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Unified expressions of all differential variational principles
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 30 (2003) 567–572 www.elsevier.com/locate/mechrescom
Unified expressions of all differential variational principles Y.C. Huang Department of Applied Physics, Beijing University of Technology, Beijing 100022, PR China Received 14 October 2001
Abstract A mathematical expression of the quantitative causal principle is given, using the expression this letter shows the unified expressions of DÕAlembert–Lagrange, virtual work, Jourdian, Gauss and general DÕAlembert–Lagrange principles of differential style, first finds the intrinsic relations among these variational principles, the conservation quantities of the above principles are first found, finally the Noether conservation charges of the all differential variational principles in the systems with the symmetry of Lie group Dm are first obtained. Ó 2003 Published by Elsevier Ltd. Keywords: Euler–Lagrange equation; Variational principle; Causal principle; Conservation quantity; Noether theorem
1. Introduction There are numerous variational principles in physics, they are classified into two kinds, i.e., they are differential and integral variational principles respectively (Pars, 1965; Cheng, 1987; Desloge, 1982). It is well known that the unified expressions of the relations among so many scrappy variational principles have not been presented up to now, and the intrinsic relations among the conservation quantities of these principles are not obtained either (Mei, 1991). It is well known that the conservation quantities of all differential variational principles do not exist in the past (Pars, 1965; Cheng, 1987; Desloge, 1982), now we first give the conservation quantities of all differential variational principles in this letter. On the other hand, in quantum field theory the causal principle demands if the square of the distance of spacetime coordinates of two boson (or fermion) operators is timelike, their commutator (or anticommutator) is not equal to zero, their measures are then coherent, for spacelike no coherent (Sterman, 1993). The dispersion relations can be deduced by the causal principle etc (Klein, 1961), and general scientific theories in physics should satisfy the elementary demand of the causal principle, and the causal relations must be quantitative, or there must be contradictory. Using the following quantitative causal principle, Ref. Huang et al. (2000, 2001, in press) give both the unified expressions of the new and old axiom systems of
E-mail address: [email protected] (Y.C. Huang). 0093-6413/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0093-6413(03)00057-0
568
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
group theories and the applications of the symmetric property to modern astrophysics and cosmology. Therefore, the causal principle is essential to research physical laws.
2. Quantitative causal principle In real physics, the quantitative actions (cause) of some quantities must lead to the corresponding equal effects, which is just a expression of the no-loss-no-gain principle in the universe (Li, 1998; Huang et al., 2001), therefore we have Quantitative causal principle: In any physical system, how much loses (cause), there must be, how much gains (result), or, quantitative actions (cause) of some quantities must result in the corresponding equal effects (result). Then, the principle may be concretely expressed as DG CG ¼ 0
ð1Þ
Eq. (1)Õs physical meanings are that the real physical result produced by any operator set D acting on G must lead to appearance of set C acting on G so that DG is equal to CG, where D and C may be different operator sets, the whole process satisfies the quantitative causal principle (QCP) so that Eq. (1)Õs right hand side satisfies QCP with the no-loss-no-gain characteristics (Huang et al., 2001, in press). Eq. (1) is thus viewed as a mathematical expression of the quantitative causal principle derived from the no-loss-no-gain principle in the universe (Huang et al., 2001). It can be seen that gauge fields of physical fundamental interactions are just connections of principal bundle (Nash and Sen, 1983), the connections are obtained by Eq. (1) when D is a connection operator, G is a section S of the bundle, C is a connection x (Chern, 1989). In fact, the Noether theorem and conservation law of energy transformation are two of a lot of applied examples of the no-loss-no-gain principle in the universe (Huang et al., 2001, in press). And Eq. (1) is useful for the following studies.
