INTERNAL REPORT(Limited Distribution)

/ / International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE SECOND VARIATION AND JACOBI EQUATIONSFOR SECOND-ORDER LAGRANGIANS

Biagio CasciaroDipartimento di Matematica, Universita di Bari,

via Orabona 5, 75125 Bari, Italy

Mauro Francaviglia1

Istituto di Fisica Matematica "J.-Louis Lagrange", Universita di Torino,via Carlo Alberto 10, 10123 Torino, Italy

and

Victor Tapia2

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

March 1995

'E-mail: FRANCAVIGLIA@DM.UNITO.IT2Permanent address: Departamento de Fisica, Universidad de Concepcion, Casilla 4009,

Conception, Chile. E-mail: VTAPIA@HALCON.DPI.UDEC.CL

T

ABSTRACT

We discuss the Jacobi equation ensuing from the second variation of second-order

actions which contain accelerations. The same is done for field theory.

1. Introduction

As is well known, the second variation of an action functional governs the behaviour of the action itselfin the neighborhood of critical sections and the Hessian of the Lagrangian defines a quadratic form whichallows to distinguish between minima, maxima and degenerate critical sections [1]. It is well known that inthe case of geodesies of a Riemannian manifold those fields which govern the transition from geodesies togeodesies (i.e., those vector fields which make the second variation to vanish identically modulo boundaryterms) are called Jacobi fields [2] and they are solutions of a second-order differential equation known asJacobi equation (of geodesies). The notion of Jacobi equation as an outcome of the second variation is infact fairly more general and formulae for the second variation and generalised Jacobi equations along criticalsections have been already considered in the Calculus of Variations from a "structural viewpoint" (see, e.g.,[3],[4]). Strangely enough, however, in most of the current literature the second variation of functionals hasbeen usually treated without resorting to general expressions; moreover, integration by parts to reduce it tomore suitable forms are generally performed by ad hoc procedures, in spite of the fact that fairly generalsuch formulae can be easily worked out. Because of this, we believe that it is important and useful to revisitthe theory of second variations (for the action functionals denned by first- and second-order Lagrangians),especially in view of a number of applications to relativistic field theories which will form the subject offorthcoming investigations [5],[6].

In a previous paper [7], in the general and global framework of first-order variational principles infibered manifolds and their jet-prolongations, the notion of generalised Jacobi equation was developed and

* \ ii mpiiniBiii f i l l ! i ( i " ' '

discussed for first-order Lagrangians. As is well known, the classical Jacobi equation for geodesies of aRiemannian manifold defines in fact the Riemann curvature tensor of g. Because of this we can say that thesecond variation S2S and the generalised Jacobi equations define the "curvature" of any given (first-order)variational principle; in the generic case, of course, this notion has very little to say; in a continuation of thatwork [8] it was shown however that this general concept of "curvature" takes a particularly significant formin the case of generalised harmonic Lagrangians, giving rise to suitable "curvature tensors" which satisfysuitable "generalised Bianchi identities". Applications to second variations of relativistic Lagrangians (i.e.,Lagrangians depending on the full curvature of a Riemannian metric), are being considered in [5] and [6].Generalised Jacobi equations for first-order Lagrangians were also considered in [9], where it was also shownhow to recast the original Euler-Lagrange equation together with the corresponding Jacobi equation as asingle variational equation generated by the first-order deformed Lagrangian; this result finds importantapplications to Riemannian Geometry [10] and extends to the higher-order case [11].

In this paper we shall consider the generalised Jacobi equations for second^rder Lagrangians. Thepaper is organised as follows. Section 2 is devoted to recall the first variation and calculate the secondvariation 82S of an action 5, defined by a Lagrangian depending at most on second-order derivatives of thefields. We then show how a number of different integrations by parts allow to recast the Hessian in moresuitable forms, which contain the Euler-Lagrange equations and define some ordinary differential equationsof the second-order which we call the generalised Jacobi equations. These are in fact the equations whichdefine along critical curves those vectorfields in the configuration space which make S2S to vanish identically(modulo boundary conditions). A description of these equations as the Euler-Lagrange equations of thefirst variation Lagrangian SL have been considered in [9]. Section 3 is devoted to field theory. Along similarlines, we evaluate the second variation S2S and the generalised Jacobi equations. In this case the generalisedJacobi equations are second-order partial differential equations. Section 4 is devoted to some applicationsand examples.

