- Email: [email protected]
On a stationarity principle for non-conservative dynamical systems
Int. J. Non-Linear Mechonrcs. @ Pergamon Press Ltd. 1978.
OOZC-7462!7810601-0139
Vol. 13, pp. 139-143. Printed m Great Britain
lOZ.WM3
ON A STATIONARITY PRINCIPLE FOR NON-CONSERVATIVE DYNAMICAL SYSTEMS T. M. ATANACKOVIC Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Yugoslavia (Received 1 September
1977; receivedfor
publieario~ 7 Februnry 1978)
Abstrac&--A stationarity principle for non-conservative, holonomic dynamical systems is formulated.
It is based on the notion of Glteaux directional derivative. Its relation to the classical and variational principle with non~ommutative variational rules is discussed.
1. INTRODUCTION
For dynamical systems that can be described by a Lagrangian, H~ilton’s principle states that the action integral, in Hamilton’s sense, has a stationary value for a natural motion when compared with adjacent motions having the same end points [l]. In order to include purely non-conservative dynamical systems into the compact Hamilton’s formalism, Vujanovic [2] used a variational principle with non-commutative variational rules. In it, the condition that the action integral of the conservative part of the system has a stationary value gives the equations of motion of the non-conservative system, if special non-commutative variational rules are prescribed. In [2] the introduction of such variational rules was based on the central Lagrangian equation while in [4] their geometrical background was discussed. It is our intention, in this note, to give an interpretation of the variational principle with non-commutative variational rules, from the functional analysis point of view. We will derive the equations of motion of conservative and non-conservative holonomic dynamical systems from the condition that the G&eaux directional derivative of the conservative part of the action integral vanishes for a certain set of directions in a suitably defined function space. This will show the connection between the classical formulation and the variational principle with non-commutative variational rules. Namely, it will be required in both principles that the Glteaux derivative of the conservative part of the action integral vanishes for a certain set of directions. This set of directions, however, is different for conservative and nonconservative dynamical systems, and depends on the non-conservative generalized forces. 2. THE
STATIONARITY
PRINCIPLE
Let us consider a rheonomic, holonomic dynamical system with y1degrees of freedom, whose kinetic energy is T = T&, q”)+ T, + q,
(1)
where?
Ti = s,f& q%“;
Tz= &A 474k4”.
(2)
In (2) qm,m = 1, . . . n, are generalized coordinates, t is the time, g,,, and gh = gSkare given functions of the generalized coordinates and time and superimposed dots denote differentiation taken with respect to time. Suppose that the generalized forces acting on the system are of two types : (i) Conservative with the potential l3(q”, t). (ii) Non-conservative Qk($“, 4”, t) k = 1,. . . n. If we define a Lagrangian of the conservative part of the system by L(q”, tj”, t) = T - l-l t Repeated indices are summed throughout. 139
(3)
140
T. M. ATANACKOW
then the equations of motion can be written as: d aL aL ----=
aqk
dt aqk
Qk;
k = l,...n.
(4)
The action integral of the conservative part of the system is 11 F= L(qk, dk, t) dt, (5) s fo where [to, tJ is an arbitrary time interval. We will use the following notation in the rest of the paper: by R we denote the space of real numbers, by C” the n-dimensional space of real valued functions that are continuous on the interval [to, tr], by C; the n-dimensional space of real valued functions that are continuous and have continuous first derivatives on the interval [r,, tr] and by e; the set @, = {f:fEC;,
f(t,) = f(tl) = 0).
(6)
Also, we put ty=Qxc
w = Cl x C”,
(7)
and define two mappings: M : W x R + C: by7 M(w, t) = L(w, t) = L(N& w:, t)
(8)
andM,:C:+Rby fI a(t) dt. (9) 5fo The Gkeaux differential of the composite mapping F = M 1 *M, at the point w = (wl, WJ in the direction * = (%r, G2), is [5] M,(a) =
F’(w, 6) = lim 1-O
F(w+AG)-F(w) A
Since L(.tiI:, &t) is continuous and has continuous arguments, F’(w, +) exists [6] and is equal to
*
derivatives with respect to all its
1 dt.