3. Unified expressions of all differential variational principles and conservation quantity When G is a functional, C is identity, D is group operator of infinitesimal continuous transformation in Eq. (1) (Djukic, 1989; Li, 1987), the general expression of variational principle of differential style is obtained as follows X DGðqðtÞ; q_ ðtÞ; tÞ GðqðtÞ; q_ ðtÞ; tÞ ¼ ðGqi Dqi þ Gq_ i Dq_ i Þ þ Gt Dt ð2Þ i
where ðrÞ
ðrÞ
ðrþ1Þ
Dqi ¼ dqi ðt0 Þ þ qi
Dt
ð3Þ
Therefore we can have DG ¼
X oG i
oqi
d oG d X oG dG Dt dqi ðt0 Þ þ dqi ðt0 Þ þ dt oq_ i dt i oq_ i dt
ð4Þ
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
When Dt is infinitesimal constant quantity, it follows that ( ) X oG d oG d X oG 0 0 dqi ðt Þ þ GDt ¼ 0 dqi ðt Þ þ oqi dt oq_ i dt oq_ i i i
569
ð5Þ
Namely, in a fixed infinitesimal time gap, the comparisons of the actual variation of the system with the other possible variations make G take limit value, and it is at the limit value that the system satisfies the quantitative causal principle. When the system satisfies the Euler–Lagrange equation, one has the conservation quantity ( ) X oG dqi ðt0 Þ þ GDt ¼ const: ð6Þ oq_ i i On the other hand, when integrating Eq. (5) during ½t1 ; t2 , and taking condition (i): qi jt¼t1 ¼ dqi jt¼t2 ¼ 0; or taking condition (ii): f gðt1 Þ ¼ f gðt2 Þ, we have X oG
d oG dqi ðt0 Þ ¼ 0 oqi dt oq_ i
i
ð7Þ
When G is taken as Lagrangian L of the holonomic system or non-holonomic system that can be transformed into holonomic system (Mei, 1987), G can be generally written as X G ¼ T0 ðq_ Þ V0 ðqÞ þ Ll ðq; q_ ; tÞ ð8Þ l
Then it follows that X i
(
) oV0 d oT0 X oLl d oLl þ dqi ðt0 Þ ¼ 0 _ dt oqi dt oq_ i oq o q i i l
ð9Þ
Defining mi € qi ¼
d oT0 ; dt oq_ i
Fi ðq; q_ ; € q; tÞ þ Ri ðq; q_ ; € q; tÞ ¼
oV0 X þ oqi l
oLl d oLl oqi dt oq_ i
ð10Þ
where Fi and Ri are general force and constraint force of the ith components of system respectively. Therefore we have X ½Fi ðq; q_ ; € q; tÞ þ Ri ðq; q_ ; € q; tÞ mi € ð11Þ qi dqi ðt0 Þ ¼ 0 i
Using arbitration of dqi , we acquire the general equation with constraint force q; tÞ þ Ri ðq; q_ ; € q; tÞ ¼ mi € qi Fi ðq; q_ ; € As considering ideal constraint X Ri ðq; q_ ; € q; tÞdqi ¼ 0
ð12Þ
ð13Þ
i
it follows that X ½Fi ðq; q_ ; € q; tÞ mi € qi dqi ðt0 Þ ¼ 0 i
ð14Þ
570
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
And we have ðrÞ
dqi ðt0 Þ ¼ dqi ðtÞ þ dq_ i ðtÞDt þ þ dqi ðtÞ 0ðrÞ
Dtr þ r!