2. Classical Mechanics

In this section we shall recall some well-known fundamental results for first-order mechanics, includingthe Jacobi equation as newly developed in [7]. Then we extend these results to second-order mechanics.For the sake of simplicity we shall use the language of fundamental calculus in Rn. All the results we shallestablish can however be easily rewritten in the intrinsic language of fiber bundles, nowadays standard forCalculus of Variations.

2.1. First-order Mechanics. The Euler-Lagrange Equation

Our starting point is the action for a generic first-order Lagrangian L

ft,S= L{t,q,q)dt. (2.1)

Jta

The problem we are interested in is to know when this action is an extreme at a given trajectory q* = q'0(t).For this one imposes that for small deformations of the trajectory q' ~ q'Q(t) the variation of the action iszero. Let us therefore consider an infinitesimal deformation

ff'=9o + f7'+o(«2) . (2.2a)where e is a smallnessparameter. For the velocities we have the corresponding relation

The deformed action, truncated at the first-order in e, is given by

(2.3)

T

where

^ = 9^ _ ± ( d£\ (2 4)

is the usual Euler-Lagrange (or variational) derivative.Setting as usual

S = SQ + eSS + ---, (2-5)

we obtain then

(2.6)

The action S is an extremal if SS = 0 for any deformation T), which certainly holds if the followingconditions hold

Conditions (2.7b) mean that the endpoints of each deformed trajectory are kept fixed. The aboveconditions just ensure extremality of the action, but they do not say anything about the stability of thesolution. As is well known, for this we must go to the next order of deformation.

2.2. First-Order Mechanics. The Jacobi Equation

In order to study the stability properties of the solutions of the Euler-Lagrange equation (2.7a) weshould consider the variation of the action under second-order deformations of the trajectory. For this letus now consider the infinitesimal deformation

together with

The truncated variation of the action is now given at second-order by

: [%<+%>] rI.Setting then

S = S0 + eSS+^2S2S+---, (2.10)

from (2.9) we inmediateiy identify the first and second variations of the action, namely

3

+ \ d t (2.Ua)

In order for 5 to be an extremal the first variation (2.11a) should be zero and this is ensured by conditions(2.7) above. In this case S2S reduces to

* * - % * ( 2 1 2 )

The sign of S2S is related to the stability properties of the solutions of the Euler-Lagrange equation(2.7a). If 52S > 0 for all deformations rf we have a minimum; if S2S < 0 for all rf we have a maximum; ifS2S ~ 0 for some rf we have a degenerate extremum. However, a closer examination of (2.12) reveals thatS2S has not a definite sign owing to the presence of the first term, which involves p' linearly and can thencetake arbitrarily large positive or negative values.

A definite sign can instead be obtained for the second term, which is a quadratic form. Therefore, werestrict ourselves to deformations in which, besides from keeping fixed the endpoints through the condition(2.7b), we also keep fixed the first derivative of their deformation, i.e. we impose the further condition

pi(t0)=pi{h) = 01 (2.13)

under which 62S reduces to

t ( 2 1 4 »

The integrand appearing in the above relation is a quadratic form called the Hessian. We shall denote it byHess and write

Hess = A i* , • ' ' • ' + 2 -T-JJTTJ rf if + -——- if if . (2.15)dq'dqi dq'dql dq'dq* K '

The stability properties of solutions of the Euler-Lagrange equations can now be investigated directly fromthe eigenvualues of Hess,

As mentioned above, the case in which S2S — 0 is particularly interesting. In this case one can infact obtain an equation governing the vectorfields which deform solutions of Euler-Lagrange equations intosolutions, generically called the "Jacobi equation". However, the quadratic form appearing in (2.14) can bechanged, without direct physical consequences on field equations, by adding total time-derivatives, possiblydepending on higher-order derivatives of 77, provided they are zero at the integration limits due to conditions(2.7b). Therefore, the Jacobi equation to which one arrives strongly depends on the terms possibly added.The simplest and most reasonable criteria would be to add total time-derivatives which do not introduceaccelerations r; or higher time-derivatives.