F’(w, I?) =
(10)
(11)
We are interested in values of F’(w, ti) for w and w belonging to two special subsets of W and iVrespectively W 2 B = {(w1,w2):w$ = Y:(wl),wl~C;}
(12)
fi 3 B; = ((~v,,+,):9’; = Pk,(&), til E PI}
(13)
where U”,( *) is a non-linear first order differential operator and Y$( *) is a non-linear first order differential operator, possibly dependent on w E B. We state now the following problem : Given U: and 95, find an element w. E B such that F’(w,,G) = 0
for all
$ E B;“.
(14)
With the suitable choice of the sets B and Br” the solution to the problem just posed could be made to satisfy the equations of motion (4). To show this, we first consider the case of conservative dynamical systems, i.e. Qk = 0. If, k = l,...n t Cjdenotes thespaceofrealvaluedfunctionsthatarecontinuousandhavecontinuousderivatives,upto order, with respect to all its arguments on the interval [to, ix).
(15) thesecond
141
On a stationarity principle for non-conservative dynamical systems
then (11) and (14) give d aL aL ----= dt agk a$
o
k = l,...n
3
(16)
that is, the condition (14)is equivalent to (4) with Qk = 0. Note that with Y’: and Y’: given by (15) the problem (14) is the basic problem of the classical variational calculus. Equations of motion for non-conservative dynamical systems can be obtained from (14) if we take (17) where 12: are arbitrary continuous functions of w E B. From (1 l), (14) and (17) we find that
(18) Choosing 1 A; = w”z----Qk, 2T2 + TI
(19)
equation (18) becomes equal to (4). The condition (14) with _Yj and _Yi given by (17) is equivalent to the variational principle with non-commutative variational rules. Therefore, stationarity of a functional in the variational principle with non-commutative variational rules is stationarity (vanishing of the Giteaux derivative) with respect to a specially selected set, B,““). The above results are stated in the following theorem. Theorem. The solution w0 = (qk, cj“) of the equations ofmotion of arheonomic, holonomic non-conservative dynamical system, gives a stationary value, with respect to the set B,““,to the functional (5). 3. NATURE
OF THE
STATIONARITY:
AN EXAMPLE
turn now to the conditions sufficient to ensure a minimum of F at the point wO.The functional F will have a local minimum at w. with respect to the set B,“” if We
F(w,+i)-F(w,)
3 0
for all
ti E B,“?
(20)
A sufficient condition for the existence of a local minimum at the point w0 is that the second GBteaux derivative (if it exists) at the point w0 in the direction % is positive F”(w,, ti, +v)> 0
ti E B;l’.
for all
(21)
Using expressions (3) and (5) for F we get,
for all
&Bp,
(22)
as the condition that ensures a local minimum of F with respect to the set Bz’. To obtain more specific results from (22) let us consider a scleronomic non-conservative dynamical system. In this case the kinetic energy is a positive definite quadratic form of the generalized velocities, so that
&d2ti2 =& 2d
2
2
d22ws2 > 0. 2
(23)
Also, from (17) we find that ($).
= -A$$
+4.
(24) We may look at (24) as the system of differential equations that determine tii for given ti2(t).
T. M. ATANACKOVIC
142
Then, taking into account that ~~~~~)= 0 [see (13) and (6)], we get the following estimate for Ml, [7XPm82)
or,
t4t
G
mt-to>,
(261
where (3”are constants. Choosing the time interval [tl -to] s~~~e~tly small we get from (22)
Thus, we con&de that in the case of a scleronomic non-~ns~rvative dynamical systems, for sufficiently small time interval [tl -to], F is in local minimum with respect to the set B,““.This is an analog of the classical result valid for conservative scleronomic dyn~ical systems f3, p. 6501. As a specific example iet us consider a two-degree-of-freedom non-conservative rheonomic system whose equations of motion are [8] jt+kx+2#-py
= 0,
jj+ky+2puJi+px
= 0. r
(28)
The action integral of the conservative part of the system is fx: = w:, y = w:, 2 = w&j = w$> F =:
s[ kl
while the non-conservative
dt
_ k(w:)2+(w32 t’ m2+(W$)2 2
2
1
’
(291
generalized forces are (Q, = Q1, Q, = Q2) Qt = -2,&i-ppy,
Q2 = -2j@-px.