ð15Þ
ðrÞ
When qi ðtÞ ¼ qi ðtÞ ðr 2 z; r 6¼ 0Þ, we obtain the formula of DÕAlembert–Lagrange principle (Rosenberg, 1977) X ½Fi ðq; q_ ; € q; tÞ mi € ð16Þ qi dqi ðtÞ ¼ 0 i *
As acceleration is zero, taking qi as * r i , Fi as F i , we get the formula of virtual work principle X* Fi ðq; q_ ; € q; tÞ dr*i ¼ 0
ð17Þ
i 0ðrÞ
ðrÞ
When qi ðtÞ ¼ qi ðtÞ ðr 6¼ 1Þ, we obtain the expression of Jourdian principle (Pars, 1965) X ½Fi ðq; q_ ; € q; tÞ mi € qi dq_ i ðtÞ ¼ 0
ð18Þ
i ðrÞ
Under the conditions q0ðrÞ ðtÞ ¼ qi ðtÞ ðr 6¼ 2Þ, we acquire the expression of Gauss principle (Rosenberg, 1977) X ½Fi ðq; q_ ; € q; tÞ mi € qi ðtÞ ¼ 0 ð19Þ qi d€ i ðrÞ
Generally, when q0ðrÞ ðtÞ ¼ qi ðtÞ ðr 2 z; r 6¼ 8kÞ, we deduce the general expression of general DÕAlembert– Lagrange principle X ðkÞ ½Fi ðq; q_ ; € q; tÞ mi € ð20Þ qi dqi ðtÞ ¼ 0 i
Expression (20) is also called general k-order variational principle. When k takes value from zero to any finite positive integer, Eq. (20) gives different variational principles, thus, for different k, we can define the different variational principle. In the past, these principles are obtained by postulating Eq. (12) existence, ðrÞ multiplying Eq. (12) with dqi ðtÞ and summing subscript i. Only DÕAlembert–Lagrange principle and Gauss principle can be obtained by variational principle and the lest action principle respectively (Pars, 1965; Rosenberg, 1977). Now we generally deduce Eq. (20) by means of the quantitative causal principle, give the unified expressions and their intrinsic relations of all variational principles of differential style. Now we generally consider conservation laws of all variational principles of differential style. In the condition (ii) above, when t1 , t2 and t in given [t01 , t02 ] are arbitrary or the system satisfies Euler– Lagrange equation, we obtain the conservation quantity X oG X oG X oG dqi þ GDt ðt1 Þ ¼ dqi þ GDt ðt2 Þ ¼ dqi þ GDt ðtÞ ¼ const: ð21Þ oq_ i oq_ i oq_ i i i i Expressions (6) and (21) are the conservation quantities of the systems, that is, the conservation quantities of the all variational principles above are derived, and Eq. (21) is invariant at different time t.
4. Noether theorem and Noether conservation charges We now use the conservation quantities to find the Noether conservation charges of the systems. When the systems have the invariant properties under the operations of Lie group D ¼ Dm , we have (Djukic, 1989; Li, 1987)
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572 ðrÞ
ðrÞ
ðrþ1Þ
Dqi ¼ dqi þ qi
ðrÞ
ðrþ1Þ
Dt ¼ dqi þ qi
ðrÞ
er sr ¼ er ðnri Þ ;
r ¼ 0; 1
571
ð22Þ
where
0 _ ot ðqðtÞ; q ðtÞ; t; aÞ
sr ¼
oar
ðnri ÞðrÞ
ðr ¼ 1; 2; . . . ; mÞ
;
ð23Þ
a¼0
0ðrÞ oqi ðq; q_ ; t; aÞ
¼
oar
;
ðr ¼ 1; 2; . . . ; m; r ¼ 0; 1Þ
ð24Þ
a¼0
in which ar ðr ¼ 1; 2; . . . ; mÞ are m linearly independent infinitesimal continuous transformation parameters of Lie group Dm , Eqs. (23) and (24) are infinitesimal generated functions under the transformation of spacetime coordinates about Lie group Dm , er ðr ¼ 1; 2; . . . ; mÞ are infinitesimal parameters corresponding to ar . Using Eqs. (5) and (21)–(24) and er Õs arbitration, we obtain Noether theorem (Djukic, 1989; Li, 1987), and the Noether conservation charges of the all differential variational principles are X oG r ðrÞ ðrþ1Þ ððni Þ qi sr Þ þ Gsr ðtÞ ¼ const: ðr ¼ 1; 2; . . . ; mÞ ð25Þ oq_ i i we thus get the conclusion that the Noether conservation charges of the all differential variational principles are invariant under the operation of Lie group Dm , and Eq. (21) is invariant at different time t. When dG ¼ 0, i.e., the system takes extreme value about time t, the system is simplified, the results of the dt above all researches still keep to be effective except not existing the terms deduced and depended by dG . dt Because QCP is more general, the applications of the principle to all integral variational principle, high order Lagrangian, field theory, the further more researches (e.g., for any high order Lagrangian with constraints etc) about this letter, the other more researches and so on will be written or have been written in the other papers (e.g., see Ref. Huang et al., 2000, 2001, 2002, in press).