For this let us first rewrite equation (2.14) in the following form

.• , d2L .. ...

with the purpose of integrating by parts only the third term. The result (see [7]) is

4

' , (2-17)to

or equivalently

8 (8L\ , , . / d2L d*L \ t.4 . 02L . , . , d ( dH( 2 1 8 )

Let us observe that the last term of (2.18) is the only quadratic function of TJ'S which can be added to theHessian without introducing accelerations.

However, other terms can be added to the Hessian changing its order in 77's. An interesting one is thefollowing

d2L , d2L ., d ( d2L * 82L

which corresponds to integrating by parts also the fourth term in (2.16).In both cases, namely (2.18) and (2.19), the time derivatives cancel at the integration limits. Therefore,

different Jacobi equations would be obtained. However, the version ensuing from (2.19) is considered as thestandard Jacobi equation:

d2L i . d*L ., d { d _ , . . _ . , . _

T h i s f o r m will b e f u r t h e r e x p l o r e d in [9].

2.3. Second-Order Mechanics. The Euler-Lagrange Equation

Our starting point is now the action

5 = f L(t, q,q,q)dt. (2.21)

Once again, the natural question is when this action is an extreme. For this we know that a given trajectoryql = ql

a(t) is an extremum if for small deformations of the trajectory the variation of the action vanishes.Let us therefore consider the infinitesimal deformation

? ' = ? j ,+er/ '+o(e2) , (2.22o)

where £ is a smallness parameter. For the velocities and the accelerations we have the corresponding relations

(2.226)

(2.22c)

The variation of the action, at the first-order in t, is given by

,• dL ... dL5 = 5 0 + f / br-7,1

where

SL

5L

are the Euler-Lagrange derivatives.Comparison of (2.23) with (2.5) shows that

dLdqi

dL

d .dl{

d ,fdL\

+ d* 1+ dPlKdq'J_ dL

~ dq{dTtl

(SL[Sqi

n SL ,. SL 4 dL A t ]

The action S is an extremal if 6S — 0 for all -q's, which is ensured by

(2.24a)

(2.246)

(2.25)

!£ = 0, (2.26a)

r,i{to)=r,i(t1) = 0, (2.266)

rf(to) = ni(h) = 0. (2.26c)

The conditions above just concern the extremality of the action, but they do not say anything about thestability of the solutions. For this we must go to the next order of deformation. (Once again, the conditions(2.26a) and (2.26b) mean that the variation is performed keeping fixed the endpoints).

2.4. Second—Order Mechanics. The Jacobi Equation

In order to study the stability of the solutions of the Euler-Lagrange equation again we should con-sider the variation of the action under second-order deformations of the trajectory. We consider then theinfinitesimal deformation

together with

q = q0 + e

T h e t runca ted variat ion of the action is now given by

12~'1

2 '

(2.276)

(2.27c)

tQ

i dL 1( dL ,.

( 2 - 2 8 )

Comparison of (2.28) with (2.10) allows to identify the first and second variation by

f1' TdL ,- dL ... dL(2.29a)

dL

(2.296)

In order for S to be an extremal the first variation (2.29a) should be zero. This is implemented by theconditions (2.26) above. In this case 62S reduces to

/ •Jta

dq'

(2.30)

The sign of S2S is related to the stability properties of the solutions of the Euler-Lagrange equation. If6 S > 0 we have a minimum; if 62S < 0 we have a maximum; if S2S = 0 we might have a degenerate criticalpoint and higher-derivatives should be checked. As for the first-order case, in order that S2S has a definitesign, we are naturally led to impose the further boundary conditions

Therefore 62S reduces to

-I"Jta

The above relation defines a quadratic form Hess, given by

whose eigenvalues are related to the stability of the Euler-Lagrange equations.

