Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/7d3/154-lagrange-s-identity-and-its-generalizations_en.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:31:11 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:07:52 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 8

ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 2, pp. 71­80. c Pleiades Publishing, Ltd., 2008.

RESEARCH ARTICLES

Lagrange's Identity and Its Generalizations
V. V. Kozlov1*
1

V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received January 14, 2008; accepted February 7, 2008

Abstract--The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in co ordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation. MSC2000 numbers: 37A60, 82B30, 82CXX DOI: 10.1134/S1560354708020019 Key words: Lagrange's identity, quasi-homogeneous function, dilations, Vlasov's equation

1. INTRODUCTION Consider a system of n interacting particles with masses m1 , ... ,mn ; their radius-vectors from a certain fixed p oint of the Euclidean space E are r1 , ... ,rn (the dimension of this space is of little imp ortance). The forces are conservative and the p otential energy V is a homogeneous function of degree m of r1 , ... ,rn . The famous Lagrange identity reads ¨ I = 4T - 2mV . The dot designates differentiation with resp ect to time, I= mi (ri ,ri ) =
2 mi ri

(1)

is the moment of inertia of the particles ab out the origin of coordinates and T= is the kinetic energy of the system. Using the integral of energy T + V = h, equation (1) can b e rearranged as ¨ I = 4h - 2(m +2)V. (3) (2) 1 2 mi (i , ri ) r

The Lagrange identity has a few imp ortant corollaries. For example, in the case of gravitational ¨ interaction m = -1 and V < 0. Hence, if h 0, then I > 0 by virtue of (3). This immediately yields the well-known theorem due to Jacobi on instability of the system of gravitating b odies with non-negative total energy: some inter-particle distances either b ecome indefinitely small or grow to infinity.
*

E-mail: kozlov@pran.ru

71


72

KOZLOV

Another imp ortant corollary concerns the case where the energy level surface (2) is compact in the phase space of the system: the time averages of the kinetic and p otential energies exist and are equal to m 2 h, T= h. (4) V= m +2 m +2 Here


1 f = lim
0

f (t) dt. these relations is that the space, which is a compact is based on the analysis of the identity (1) is replaced (5)

The relations (4) were obtained by Clausius. An imp ortant feature of time averages of T and V coincide with the averages over the whole energy manifold. A statistical reformulation of the Clausius theorem [1] Gibbs ensembles. In the generic case when the forces F1 , ... ,Fn are not conservative with the following virial relation: ¨ I = 4T +2 If the forces are conservative (Fi ,ri ).

V ri and V is a homogeneous function of degree m, then, according to the Euler formula, Fi = - V ,ri ri = mV .

In this case equation (5) follows from (1). Our ob jective is to obtain natural generalizations of the Lagrange identity to the case of · constrained systems, · quasi-homogeneous p otential energy, · continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. 2. VIRIAL RELATION FOR CONSTRAINED SYSTEMS Consider a collection of material p oints sub jected to (generally non-integrable) constraints linear in velocities (a1 , r1 )+ ... +(an , rn ) = b. (6) Here a1 , ... ,an , (b) are vector-functions of particle p ositions and time. There can b e a few such relations. The dynamics of such a system is describ ed by the d'Alemb ert­Lagrange principle: ¨ (mi ri - Fi ,ri ) = 0, (a1 ,r1 )+ ... +(an ,rn ) = 0. (7)

for all p ossible displacements r1 , ... ,rn that satisfy the following homogeneous linear equations (8) Let r0 b e the radius-vector of a p oint O E . In the general case this p oint moves in space and therefore r0 is a function of time t. The radius-vector from O to a particle i with mass mi is ri - r0 . The vector to the center of mass of the system of material p oints reads = Let us give some definitions.
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008

mi ri . mi


LAGRANGE'S IDENTITY AND ITS GENERALIZATIONS

73

Definition 1. The kinetic energy of the system relative the point O is the quadratic form 1 mi i , i . T0 = 2 Obviously, this form is independent of the choice of the origin of coordinates. Definition 2. The moment of inertia of the system about the point O is I0 = mi (i ,i ).