5. Summary and conclusions This letter gives a mathematical expression of the quantitative causal principle, presents the unified expressions of DÕAlembert–Lagrange, virtual work, Jourdian, Gauss and general DÕAlembert–Lagrange principles of differential style, first finds the conservation quantities of the all variational principles above, the intrinsic relations among the all differential variational principles are first exposed, and it is first found that the conservation quantities of the all variational principles above, furthermore their Noether conservation charges of the all differential variational principles are first shown up under the operation of Lie group Dm . The above researches make the expressions of the past scrappy differential variational principles be unified into the relative consistent system of the all differential variational principles in terms of the quantitative causal principle, which is essential for researching the intrinsic relations among the past scrappy differential variational principles and their Noether theorems and for further making their logic simplification and clearness.
Acknowledgements Author is grateful to Prof. Z.P. Li and Prof. F.X. Mei for discussions.
572
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
References Cheng, B., 1987. Analytical Dynamics. Beijing University Press, Beijing. p. 346. Chern, S.S., 1989. Vector bundles with a connection. In: Chern, S.S., Chern, W.H. (Eds.), Studies in Global Differential Geometry, Mathematical Association of America. In: Lecture on Differential Geometry. Beijing University Press, Beijing, p. 103. Desloge, E.A., 1982. Classical Mechanics, vol. I, II. John Wiley and Sons, New York. Djukic, D.S., 1989. Int. J. Non-linear Mech. 8, 479. Huang, Y.C., et al., 2000. On Symmetries and Quantitative Causal Principle, Science Times (In Chinese ), 4, 17. Huang, Y.C. et al., 2001. Unified origin of variational principle and interacting local principles in terms of quantitative causal principle. J. Nature 23 (4), 227 (in Chinese). Huang, Y.C. et al., 2002. Phys. Lett. A 299, 644, 306 (2003) 369. Huang Y.C., et al., Introduction to Astrophysics and Cosmology. Science Press, in press. Klein, L., 1961. Dispersion Relations and Abstract Approach to Field Theory. In: International Science Review Series, vol. l. Gordon and Breach Publishers Inc., New York. p. 147. Li, H., 1998. Zhuanfalun. Ost-West-Verlag, Bad Pyrmont. p. 140 (in German). Li, Z.P., 1987. Int. J. Theor. Phys. 26, 853. Mei, F.X., 1987. Studies in Non-holonomic Dynamics. Beijing Institute of Techonology Press, Beijing. p. 226. Mei, F.X., 1991. Advanced Analytical Mechanics. Beijing Institute of Technology Press, Beijing. p. 59. Nash, C., Sen, S., 1983. Topology and Geometry for Physicists. Academic Press, London. p. 174. Pars, L.A., 1965. A Treatise on Analytical Dynamics. Heinemann Educational Books Ltd, London. Rosenberg, R.M., 1977. Analytical Dynamics of Discrete System. Plenum, New York. Sterman, G., 1993. An Introduction to Quantum Field Theory. Cambridge University Press, New York. p. 35.