7

(2.31a)(2.31a)

(2.32)

(2.33)

«. ,M if. t|l - IE- | J V*

As mentioned above, the case in which 62S - 0 is particularly interesting, since it generates the Jacobiequation which governs the vectorfields transforming solutions into nearby solutions.

As for first-order mechanics, the quadratic form appearing in (2.32) can be changed by adding totaltime-derivatives provided they are zero at the integration limits due to conditions (2.26b). Therefore, theJacobi equation to which one arrives strongly depends on the terms added. One criteria would be to addtotal time-derivatives which do not introduce variables other than (77,77,7). For this let us first rewriteequation (2.32) in the form

d2L

{2M)

According to the above prescription, the only terms which can be integrated by parts are the fourth,fifth, seventh and eighth ones. The possible outcomes are summarised in the following table

d fdL d (dl\\ i , ( 82L 32L \ ,- ., n d2L ,-..,Hess =WW{w))T}7f + [WW WW) r}rf+2 WWn *

d2L ,t , n 82L ., ... 82L .... ... d , _ _ , , ,

dH d (JPL\\ j ^ , (n d2L

W ^ + ^ + ( ^ ) (2'356)

d (dL i ^ . j isi

^ ^ ^ ^ ) 1 ( 2 ' 3 5 c )

d2L ,- * „ d2L ,- .- n 82L , ... f d2L d ( d2L

d

Apart from first order derivatives, we can add second-order derivatives without introducing extra vari-ables into the Hessian. The only two possibilities are the following

d2 ( 32L t A d? ( d2L

{ ) {d2L d2L

( n {dt2 {dq'dqi n *} ~ dt2 \dq*dqj) V * dt

d ( dL

dt {dfdi

A simpler possibility is to try to combine all the above integrations by parts in a single decompositiondisplaying better properties. The result similar to (2.19) is given by

d fSL\ , ,- ( d2L d2L

d2L ..( . d ( d SL ; , d2L

(2.37)

However, other terms can be added to the Hessian changing its functional dependence on derivatives ofTJ. A physically and mathematically relevant one is the following

- [ 9H _• 82L ^ss~ [Ww^ + WW*

d2 f d2L

d\( d2L • 82L . d2L ^ d ( d2L d

dt [ydqidqi * dqidqi * + dtfdqi * dt {dfdj * dtfdi *

82L ; d2L ., d2L ..A .,] . .f + f + f ) \ (2"38)

In all these cases the time derivatives cancel at the integration limits. Therefore different Jacobi equa-tions would be obtained. However, in analogy with first-order mechanics, version (2.32) will be consideredas the standard Jacobi equation, namely

d2L i

dq{dqi **d2L ., d2L ..

1 we? "d (dt \

I d 2 I' dt2 \

' d2L ,

' d2L .^dq'dqi ̂

' i dL n*

i , d2L .,dtfdq'i '

82L

d2Ldq'dqi

rj>\ - 0 . 2.39

3. Fifild Theory

Now we recall the fundamental results for first-order field theory, including the Jacobi equation asdeveloped in [7], with the purpose of extending these results to second-order field theory.

3.1. First-Order Field Theory. The Euler-Lagrange Equation

We now consider the generalisation of the above results to field theory. We consider the action

9

T IT —liiiiiiiniiimirrtiiiITn

= / C(x, <f>, d<t>) tPz (3.1)

and we assume that <f>A = <j>A{x) is a section along which S has an extremum. The conditions for <j>A(x) tobe an extremum is that for small deformations of the functions 4>A(x) the variation of the action vanishes.Let us therefore consider an infinitesimal deformation

4A=4A + enA + o(c7), (3.2a)

where c is a smallness parameter. For the derivatives of the fields we have the induced relation