Definition 3. We say that the constraints (6) imposed on the system al low for infinitesimal dilations about the point O if the vectors r1 = 1 , ... ,rn = n , are virtual displacements (i.e. satisfy equations (8)). All dilation (or similarity) transformations i i , R+ , form a one-parametric group. Their differentials with resp ect to for = 1 (i.e. at the group identity) coincide with (9). Theorem 1. Suppose that at any time t the constraints imposed on the system al low for infinitesimal dilations about the point O and (¨0 , - r0 ) = 0, r then the fol lowing virial relation holds ¨ I0 = 4T0 +2 (Fi ,i ). (11) (10) R, (9)

In its formulation, this theorem resembles the fundamental theorems of dynamics: balance of linear (angular) momentum along (ab out) a moving axis (obtained in [2]). Let us consider some sp ecial cases when the condition (10) holds true. r 1 . Uniform rectilinear motion of the p oint O (¨0 = 0). 2 . The p oint O coincides with the center of mass of the system (r0 = ). It should b e noted that Lagrange himself formulated the identity (1) as well referred to the center of mass of the interacting particles. 3 . The radius-vector from O to the center of mass is constant b oth in magnitude and direction and orthogonal to the velocity of the p oint O. Generally sp eaking, the difference - r0 is equal to 0 ­ the radius-vector from O to the center of mass. Hence, the condition (10) can b e rewritten as (¨0 ,0 ) = 0. r If the forces F1 , ... ,Fn are conservative and the p otential energy is a homogeneous function of 1 , ... ,n , then the Lagrange identity of the form (1) follows from (11). To prove Theorem 1 substitute the virtual displacements (9) into equation (7): ¨ (mi ri ,i ) = This equation can b e rewritten as ¨ mi (i ,i )+ mi (¨0 ,i ) = r (Fi ,i ). (12) (Fi ,i ).

r Obviously, the condition mi (¨0 ,i ) = 0 is equivalent to (10). On some simple rearrangement, (11) follows from (12). It turns out that the condition (10) is a criterion of applicability of the theorem on the time rate of change of kinetic energy in a moving coordinate frame with origin at O.
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008


74

KOZLOV

Theorem 2. If at al l times t real relative displacements of the system di = i dt lie among its virtual displacements (i. e. satisfy (8)) while (10) holds true, then dT0 = (Fi ,di ).

Proof is based on replacing ri in the d'Alemb ert­Lagrange equation (7) with the relative velocities i and trivial manipulations that ensue. Supp ose that the assumptions of Theorem 2 are fulfilled and the forces are conservative: Fi = V0 , i 1 i n; V0 = V0 (1 , ... n ). (13)

Then the equations of motion admit the generalized integral of energy T0 + V0 = h0 = const. In particular, if V0 is a homogeneous function of degree m, then, by Theorem 1, we get the generalized Lagrange identity (3): ¨ I0 = 4h0 - 2(m +2)V0 . Let us make a simple observation. Supp ose that in the original fixed frame of reference the forces are conservative and the p otential energy V is a function of inter-particle distances |ri - rj | only; then the relations (13) are satisfied and V0 = V |
r

.

We have |ri - rj | = |i - j | while differentiation with resp ect to ri and with resp ect to i yield the same result b ecause ri = i + r0 (t). As the conclusion of this section, consider some prop erties of constraints that allow for infinitesimal dilations. For simplicity, put r0 = 0. A. Consider constraints of the form (6) which now are assumed to b e integrable and represented as f (r1 , ... ,rn ,t) = 0. Here f is a homogeneous function of r1 , ... ,rn for all t. Then aj = f , rj b= f t (14)

and, by Euler's formula for homogeneous functions, (ai ,ri ) = 0 on the non-stationary conical surface (14). B. Let us give an example of non-integrable constraint that allows for infinitesimal dilations. To this end, consider a single particle in the three-dimensional Euclidean space. The constraint is given by the equation (a, r ) = 0, a = [r, ], (15)

where is a smooth vector-function of r = (x1 ,x2 ,x3 ) and [ , ] denotes the standard vector product. Since ([r, ],r ) 0, the constraint (15) allows for infinitesimal dilations. It remains to show that can b e chosen in such a way that the constraint (15) is indeed non-integrable, that is, (rot a, a) = 0. One can easily check that this condition is satisfied with = (1,x3 , 0).
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008