Mechanics Research Communications 30 (2003) 567–572 www.elsevier.com/locate/mechrescom
Unified expressions of all differential variational principles Y.C. Huang Department of Applied Physics, Beijing University of Technology, Beijing 100022, PR China Received 14 October 2001
Abstract A mathematical expression of the quantitative causal principle is given, using the expression this letter shows the unified expressions of DÕAlembert–Lagrange, virtual work, Jourdian, Gauss and general DÕAlembert–Lagrange principles of differential style, first finds the intrinsic relations among these variational principles, the conservation quantities of the above principles are first found, finally the Noether conservation charges of the all differential variational principles in the systems with the symmetry of Lie group Dm are first obtained. Ó 2003 Published by Elsevier Ltd. Keywords: Euler–Lagrange equation; Variational principle; Causal principle; Conservation quantity; Noether theorem
1. Introduction There are numerous variational principles in physics, they are classified into two kinds, i.e., they are differential and integral variational principles respectively (Pars, 1965; Cheng, 1987; Desloge, 1982). It is well known that the unified expressions of the relations among so many scrappy variational principles have not been presented up to now, and the intrinsic relations among the conservation quantities of these principles are not obtained either (Mei, 1991). It is well known that the conservation quantities of all differential variational principles do not exist in the past (Pars, 1965; Cheng, 1987; Desloge, 1982), now we first give the conservation quantities of all differential variational principles in this letter. On the other hand, in quantum field theory the causal principle demands if the square of the distance of spacetime coordinates of two boson (or fermion) operators is timelike, their commutator (or anticommutator) is not equal to zero, their measures are then coherent, for spacelike no coherent (Sterman, 1993). The dispersion relations can be deduced by the causal principle etc (Klein, 1961), and general scientific theories in physics should satisfy the elementary demand of the causal principle, and the causal relations must be quantitative, or there must be contradictory. Using the following quantitative causal principle, Ref. Huang et al. (2000, 2001, in press) give both the unified expressions of the new and old axiom systems of
E-mail address: [email protected] (Y.C. Huang). 0093-6413/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0093-6413(03)00057-0
568
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
group theories and the applications of the symmetric property to modern astrophysics and cosmology. Therefore, the causal principle is essential to research physical laws.
2. Quantitative causal principle In real physics, the quantitative actions (cause) of some quantities must lead to the corresponding equal effects, which is just a expression of the no-loss-no-gain principle in the universe (Li, 1998; Huang et al., 2001), therefore we have Quantitative causal principle: In any physical system, how much loses (cause), there must be, how much gains (result), or, quantitative actions (cause) of some quantities must result in the corresponding equal effects (result). Then, the principle may be concretely expressed as DG CG ¼ 0
ð1Þ
Eq. (1)Õs physical meanings are that the real physical result produced by any operator set D acting on G must lead to appearance of set C acting on G so that DG is equal to CG, where D and C may be different operator sets, the whole process satisfies the quantitative causal principle (QCP) so that Eq. (1)Õs right hand side satisfies QCP with the no-loss-no-gain characteristics (Huang et al., 2001, in press). Eq. (1) is thus viewed as a mathematical expression of the quantitative causal principle derived from the no-loss-no-gain principle in the universe (Huang et al., 2001). It can be seen that gauge fields of physical fundamental interactions are just connections of principal bundle (Nash and Sen, 1983), the connections are obtained by Eq. (1) when D is a connection operator, G is a section S of the bundle, C is a connection x (Chern, 1989). In fact, the Noether theorem and conservation law of energy transformation are two of a lot of applied examples of the no-loss-no-gain principle in the universe (Huang et al., 2001, in press). And Eq. (1) is useful for the following studies.