(301
Using (29) in (22) we see that F”(wO,6, +) > 0 if k c 0 for any to and tl. Thus if x0 and y, are the solutions to the equations (28) then F =
ft q&+j$ 2 S[to
k$+y;
___ 2
dt (31) 1 is in locaf burns with respect to the set B,WO, The elements ti = (S:, $f, S$, 6;) of the set BP satisfy the following relations
Using the notion of the G&teaux directional derivative a stationarity principle for nonconservative dynamical systems is formulated” It is shown that this station~ity principle is equivalent to the variational principle with non-commutative variational rules. However, the present approach extends the earlier treatment since the criteria for the existence of a minimum, with respect to a specially selected s&, is developed. This criteria is app~ed to the scleronomic non-conservative dynamical systems, for which the existence of a minimum, for the sufficiently small time interval is proved. Acknowledgements-The
author is grateful to Dr Djukic and Prof. W. F. Ames for constructive review of the
manuscript. REFERENCES 1, J. I_..Synge and B. A. GrifSth, Princjples [email protected], New York (1959). 2. B. Vujanovic, A variational principle for non-conservative d~arni~i systems, Z. angew. Math. Me&. 55, 321-331 (1975).
On a stationarity principle for non-conservative
dynamical systems
143
3. A. I. Lure, Analytical Mechanics. G.I.F.M.L., Moscow (1961). 4. Dj. Djukic and B. Vujanovic, On some geometrical aspects of classical nonconservative mechanics. J. Mach. Phys. 16,2099-2102 (1975). 5. L. B. Rall, Non-linear Functional Analysis and Applications. Academic Press, New York (1971). 6. A. D. Ioffe and B. M. Tihomirov. The.Theory ofExrremalProblems. Nauka, Moscow (1974). 7. J. K. Hale, Ordinary Dtfirential Equations. Wiley-Interscience, New York (1969). 8. Dj. S. Djukic and B. Vujanovic, Noether’s theory in classical non-conservative mechanics, Acta Mech. 23, 17-27 (1975).
On formule un principe de stationnarit.6 pour des syst&nes dynamiques non conservatifs holonomes. I1 est base sur la notion de deriv6es directionnelles de Gateaux. On discute ses relations avec le principe classique et variationnel avec des rggles variationnelles non comnutatives. Zusaimnenfassung: Ein Stationaritatsprinzip fur nichtkonservative. holonomische, dynamische Systeme wird formuliert. Es baut auf dem Begriff der gerichteten Ableitung nach Gateaux auf. Sein Verhaltnis zu dem klassischen und Variationsprinzip mit nichtkomnutativen Variationsregeln wird diskutiert.
OOZC-7462!7810601-0139
Vol. 13, pp. 139-143. Printed m Great Britain
lOZ.WM3
ON A STATIONARITY PRINCIPLE FOR NON-CONSERVATIVE DYNAMICAL SYSTEMS T. M. ATANACKOVIC Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Yugoslavia (Received 1 September
1977; receivedfor
publieario~ 7 Februnry 1978)
Abstrac&--A stationarity principle for non-conservative, holonomic dynamical systems is formulated.
It is based on the notion of Glteaux directional derivative. Its relation to the classical and variational principle with non~ommutative variational rules is discussed.
1. INTRODUCTION
For dynamical systems that can be described by a Lagrangian, H~ilton’s principle states that the action integral, in Hamilton’s sense, has a stationary value for a natural motion when compared with adjacent motions having the same end points [l]. In order to include purely non-conservative dynamical systems into the compact Hamilton’s formalism, Vujanovic [2] used a variational principle with non-commutative variational rules. In it, the condition that the action integral of the conservative part of the system has a stationary value gives the equations of motion of the non-conservative system, if special non-commutative variational rules are prescribed. In [2] the introduction of such variational rules was based on the central Lagrangian equation while in [4] their geometrical background was discussed. It is our intention, in this note, to give an interpretation of the variational principle with non-commutative variational rules, from the functional analysis point of view. We will derive the equations of motion of conservative and non-conservative holonomic dynamical systems from the condition that the G&eaux directional derivative of the conservative part of the action integral vanishes for a certain set of directions in a suitably defined function space. This will show the connection between the classical formulation and the variational principle with non-commutative variational rules. Namely, it will be required in both principles that the Glteaux derivative of the conservative part of the action integral vanishes for a certain set of directions. This set of directions, however, is different for conservative and nonconservative dynamical systems, and depends on the non-conservative generalized forces. 2. THE
STATIONARITY
PRINCIPLE
Let us consider a rheonomic, holonomic dynamical system with y1degrees of freedom, whose kinetic energy is T = T&, q”)+ T, + q,
(1)
where?
Ti = s,f& q%“;
Tz= &A 474k4”.