LAGRANGE'S IDENTITY AND ITS GENERALIZATIONS

75

3. QUASI-HOMOGENEOUS POTENTIALS Let us return to unconstrained (free) systems and generalize our previous argument. Let now E stand for n-dimensional Euclidian space with Cartesian coordinates (x1 , ... ,xn ) = x; this is our configuration space. The kinetic energy is a quadratic form 1 1 gij xi xj , G = gij T = (Gx, x) = 2 2 with constant coefficients; F1 , ... ,Fn are the generalized forces applied to the system. The equation of motion is Gx = F, ¨ F = (F1 , ... ,Fn )T . Taking the scalar product of this equation with the vector Ax, where A is some linear op erator, gives (Gx, Ax) = (F, Ax). ¨ Assuming that the matrix AT G = B is symmetric, we can rearrange the left-hand side as follows: ¨ (Bx, x)¨ = 2(B x, x)+ 2(B x, x) = 2(B x, x)+ 2(F, Ax). Let us introduce another "kinetic energy" K= and associated moment of inertia J = (Bx, x). Then (16) can b e represented in the form of a generalized virial relation ¨ J = 4K +2(F, Ax). (17) Obviously, A = G-1 B . In this connection, recall that any matrix can b e represented as the product of two symmetric matrices (see, for example, [3]). These trivial calculations generalize the relation (5). Indeed, as E n take the direct product of several Euclidian spaces so that the coordinates of a p oint from E n are the Cartesian coordinates of the particles. As the inertia tensor G take diag(m1 ,m1 ,m1 ,m2 ,m2 ,m2 , ... ), and let A be the unit operator. Supp ose that the generalized forces are conservative F = -V / x. Lemma 1. If the spectrum of A lies in the left (right) complex half-plane, then for any m = 0 there exist potentials V : E \{0} R such that V ,Ax x = mV . (18) 1 (B x, x) 2 (16)

This equation is a generalization of Euler's homogeneous function theorem: Euler's theorem itself results when A is the identity op erator. To prove the lemma consider the system of linear differential equations dx = Ax; d the phase flow is a family of maps: x eA x, R. (20) Supp ose that a smooth function V : E R satisfies the condition V (eA x) = em V (x).
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008

(19)


76

KOZLOV

Then it also satisfies (18). The proof consists in differentiating this equation with resp ect to and putting then = 0. It remains to show that functions of the from (20) really exist. Since the sp ectrum of A lies in the left (right) complex half-plane, there is an ellipsoid S = {x E : (x, x) = 1} such that each phase curve of (19) (except for the equilibrium x = 0) intersects S at exactly one p oint. More exactly, by Lyapunov's theorem, there exists a p ositive definite quadratic form f = (x, x) such that its derivative under the flow (19) is a p ositive (negative) definite quadratic form. For example, put V |S = 1. Then, using (20), one can extend V to the p oints on the solution of (19) that passes through x S . It can b e easily verified that V is well defined in E \{0} and satisfy (20) everywhere. The lemma is proved. Supp ose that the sp ectrum of A lies in the left half-plane. Then for m > 0 the p otential just constructed b ecomes continuous at x = 0 if we put V (0) = 0. For m < 0 the p otential has a singularity at the origin, and for m = 0, by virtue of (20), the function V is a first integral of the linear system (19). An imp ortant example are quasi-homogeneous functions that satisfy the condition V ( 1 x1 , ... , The p ositive numb er ating these relations Equation (21) is redu numb ers. If the p otential V
n

xn ) = m V (x1 , ... ,xn ),

> 0.

(21)

s 1 , ... ,n are called exponents of quasi-homogeneity or weight. Differentiwith resp ect to and putting = 1 gives (18), where A = diag(1 ... ,n ). ced to (20) if we put = e . The sp ectrum of A consists of n real non-negative satisfies (18), then (17) turns into the generalized Lagrange identity ¨ J = 4K - 2mV . (22)