3. Unified expressions of all differential variational principles and conservation quantity When G is a functional, C is identity, D is group operator of infinitesimal continuous transformation in Eq. (1) (Djukic, 1989; Li, 1987), the general expression of variational principle of differential style is obtained as follows X DGðqðtÞ; q_ ðtÞ; tÞ GðqðtÞ; q_ ðtÞ; tÞ ¼ ðGqi Dqi þ Gq_ i Dq_ i Þ þ Gt Dt ð2Þ i
where ðrÞ
ðrÞ
ðrþ1Þ
Dqi ¼ dqi ðt0 Þ þ qi
Dt
ð3Þ
Therefore we can have DG ¼
X oG i
oqi
d oG d X oG dG Dt dqi ðt0 Þ þ dqi ðt0 Þ þ dt oq_ i dt i oq_ i dt
ð4Þ
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
When Dt is infinitesimal constant quantity, it follows that ( ) X oG d oG d X oG 0 0 dqi ðt Þ þ GDt ¼ 0 dqi ðt Þ þ oqi dt oq_ i dt oq_ i i i
569
ð5Þ
Namely, in a fixed infinitesimal time gap, the comparisons of the actual variation of the system with the other possible variations make G take limit value, and it is at the limit value that the system satisfies the quantitative causal principle. When the system satisfies the Euler–Lagrange equation, one has the conservation quantity ( ) X oG dqi ðt0 Þ þ GDt ¼ const: ð6Þ oq_ i i On the other hand, when integrating Eq. (5) during ½t1 ; t2 , and taking condition (i): qi jt¼t1 ¼ dqi jt¼t2 ¼ 0; or taking condition (ii): f gðt1 Þ ¼ f gðt2 Þ, we have X oG
d oG dqi ðt0 Þ ¼ 0 oqi dt oq_ i
i
ð7Þ
When G is taken as Lagrangian L of the holonomic system or non-holonomic system that can be transformed into holonomic system (Mei, 1987), G can be generally written as X G ¼ T0 ðq_ Þ V0 ðqÞ þ Ll ðq; q_ ; tÞ ð8Þ l
Then it follows that X i
(
) oV0 d oT0 X oLl d oLl þ dqi ðt0 Þ ¼ 0 _ dt oqi dt oq_ i oq o q i i l
ð9Þ
Defining mi € qi ¼
d oT0 ; dt oq_ i
Fi ðq; q_ ; € q; tÞ þ Ri ðq; q_ ; € q; tÞ ¼
oV0 X þ oqi l
oLl d oLl oqi dt oq_ i
ð10Þ
where Fi and Ri are general force and constraint force of the ith components of system respectively. Therefore we have X ½Fi ðq; q_ ; € q; tÞ þ Ri ðq; q_ ; € q; tÞ mi € ð11Þ qi dqi ðt0 Þ ¼ 0 i
Using arbitration of dqi , we acquire the general equation with constraint force q; tÞ þ Ri ðq; q_ ; € q; tÞ ¼ mi € qi Fi ðq; q_ ; € As considering ideal constraint X Ri ðq; q_ ; € q; tÞdqi ¼ 0
ð12Þ
ð13Þ
i
it follows that X ½Fi ðq; q_ ; € q; tÞ mi € qi dqi ðt0 Þ ¼ 0 i
ð14Þ
570
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
And we have ðrÞ
dqi ðt0 Þ ¼ dqi ðtÞ þ dq_ i ðtÞDt þ þ dqi ðtÞ 0ðrÞ
Dtr þ r!
ð15Þ
ðrÞ
When qi ðtÞ ¼ qi ðtÞ ðr 2 z; r 6¼ 0Þ, we obtain the formula of DÕAlembert–Lagrange principle (Rosenberg, 1977) X ½Fi ðq; q_ ; € q; tÞ mi € ð16Þ qi dqi ðtÞ ¼ 0 i *
As acceleration is zero, taking qi as * r i , Fi as F i , we get the formula of virtual work principle X* Fi ðq; q_ ; € q; tÞ dr*i ¼ 0
ð17Þ
i 0ðrÞ
ðrÞ
When qi ðtÞ ¼ qi ðtÞ ðr 6¼ 1Þ, we obtain the expression of Jourdian principle (Pars, 1965) X ½Fi ðq; q_ ; € q; tÞ mi € qi dq_ i ðtÞ ¼ 0
ð18Þ
i ðrÞ
Under the conditions q0ðrÞ ðtÞ ¼ qi ðtÞ ðr 6¼ 2Þ, we acquire the expression of Gauss principle (Rosenberg, 1977) X ½Fi ðq; q_ ; € q; tÞ mi € qi ðtÞ ¼ 0 ð19Þ qi d€ i ðrÞ
Generally, when q0ðrÞ ðtÞ ¼ qi ðtÞ ðr 2 z; r 6¼ 8kÞ, we deduce the general expression of general DÕAlembert– Lagrange principle X ðkÞ ½Fi ðq; q_ ; € q; tÞ mi € ð20Þ qi dqi ðtÞ ¼ 0 i
Expression (20) is also called general k-order variational principle. When k takes value from zero to any finite positive integer, Eq. (20) gives different variational principles, thus, for different k, we can define the different variational principle. In the past, these principles are obtained by postulating Eq. (12) existence, ðrÞ multiplying Eq. (12) with dqi ðtÞ and summing subscript i. Only DÕAlembert–Lagrange principle and Gauss principle can be obtained by variational principle and the lest action principle respectively (Pars, 1965; Rosenberg, 1977). Now we generally deduce Eq. (20) by means of the quantitative causal principle, give the unified expressions and their intrinsic relations of all variational principles of differential style. Now we generally consider conservation laws of all variational principles of differential style. In the condition (ii) above, when t1 , t2 and t in given [t01 , t02 ] are arbitrary or the system satisfies Euler– Lagrange equation, we obtain the conservation quantity X oG X oG X oG dqi þ GDt ðt1 Þ ¼ dqi þ GDt ðt2 Þ ¼ dqi þ GDt ðtÞ ¼ const: ð21Þ oq_ i oq_ i oq_ i i i i Expressions (6) and (21) are the conservation quantities of the systems, that is, the conservation quantities of the all variational principles above are derived, and Eq. (21) is invariant at different time t.