The variation of the action, at the first order in e, is given by

where

IL = ̂ £-jL(J£-\ (34)S<j>A d<f>A dx" {d^^J v '

is the usual Euler-Lagrange derivative.Setting as usual

S=S0 + eSS+--- , (3.5)

we infer from (3.3) that

The action S has an extremal if 55 = 0 for all i)A, which can be classically satisfied by the followingequations and boundary conditions

= 0. (3.76)an

Condition (3.7b) means that the deformation is fixed at the boundary. As for mechanics, the conditionsabove just concern the extremality of the action. They do not say anything about the stability of thesolutions and to this purpose we must go to the next order of deformation.

3.2. First Order Field Theory. The Jacobi Equation

In order to study the stability properties of the solutions of the Euler-Lagrange equation we should con-sider the variation of the action under second-order deformations. For this we consider now the infinitesimaldeformation

10

along with its differentials

The variation of the action is given, at order e2, by

(3.86)

5 = So + c

+ iJAw'

Jn [&<t>A

S-^K*ac_.

A B

As before, if we set

we identify from (3.9) the first and second variations of the action; namely:

(3.9)

(3.10)

SS

S2S

•L-L dC

(3.11a)

IA 82C d2Cd<t>Ad<f>B d<t>Ad4>

(3.116)

In order for S to be an extremal the first variation (3.11a) should be zero. This is usually implemented bythe conditions (3.7) above, under which 62S reduces to

6>S= £Jan d<t>A

L

SA d<j>Ad<fi(3.12)

The sign of S2S is related to the stability properties of the solutions of the Euler-Lagrange equation.Once again, in order for S2S to have a definite sign we impose the further boundary condition

= 0.

Therefore S2S reduces to

S2S= fJn

d2£84>Ad<j>B

(3.13)

(3.14)

The above relation defines a quadratic form, i.e. the Hessian of £, whose eigenvalues are related to thestability of the Euler-Lagrange equations. As already discussed in [7] an equation called the Jacobi equationcan be obtained by a suitable integration by parts.

11

However, the quadratic form appearing in (3.14) can be changed by adding surface terms which, due toconditions (3.7b), are zero on the boundary. Therefore, the Jacobi equation to which one arrives stronglydepends on the terms we add at the boundary. One possibility is to add surface terms which do not changethe functional dependence on derivatives of TJ. AS for mechanics, eq. (3.14) is more conveniently rewrittenas follows

-L d2C A B d2C A B

*

(3.15)

with the purpose of integrating by parts only the third one, without introducing extra derivatives of TJ. Asalready shown in [7], the result is

if r)a dZ^ , (3.16)

or, equivalently

i ^ _ nA nB , ( 92C u-L. l > „P

r^n*)- (3.17)

Let us remark that the last term is the only quadratic function of TJ'S which can be added to the Hessianwithout changing its order in derivatives of -q.

However, other terms can be added to the Hessian changing the dependence on only [r\,df]). Aninteresting one is the following

d2£ B d f d £ B d*C ^

( 3 1 8 )

In both cases (3.17) and (3.18) the surface terms cancel at the boundary. Therefore different Jacobiequations would be obtained. However, the version obtained by setting the l.h.s. of (3.18) equal to zero,namely

d2C B d ( d2C B d2£ \ _v =0'

is considered as the standard Jacobi equation. This form will be further explored in [9].

3.3. Second—Order Field Theory; the Euler-Lagrange Equation

In this case the starting point is the action

12

T

s = (3.20)

The natural question again is when this action is an extremum. For this let us once more assume that4>A = 4>A{x) is the section over which S is an extremum. The conditions for 4>A(x) to be an extremum is thatfor small deformations of the section 4>A(x) the variation of the action be zero. Let us therefore consider theinfinitesimal deformation

4>A ~ 4>A + f y A + ° ( t 2 ) i (3.21a)

where £ is a smallness parameter. For the first-order and second-order derivatives we have similar relations

<!>*,> = {<t>A)li+ti)Ali + o{e2), (3.216)