An imp ortant corollary of this formula is as follows. Theorem 3. Let G = diag(m1 , ... ,mn ) and the potential energy V be a quasi-homogeneous function of degree m > 0. Then any motion t x(t) with negative total energy goes to infinity as t + or t -. Proof. Under these assumptions the matrix AT G is symmetric and p ositive definite, moreover 1 mi i x2 . i 2 The energy is preserved and negative T + V = h < 0. Hence, the motion is confined to the domain ¨ V (x) h < 0. In view of (22), J 2m|h| > 0 and therefore J (t) as t ±. This completes the proof. J= mi i x2 , i K= Corollary 1. If the quasi-homogeneous potential energy does not have a local minimum at x = 0, then the equilibrium x = 0 is unstable. Indeed, any neighb orhood of the equilibrium x = 0, x = 0 contains p oints at which the total energy is negative. There is one more family of quasi-homogeneous functions which, sp eaking formally, does not come within the definition (21). These are logarithmic homogeneous functions: V (x1 , ... ,xn ) = V (x1 , ... ,xn )+ c ln ; > 0, c = const. (23) This can reformulated as follows: the exp onential of V is a homogeneous function of degree c. Here is a good example: V (r1 , ... ,rn ) =
i
ij ln rij , ij .
2008

where rij is the distance from the particle mi to the particle mj . In this example c =
REGULAR AND CHAOTIC DYNAMICS Vol. 13

No. 2


LAGRANGE'S IDENTITY AND ITS GENERALIZATIONS

77

Differentiating (23) with resp ect to and putting = 1 give V xi = c. xi It is easy to prove that for systems with logarithmic p otential the necessary condition of stability b ecomes c 0. Indeed, if c < 0, then ¨ I = 4T - 2c 2|c| > 0. This means that the particles scatter away from each other b oth in the future t + and in the past t -. 4. CONTINUUM OF INTERACTING PARTICLES Let us show how to extend the Lagrange identity (with necessary precautions taken) to the case of continuum of interacting particles. Individual masses should b e replaced with a spatial distribution with integrable density (x, v , t), where x is a p oint in the n-dimensional Euclidian space and v denotes its velocity. The following normalizing condition is assumed to hold (x, v , t) dn xdn v = 1,


= E â Rn .

(24)

This relation is valid at any time t b ecause particles neither disapp ear from or app ear in the space. Let W b e the density of the p otential describing particle's pair-wise interaction; w is a function of inter-particle distances. The dynamics of the continuum is governed by the kinetic Vlasov equation + t where F =- x


,v + x

,F v

= 0,

(25)

W (|x - y |)(y, u, t) dn ydn u.

This equation plays an imp ortant role in kinetics and esp ecially in the plasma theory (see, for example, [4]). Assume that the Cauchy problem (initial value problem) for the integro-differential Vlasov equation has a unique solution in some appropriately chosen function spaces. This alone is a far non-trivial problem which has not b een yet solved completely (in this connection, see [5, 6]). Introduce the moment of inertia of the system of particles ab out the origin I (t) = the system's kinetic energy T (t) = 1 2 (x, v , t)v 2 dn xdn v (x, v , t)x2 dn xdn v, (26)

and the p otential energy of interaction 1 (x, v , t)(y, u, t)W (|x - y |) dn xdn vdn ydn u. (27) V (t) = 2 Of course, we make additional assumptions that these integrals converge and are smooth functions of time. If we omit the factor 1 in (27), then the p otential energy due to the interaction b etween 2 any two particles is counted twice. In the case of individual particles this phenomenon was already known to Lagrange. The model we consider does not incorp orate "self-action effect": any particle does not influence itself. Using (25) and the Gauss­Ostrogradski formula one can prove the conservation of the total energy T + V = h = const. (28) Though this fact is obvious from the physical p oint of view.
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008


78

KOZLOV

Theorem 4. If W is a homogeneous function of degree m, then the Lagrange identity (1) holds true. Proof. Using (25), we find the time derivative of (26) to b e I= + x
2

2 x dx dv = - ,v x2 dx dv t x , W (|x - y |)(y, u, t) dy du dx dv . v x

With the help of the Gauss­Ostrogradski formula, the first integral can b e rearranged as 2 (x, v )(x, v , t) dx dv , (29)

while the second integral is zero (it suffices to integrate with resp ect to v ). Similarly, the second derivative of the integral (26) is -2 +2 (x, v ) , v x ,v (x, v ) dx dv x W (|x - y |)(y, u, t) dy du dx dv .

Applying again the Gauss­Ostrogradski formula yields for the first integral 2 The second integral b ecomes 2 (x, v , t) x, x W (|x - y |)(y, u, t) dy du dx dv . (30) (v, v ) dx dv = 4T.