(2)
In (2) qm,m = 1, . . . n, are generalized coordinates, t is the time, g,,, and gh = gSkare given functions of the generalized coordinates and time and superimposed dots denote differentiation taken with respect to time. Suppose that the generalized forces acting on the system are of two types : (i) Conservative with the potential l3(q”, t). (ii) Non-conservative Qk($“, 4”, t) k = 1,. . . n. If we define a Lagrangian of the conservative part of the system by L(q”, tj”, t) = T - l-l t Repeated indices are summed throughout. 139
(3)
140
T. M. ATANACKOW
then the equations of motion can be written as: d aL aL ----=
aqk
dt aqk
Qk;
k = l,...n.
(4)
The action integral of the conservative part of the system is 11 F= L(qk, dk, t) dt, (5) s fo where [to, tJ is an arbitrary time interval. We will use the following notation in the rest of the paper: by R we denote the space of real numbers, by C” the n-dimensional space of real valued functions that are continuous on the interval [to, tr], by C; the n-dimensional space of real valued functions that are continuous and have continuous first derivatives on the interval [r,, tr] and by e; the set @, = {f:fEC;,
f(t,) = f(tl) = 0).
(6)
Also, we put ty=Qxc
w = Cl x C”,
(7)
and define two mappings: M : W x R + C: by7 M(w, t) = L(w, t) = L(N& w:, t)
(8)
andM,:C:+Rby fI a(t) dt. (9) 5fo The Gkeaux differential of the composite mapping F = M 1 *M, at the point w = (wl, WJ in the direction * = (%r, G2), is [5] M,(a) =
F’(w, 6) = lim 1-O
F(w+AG)-F(w) A
Since L(.tiI:, &t) is continuous and has continuous arguments, F’(w, +) exists [6] and is equal to
*
derivatives with respect to all its
1 dt.
F’(w, I?) =
(10)
(11)
We are interested in values of F’(w, ti) for w and w belonging to two special subsets of W and iVrespectively W 2 B = {(w1,w2):w$ = Y:(wl),wl~C;}
(12)
fi 3 B; = ((~v,,+,):9’; = Pk,(&), til E PI}
(13)
where U”,( *) is a non-linear first order differential operator and Y$( *) is a non-linear first order differential operator, possibly dependent on w E B. We state now the following problem : Given U: and 95, find an element w. E B such that F’(w,,G) = 0
for all
$ E B;“.
(14)
With the suitable choice of the sets B and Br” the solution to the problem just posed could be made to satisfy the equations of motion (4). To show this, we first consider the case of conservative dynamical systems, i.e. Qk = 0. If, k = l,...n t Cjdenotes thespaceofrealvaluedfunctionsthatarecontinuousandhavecontinuousderivatives,upto order, with respect to all its arguments on the interval [to, ix).
(15) thesecond
141
On a stationarity principle for non-conservative dynamical systems
then (11) and (14) give d aL aL ----= dt agk a$
o
k = l,...n
3
(16)
that is, the condition (14)is equivalent to (4) with Qk = 0. Note that with Y’: and Y’: given by (15) the problem (14) is the basic problem of the classical variational calculus. Equations of motion for non-conservative dynamical systems can be obtained from (14) if we take (17) where 12: are arbitrary continuous functions of w E B. From (1 l), (14) and (17) we find that
(18) Choosing 1 A; = w”z----Qk, 2T2 + TI
(19)
equation (18) becomes equal to (4). The condition (14) with _Yj and _Yi given by (17) is equivalent to the variational principle with non-commutative variational rules. Therefore, stationarity of a functional in the variational principle with non-commutative variational rules is stationarity (vanishing of the Giteaux derivative) with respect to a specially selected set, B,““). The above results are stated in the following theorem. Theorem. The solution w0 = (qk, cj“) of the equations ofmotion of arheonomic, holonomic non-conservative dynamical system, gives a stationary value, with respect to the set B,““,to the functional (5). 3. NATURE
OF THE
STATIONARITY:
AN EXAMPLE
turn now to the conditions sufficient to ensure a minimum of F at the point wO.The functional F will have a local minimum at w. with respect to the set B,“” if We
F(w,+i)-F(w,)
3 0
for all
ti E B,“?
