, we have Mn y(x) Mn x(t ),y(t ) J t F(x, y, y )dx F x(t), y(t), x y x 1 t J (x, y, x, y) t 1 and t satsfes the followng EL equatons. 0 and y d y dt t 1 should not depend on t explctly. y 0 G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 1 dt y xdt

22 A comment We saw that change of varables or parametrc form do not alter the form of Euler Lagrange equatons. It s very useful n a number of stuatons. Parametrc form s especally useful when y(x) s to denote a closed curve. It s also useful n dealng wth dynamcs problems too. There s a more general theorem related to nvarance of Euler Lagrange theorem. It s called Noether s theorem. Noether s theorem s related to the frst ntegrals we dscussed earler n ths lecture. It leads to conserved quanttes. Proved by German mathematcan Emmy Noether, ths theorem was prased by Ensten for ts penetratng thnkng. It s used wdely n mathematcal physcs. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton Noether s theorem next

23 Invarance under transformatons Consder ˆx (x, y, y ) ŷ (x, y, y ) Mn J y(x) F x, dy dx Mn Ĵ F ˆx, ŷ, dŷ, y, dˆx dx y, ŷ(x) dˆx x 1 ˆ ˆx 1 If J Ĵ, we say that the functonal s nvarant under the transformaton shown above. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 3

24 Noether s theorem Consder ˆx (x, y, y, ) ŷ (x, y, y, ) A one parameter transformaton. dy Ĵ ˆx, ŷ, dŷ If J F x, y, dˆx dx dx J F y, dˆx dx x 1 we say that the functonal s nvarant under the transformaton shown above. Then, ˆ ˆx 1 Fy F yf y 0 0 constant G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 4

25 Noether s theorem (case of many functons) If Consder xˆ ( x, y, y,, y, y, y,, y, ) 1 n 1 yˆ ( x, y, y,, y, y, y,, y, ), 1,,,n 1 n 1 n x dy ˆ dyˆ J F x, y, dx J F xˆ, yˆ, dxˆ dx dxˆ x x ˆ 1 1 we say that the functonal s nvarant under the transformaton shown above. Then, x ˆ n n Fy F yf y constant G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 5

26 An applcaton of Noether s theorem Consder a system of n partcles wth poston coordnates: x (t), y (t), z ( t) ( 1,,, n) n 1 KE m x y z 1 Let the potental energy be = PE U ( x 1, y 1, z 1,, xn, yn, zn ) The knetc energy of such a system = Consder the Hamltonan = Consder x x cos y * * sn y y x y z * z sn cos t 1 H KE PE dt t 0 A one parameter famly of transformatons for the rotaton of the system of partcles about the z axs. Suppose that H s nvarant under the above transformaton. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 6

27 Compare wth the generc transformaton. xˆ ( x, y, y, ) yˆ ( x, y, y, ) (t,,y,,,y,, ) x (t, x,y, z, x,y, z, ) y (t, x,y, z, x,y, z, ) z (t, x,y, z, x,y, z, ) * t x z x z * 1 * * 3 t t No transformaton n the ndependent varable. x x cos y * y x sn y z * * z sn cos Now, as per Noether s theorem, we have n Fy F y constant Fy n KE KE KE 1 3 constant 1 x 0 y 0 z 0 0 (contd.) G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 7 Note that t 0

28 Noether s theorem applcaton (contd.) Note that x x * y * 0 0 y x and 0 0 * z 0 0 n (contd.) KE KE KE 3 n constant 1 1 x 0 y 0 z 0 mxy myx constant p r 1 1 where p mx,my,m z ad n r x, y, z Lnear momentum vector n Poston vector constant Conservaton of angular momentum! G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 8

29 Why s Noether s theorem mportant? Because t lets us fnd conserved quanttes for any calculus of varatons problems leadng to frst ntegrals. It can be extended to multple functons. It can be extended to multple dervatves. In mechancs, conservaton of energy, conservaton of lnear momentum, and conservaton of angular momentum, etc., follow from Noether s theorem. The prevous example llustrated the for conservaton of angular momentum. G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 9

30 The end note Frst ntegrals for varous forms of functonals Frst nteg grals and nvaranc ce of Eule er Lagrange equaton ns Ways to smplfy Euler Lagrange equatons and thereby solve them analytcally. Change of varables does not alter the form of Euler Lagrange equatons. Parametrc form too does not alter the form of the El equatons. Invarant transformatons and conserved quanttes usng Noether s theorem Thanks G. K. Ananthasuresh, IISc NPTEL course: Varatonal Methods and Structural Optmzaton 30