Since W is a homogeneous function of |x - y | = we get x, W x + y, W y = mW. (31) (xi - yi )2
1 2

,

Interchanging the groups of variables x, v and y, u does not affect the value of the integral (30). Using then (31), we reduce the integral (30) to the form m This proves the theorem. Under an additional assumption that the derivative I is b ounded, the Clausius formulas (4) can b e deduced from Theorem 4 and equation (28). However, unlike the finite-dimensional case, the b oundedness of I is not so easy to verify. Let us p oint out some sufficient conditions for b oundedness . of I If the moment of inertia I itself is b ounded and the p otential energy is non-negative (e. g. W 0), then |I (t)| is b ounded on the whole t-axis. Indeed, using the Cauchy inequality 2|(x, v )| (x, x)+(v, v ), it follows from (29) that |I | I +2T.
Vol. 13 No. 2 2008

(x, v , t)(y, u, t)W (|x - y ) dx dv dy du = 2mV .

REGULAR AND CHAOTIC DYNAMICS


LAGRANGE'S IDENTITY AND ITS GENERALIZATIONS

79

It remains to note that from (28) and the inequality V 0 one immediately obtains the b oundedness of the kinetic energy (T h). It should b e noted that the fact that W is a homogeneous function of degree m does not seem to entail the homogeneity of the p otential energy (27) b ecause the expression for V contains the density, which also dep ends on particle coordinates. However, there is one more thing to consider: the kinetic Vlasov equation (25) is invariant under the action of the similarity group t
2-m 2

t,

x x,

v 2 v,

m

-

m+2 2

n

.

(32)

Moreover, such transformations do not alter the normalizing condition (24), whose physical meaning is the preservation of the total mass of the particles. Under the transformation (32) the integral quantities I, T and V are multiplied by 2 ,m and m , resp ectively. In particular, this means that the total p otential energy of the continual system is a homogeneous functional of degree m as required. In conclusion consider continuum of particles in the n-dimensional Euclidian space E n = {x}. The particles attract each other with an elastic p otential whose density W is k k |x - y |2 = (xi - yi )2 . 2 2 Here k = const is the coefficient of elasticity and W is a homogeneous function of degree two m = 2. In view of the normalizing condition (24), the radius-vector to the center of mass is = x(x, v , t) dx dv .

This is a function of time. Let us show that = 0, meaning that the center of mass is in the state ¨ of a uniform rectilinear motion. Of course, this is so whatever the interaction p otential is chosen. The velocity of the center of mass is =- x ,v - x , v x W(y, u, t) dy du dx dv = v dx dv .

Then, using the Gauss­Ostrogradski formula, we see that =- ¨ b ecause the integral W dy du is indep endent of v . Without loss of generality assume that = 0. Adopting this assumption means that we move to an inertial frame of reference that travels with the center of mass. With this assumption and by virtue of (24) the total p otential energy (27) for the elastic interaction now reads k 4 (x, v , t)(y, u, t)x2 dx dv dy du + - k 2 xi (x, v , t) dx dv k 4 (x, v , t)(y, u, t)y 2 dx dv dy du yi (y, u, t) dy du = k I. 2 v ,v - x , v x W(y, u, t) dy du dx dv = 0

Then, by Theorem 4 and using (3), we finally get ¨ I = 4h - 4kI . Therefore, the moment of iner of a continuum of oscillators ab out the center of mass oscillates tia harmonically with frequency 2 k , and its average is h/k. This work was supp orted by the Grant NWO-RFBR (047.011.2004.059).
REGULAR AND CHAOTIC DYNAMICS Vol. 13 No. 2 2008


80

KOZLOV

REFERENCES
1. Kozlov, V.V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Mo dels of Thermostat, Rus. J. Nonlin. Dyn., 2006, vol. 3, no. 2, pp. 123­140. 2. Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 28­33. 3. Horn, R. and Johnson, C., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1985. 4. Vedenyapin, V.V., Boltzmann and Vlasov Kinetic Equation, Moscow: Fizmatlit, 2001. 5. Maslov, V.P., Equations of Self-Consistent Field, in Current Problems in Mathematics, Vol. 11, Akad. Nauk SSSR Vseso juz. Inst. Nauchn. i Tehn. Informacii (VINITI), Moscow, 1978, pp. 153­234. 6. Dobrushin, R.L., Vlasov Equations, Funktsional. Anal. i Prilozhen., 1979, vol. 13, no. 2, pp. 48­58.

REGULAR AND CHAOTIC DYNAMICS

Vol. 13

No. 2

2008