4. Noether theorem and Noether conservation charges We now use the conservation quantities to find the Noether conservation charges of the systems. When the systems have the invariant properties under the operations of Lie group D ¼ Dm , we have (Djukic, 1989; Li, 1987)
Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572 ðrÞ
ðrÞ
ðrþ1Þ
Dqi ¼ dqi þ qi
ðrÞ
ðrþ1Þ
Dt ¼ dqi þ qi
ðrÞ
er sr ¼ er ðnri Þ ;
r ¼ 0; 1
571
ð22Þ
where
0 _ ot ðqðtÞ; q ðtÞ; t; aÞ
sr ¼
oar
ðnri ÞðrÞ
ðr ¼ 1; 2; . . . ; mÞ
;
ð23Þ
a¼0
0ðrÞ oqi ðq; q_ ; t; aÞ
¼
oar
;
ðr ¼ 1; 2; . . . ; m; r ¼ 0; 1Þ
ð24Þ
a¼0
in which ar ðr ¼ 1; 2; . . . ; mÞ are m linearly independent infinitesimal continuous transformation parameters of Lie group Dm , Eqs. (23) and (24) are infinitesimal generated functions under the transformation of spacetime coordinates about Lie group Dm , er ðr ¼ 1; 2; . . . ; mÞ are infinitesimal parameters corresponding to ar . Using Eqs. (5) and (21)–(24) and er Õs arbitration, we obtain Noether theorem (Djukic, 1989; Li, 1987), and the Noether conservation charges of the all differential variational principles are X oG r ðrÞ ðrþ1Þ ððni Þ qi sr Þ þ Gsr ðtÞ ¼ const: ðr ¼ 1; 2; . . . ; mÞ ð25Þ oq_ i i we thus get the conclusion that the Noether conservation charges of the all differential variational principles are invariant under the operation of Lie group Dm , and Eq. (21) is invariant at different time t. When dG ¼ 0, i.e., the system takes extreme value about time t, the system is simplified, the results of the dt above all researches still keep to be effective except not existing the terms deduced and depended by dG . dt Because QCP is more general, the applications of the principle to all integral variational principle, high order Lagrangian, field theory, the further more researches (e.g., for any high order Lagrangian with constraints etc) about this letter, the other more researches and so on will be written or have been written in the other papers (e.g., see Ref. Huang et al., 2000, 2001, 2002, in press).
5. Summary and conclusions This letter gives a mathematical expression of the quantitative causal principle, presents the unified expressions of DÕAlembert–Lagrange, virtual work, Jourdian, Gauss and general DÕAlembert–Lagrange principles of differential style, first finds the conservation quantities of the all variational principles above, the intrinsic relations among the all differential variational principles are first exposed, and it is first found that the conservation quantities of the all variational principles above, furthermore their Noether conservation charges of the all differential variational principles are first shown up under the operation of Lie group Dm . The above researches make the expressions of the past scrappy differential variational principles be unified into the relative consistent system of the all differential variational principles in terms of the quantitative causal principle, which is essential for researching the intrinsic relations among the past scrappy differential variational principles and their Noether theorems and for further making their logic simplification and clearness.
Acknowledgements Author is grateful to Prof. Z.P. Li and Prof. F.X. Mei for discussions.
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Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572
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