(20)
A sufficient condition for the existence of a local minimum at the point w0 is that the second GBteaux derivative (if it exists) at the point w0 in the direction % is positive F”(w,, ti, +v)> 0
ti E B;l’.
for all
(21)
Using expressions (3) and (5) for F we get,
for all
&Bp,
(22)
as the condition that ensures a local minimum of F with respect to the set Bz’. To obtain more specific results from (22) let us consider a scleronomic non-conservative dynamical system. In this case the kinetic energy is a positive definite quadratic form of the generalized velocities, so that
&d2ti2 =& 2d
2
2
d22ws2 > 0. 2
(23)
Also, from (17) we find that ($).
= -A$$
+4.
(24) We may look at (24) as the system of differential equations that determine tii for given ti2(t).
T. M. ATANACKOVIC
142
Then, taking into account that ~~~~~)= 0 [see (13) and (6)], we get the following estimate for Ml, [7XPm82)
or,
t4t
G
mt-to>,
(261
where (3”are constants. Choosing the time interval [tl -to] s~~~e~tly small we get from (22)
Thus, we con&de that in the case of a scleronomic non-~ns~rvative dynamical systems, for sufficiently small time interval [tl -to], F is in local minimum with respect to the set B,““.This is an analog of the classical result valid for conservative scleronomic dyn~ical systems f3, p. 6501. As a specific example iet us consider a two-degree-of-freedom non-conservative rheonomic system whose equations of motion are [8] jt+kx+2#-py
= 0,
jj+ky+2puJi+px
= 0. r
(28)
The action integral of the conservative part of the system is fx: = w:, y = w:, 2 = w&j = w$> F =:
s[ kl
while the non-conservative
dt
_ k(w:)2+(w32 t’ m2+(W$)2 2
2
1
’
(291
generalized forces are (Q, = Q1, Q, = Q2) Qt = -2,&i-ppy,
Q2 = -2j@-px.
(301
Using (29) in (22) we see that F”(wO,6, +) > 0 if k c 0 for any to and tl. Thus if x0 and y, are the solutions to the equations (28) then F =
ft q&+j$ 2 S[to
k$+y;
___ 2
dt (31) 1 is in locaf burns with respect to the set B,WO, The elements ti = (S:, $f, S$, 6;) of the set BP satisfy the following relations
Using the notion of the G&teaux directional derivative a stationarity principle for nonconservative dynamical systems is formulated” It is shown that this station~ity principle is equivalent to the variational principle with non-commutative variational rules. However, the present approach extends the earlier treatment since the criteria for the existence of a minimum, with respect to a specially selected s&, is developed. This criteria is app~ed to the scleronomic non-conservative dynamical systems, for which the existence of a minimum, for the sufficiently small time interval is proved. Acknowledgements-The
author is grateful to Dr Djukic and Prof. W. F. Ames for constructive review of the
manuscript. REFERENCES 1, J. I_..Synge and B. A. GrifSth, Princjples [email protected], New York (1959). 2. B. Vujanovic, A variational principle for non-conservative d~arni~i systems, Z. angew. Math. Me&. 55, 321-331 (1975).
On a stationarity principle for non-conservative
dynamical systems
143
3. A. I. Lure, Analytical Mechanics. G.I.F.M.L., Moscow (1961). 4. Dj. Djukic and B. Vujanovic, On some geometrical aspects of classical nonconservative mechanics. J. Mach. Phys. 16,2099-2102 (1975). 5. L. B. Rall, Non-linear Functional Analysis and Applications. Academic Press, New York (1971). 6. A. D. Ioffe and B. M. Tihomirov. The.Theory ofExrremalProblems. Nauka, Moscow (1974). 7. J. K. Hale, Ordinary Dtfirential Equations. Wiley-Interscience, New York (1969). 8. Dj. S. Djukic and B. Vujanovic, Noether’s theory in classical non-conservative mechanics, Acta Mech. 23, 17-27 (1975).
On formule un principe de stationnarit.6 pour des syst&nes dynamiques non conservatifs holonomes. I1 est base sur la notion de deriv6es directionnelles de Gateaux. On discute ses relations avec le principe classique et variationnel avec des rggles variationnelles non comnutatives. Zusaimnenfassung: Ein Stationaritatsprinzip fur nichtkonservative. holonomische, dynamische Systeme wird formuliert. Es baut auf dem Begriff der gerichteten Ableitung nach Gateaux auf. Sein Verhaltnis zu dem klassischen und Variationsprinzip mit nichtkomnutativen Variationsregeln wird diskutiert.