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ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 5, pp. 431-435. c Pleiades Publishing, Ltd., 2008.

NONHOLONOMIC MECHANICS

Gauss Principle and Realization of Constraints
V. V. Kozlov*
V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received July 13, 2008; accepted Septemb er 9, 2008

Abstract--The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints. MSC2000 numbers: 70F25, 37J60 DOI: 10.1134/S1560354708050055 Key words: Gauss principle, constraints, anisotropic friction

1. INTRODUCTION Let x = (x1 , . . . , xn ) b e local coordinates of mechanical system with n degrees of freedom which is under the action of generalized forces F = (F1 , . . . , Fn ). We assume as usual that the forces F are known functions of velocities x, coordinates x and time t. Let T= 1 2 aij (x)xi xj

b e the kinetic energy of the system, where aij = aj i . The equations of motion in the Lagrange form are T d T - = F. dt x x In these equations we isolate summands dep ending on accelerations: Ax = , Е A = aij . (1) Here is a known function of x, x and t. Now supp ose that our system is under the action of additional forces Q. Then equation (1) b ecomes: Ax = + Q. Е (2)

By analogy with the Gauss principle we shall give some definitions. Let t0 b e an initial moment of time. Consider only smooth paths t x(t) such that x(t0 ) = x0 and x(t0 ) = x0 . A path is actual (resp ectively free) motion, if it satisfies equation (2) (resp ectively (1)). A path x(ћ) is called a virtual motion if (Q, x - xr )|t0 = 0, ЕЕ (3) where xr is the acceleration of an actual motion. Е Since the p osition of the system and velocities are fixed the acceleration difference x - xr is ЕЕ transformed under changes of local coordinates as a (contravariant) vector. A force is a covector, therefore the left side of relation (3) is invariant under variable changes: this is the pairing of a
*

E-mail: kozlov@pran.ru

431


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KOZLOV

covector and a vector. From (3) it immediately follows that an actual motion is a virtual one. Condition (3) is equivalent to the following (Q, x) = (Q, A-1 ( + Q)). Е (4) If Q = 0, then the acceleration of a virtual motion can b e arbitrary. We can illustrate these definitions by the following example. Consider a dynamical system with the following constraint g = 0. (5) x According to the method of Lagrange multipliers we can consider this system as a free one but with the additional force g Q= . x Evidently, the initial conditions x(t0 ) = x0 and x(t0 ) = x0 should satisfy (5). The multiplier can b e found from equation (2) and the following relation g(x, x, t) = 0, g ,x Е x =- g - t g ,x . x (6)

According to the mechanical theory of systems with constraints a path t x(t) (with a fixed initial p oint) is virtual if it satisfies (5). Its acceleration satisfies relation (6). Let us demonstrate that if (t0 ) = 0 then relation (6) is equivalent to (4). Indeed, =- Substituting Q constraint does Note that th of systems with g + t g ,x + x g -1 ,A x A-
1

g g , x x

-1

.

(7)

g = x into (4) and using (7) one gets (6). If = 0, then at this moment the not act on the system. e ab ove definitions of actual and free motions corresp ond to the mechanical theory constraints.

2. GENERALIZED GAUSS PRINCIPLE Let the accelerations of virtual and free motion at the moment t0 b e xm and xf resp ectively. Е Е Similar to the Gauss approach we introduce the enforcement 1 (A(xm - xf ), (xm - xf )) Е Е Е Е 2 as a measure of deviation of the virtual motion from the free one. The next theorem is a generalization of the Gauss principle. Zm,f = Theorem 1. The virtual motion is an actual one if its deviation from the free motion is minimal. Proof. Indeed, Zm,f = 1 (A[(xm - xr ) + (xr - xf )], (xm - xr ) + (xr - xf )) Е Е Е Е Е Е Е Е 2 = Zm,r + Zr,f + (A(xm - xr ), (xr - xf )). Е Е Е Е (xm - xr , A(xr - xf )) = (Q, xm - xr ) = 0 Е Е Е Е Е Е due to (3). Now from (8) one gets Zr,
f

(8)

However, the last summand vanishes since

Zm,
r

f

since the op erator A is p ositive and therefore Zm,

0.
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GAUSS PRINCIPLE AND REALIZATION OF CONSTRAINTS

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Remark. Obviously, this theorem is valid for Q = 0. Sp ecifically it contains the classical Gauss principle for constrained systems as a particular case. In order to illustrate the general definition of virtual motions, consider the following instructive example. Let V , x Rn Е x b e the equation of motion in the p otential V . This equation has the first integral x=- Е (9)

x2 + V (x) = h = const, 2 which can b e considered as a nonlinear in velocity constraint. Free motions satisfy the equation V x = 0 and coincide with inertial motions. Thus, equation (9) coincides with (2), where Q = - Е x and A is the unity op erator. An analog of (6) is the equation (x, x) = - Е V ,x . x (10)

If one considers this relation as a definition of virtual motions, then the generalized Gauss principle leads to the equation x=- Е ,x x, (x, x)
V x

which also has the integral of energy, but differs from the initial equation (9). Actually due to (4) one should replace the equation for virtual accelerations (10) by the following equation - V ,x Е x = V V , . x x

Now our general theorem leads to the correct equation (9). 3. GAUSS PRINCIPLE AND REALIZATION OF NONHOLONOMIC CONSTRAINTS Generally, the theorem from Section 2 does not dep end up on the presence of constraints. On the other hand, nonholonomic constraints can b e realized by large anisotropic external forces of viscous friction (see [1, 2]). We shall show that in this limit Theorem 1 transforms into the classical Gauss principle. Let N N = (a, x)2 2 b e the Rayleigh dissipation function. Here a(x) is a smooth covector field and N is a large parameter. Below we shall put N . In order to consider dissipative forces one should replace the equation of 'free' motion (1) by the equation Ax = - Е N = - N (a, x)a. x xN (0) = x0 , (11)

Let t xN (t) b e its solution with initial conditions xN (0) = x0 , while (a(x0 ), x0 ) = 0.
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It is shown in [1, 2] that in this situation the following limits
N

lim xN (t) = x(t), ^

N

lim xN (t) = xћ (t), ^

exist on every finite interval of time [t0 , t0 + ]. Also, it holds x(0) = x0 , xћ (0) = x0 and ^ ^ N (a(x), x)
xN (t)

-(t),

where is a Lagrange multiplier. The limit function x(ћ) satisfies the constrained equation of motion ^ Ax = + a, Е (a, x) = 0. Evidently, equation (11) has the form (2) for

(12)

Free motions are defined by equation (1) as for constrained systems. To complete our discussion it remains to show that for all virtual accelerations equation (4) transforms into the equation of the form (6) (a, x) = - Е a x, x x

Q = -N (a, x)a.

as N . To do so one should put g = (a, x) in the example from Section 1 and use continuity arguments that is almost evident. Hence, the common Gauss principle for systems with nonholonomic constraints is a natural limiting case of a more general Theorem 1. ACKNOWLEDGMENTS This work was supp orted by the RFBR grant (08-01-00025-a) and the program "State Supp ort for Leading Scientific Schools" (NSh-691.2008.1). REFERENCES
1. Karapetyan, A.V., On Realization of Nonholonomic Constrains and on Stability of the Celtic Stones, J. Appl. Math. Mech., 1981, vol. 45, no. 1. pp. 45-51. 2. Brendelev, V.N., On Realization of Constraints Nonholonomic Mechanics, J. Appl. Math. Mech., 1981, vol. 45, no. 3, pp. 481-487.

REGULAR AND CHAOTIC DYNAMICS

Vol. 13

No. 5

2008