<t>A»» = {4>A)til, + zVA^ + o{(2). (3.21c)

The variation of the action, at first-order in e, is given by

S = So + L d£d<f>A

SL

'nWASL

dnnA +

dL(3.22)

where

SL dL

SLd<j>A

dL

d2 dL

d<t>dL

are the Euler-Lagrange derivatives.Comparison of (3.22) with (3.5) shows that

(3.23a)

(3.236)

SS

= f T^J VA d"x+ I \Jn HA Ian L

1 ' d4>\

m J i ^^ A dL

an< * £ , , (3.24)

The action S has an extremum if SS = 0 for all IJ'S, which is classically ensured by requiring:

6<f>'

= 0,

= 0.

(3.25a)

(3.256)

(3.25c)

The conditions above concern the extremum of the action. They do not say anything about the stabilityof the solution and for this we must go to the next order of deformation. (Again, conditions (3.25b) and(3.25c) mean that the boundary is kept fixed).

13

i4

3.4. Second-Order Field Theory. The Jacobi Equation

In order to study the stability of the solutions to the Euler-Lagrange equations we should consider thevariation of the action under second-order deformations of the sections. For this let us now consider theinfinitesimal deformation

<f>A = 4>Q +e r]A + -e2pA +o(c3) , (3.26a)

together with

<t>Av, - (<£(?)„ + f *lAp. + 77 <? PAll + o ( f 3 ) , (3.266)

? V + \*2 pA + °(f3) (3-26c)VThe truncated variation of the action is given by

- 9 • f / ( 5C nA

Jn \HA

L

A 6C A M A

~A ~B j_ o " ^ nA B , ? " >~ ^AJB

8C , 8C

•J t. „

d2C d2 C

* B + 2

82C A B 82C A B d2£ A B

(3.27)

As before, by comparison with (3.10) we identify the the first and second variation by

(3.286)

In order for S to be an extremum, (3.28a) should be zero and this is ensured by the conditions (3.25)above. In this case S2S reduces to

d2£ d2C d2C 1

(3.29)

14

The sign of 62S is related to the stability properties of the Euler-Lagrange equations. If S2S > 0 wee a minimum; if S2S < 0 we have a maximum; if S2S — 0 we might have a degenerate critical point.have a minimum; if S2S < 0 we nave a maximum; it d*£> — U we might have a degenerate critical pOnce again, in order for S2S to have a definite sign we need to impose the further conditions

A

= o ,an

= 0,dil

(3.30a)

(3.306)

under which S2S reduces to

=L{ d<t>Ad<f> d<j>Ad4>B„„

• ( 3 3 1 )

The above relation defines essentially a quadratic form, the Hessian, whose eigenvalues are related to thestability of the Euler Lagrange equation.

Integration by parts allows then to obtain the Jacobi equation, as developed in [7].However, the quadratic form appearing in (3.31) can be changed by adding surface terms which, due to

conditions (3.25), are zero on the integration boundary. Therefore, the form of the Jacobi equation whichone arrives to strongly depends on the terms added. One criteria would be to add surface terms which donot change the dependence on [r),dr],d2r]). For this let us rewrite eq. (3.31) in the form

A B A B A B

A B A B A B

a2 f a2 p ffl CcTx. (3.32)

The only terms which can be integrated by parts are the fourth, fifth, seventh and eighth terms. Thefour possible outcomes are summarized in the following table

ss ~ d4>B

82C

d ( 3C \ \ A B f

F$- f

A B1 n

82CA B

82CA B

( 3-3 3 f l )

Hess = 84>Ad<f>B

d2C

d2C

A B

d<t>A

15

A B

V 1

A B d ( d2c A B \

„ d2£ A B , (n B2£ 8 d ( dC \ \ A B d2£ A B

d 2 c d 2 c \ A B d2c A B

d2C A B n d2LA B n d2L A B

d2c d ( d2c \ \ A B f d2c d2c \ A B

82C A B d ( d C A B \

Apart from first-order divergencies, we can also add second-order divergencies to the Hessian, alonglines similar to those discussed in Section 2. The only two possibilities which do not introduce derivativeshigher than fourth-order ones are:

._^C A *\ _ _*_ ( 82£ \ , n ^ A d (_&£ \ , -B dV + dx*u<p Q(p / dx^dx y u(p 0<p j fljr y 0(p c

„ d2£

1/

d2 ( d2C A B\ d2 f 82C \ A B d f d2C d2C \ A B

( d2C d2 C \ d2C

A simpler possibility is to combine all the above integration by parts in a single decomposition displayingbetter properties. The result similar to (2.36) is given by

Hess = JL. (-^\ VA r)B + ( fc

B B-^r) iA vBp + ( fC B - aj.fJlB 1 nAu vBv

d2£ d2£ \ A R d2 £ A n

A Bd ( a sc A B B2£ A B

However, other terms can be added to the Hessian changing the dependence on only {t},dr},d2r)). Aninteresting one is the following further expression:

- [ ^ BHess ~ [ n

16

d2£g d£ B

V + n

d2 ( 82£ B 82£

{ +

d d2£ B 82£

d2£ „ d2£ R d2£nB (3.36)

In all the above cases the surface terms cancel at the integration boundary. Therefore, different Jacobiequations would be obtained.

The equation ensuing from S2S = 0 and (3.36) above when the boundary conditions (3.13) and (3.7b)hold together with their derivatives and together with the field equations (3.25), namely the equation obtainedby setting the l.h.s. of (3.36) equal to zero

d2c B Bv n x + d<t>Aa<t>BB

Xf>B d2c B d2£ B

V + r T ) + nd<j>A

"B ^ U^BX VB» + d4>AUg<i>Bx »*>«,) = 0, (3.37)

will be called the "standard Jacobi equation" for £.

4. Examples and Applications

Here we consider two simple Lagrangians involving second-order derivatives and we evaluate their Jacobiequations.

4.1. The Riewe Lagrangian

The Riewe Lagrangian (see [12]) was introduced to describe a classically spinning particle. It is givenby

1 .9 1 m „ 7

where q is a three-dimensional vector. The equations of motion are given by

0- (4-2)

The solutions of the above equation of motion are given by

<}' = 4 + vo l + a< c o s ( w 0 + bi sin(w t). (4.3)

These solutions describe a helical motion along a ellipsoid of semiaxis a1 and b' oriented in the direction of

17

The corresponding Jacobi equation is given by

ff + \ tf' = 0 . (4.4)

4.2. The Shadwick Lagrangian

The Shadwick Lagrangian is given by:

where 0 is a scalar field (see [13]). The field equations are given by

— = 3 (<f>00 <j>lx - ^ j j j ) = 0 . (4.6)

The corresponding Jacobi equation is given by

5. Conclus ions

In a previous paper two of us discussed the various (generalized) Jacobi equations for firt-order fieldtheory. We have presented here a natural and direct generalisation of these results to the case of second-order Lagrangians, which are important not only for the sake of completeness but also and especilly in viewof applications to relativistic field theories (whereby gravitational Lagrangians depend effectively on second-order derivatives of a metric). Concrete applications to relativistic field theories will form the subject offorthcoming investigations.

Acknowledgements

One of us (V.T.) has been partially supported by Direccion de Investigacion, Universidad de Concepcion,under contract 94.11.08-1, by a Visiting Professorship of Italian GNFM-CNR and by a Visiting Professorshipof the International Centre for Theoretical Physics, Trieste, Italy. One of us (M.F.) acknowledges support ofGNFM-CNR and of MURST (Nat. Res. Proj. "Metodi Geometrici e Probabilistic in Fisica Matematica").One of us (B.C.) acknowledges support of GNSAGA-CNR and of MURST (Nat. Res. Proj. "Geometriadelle Varieta Differenziabili"). V.T. would like to thank Professor Abdus Salara, the International AtomicEnergy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

References

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