MECHANICS

Mechanics Research Communications 30 (2003) 567–572

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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 D�Alembert–Lagrange, virtual work, Jourdian, Gauss and general D�Alembert–Lagrange prin-ciples of differential style, first finds the intrinsic relations among these variational principles, the conservation quan-

tities 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.

� 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 wefirst 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 anticom-

mutator) 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: ychuang@bjut.edu.cn (Y.C. Huang).

0093-6413/$ - see front matter � 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 correspondingequal effects (result).

Then, the principle may be concretely expressed as

DG � CG ¼ 0 ð1Þ

Eq. (1)�s physical meanings are that the real physical result produced by any operator set D acting on Gmust lead to appearance of set C acting on G so that DG is equal to CG, where D and C may be differentoperator sets, the whole process satisfies the quantitative causal principle (QCP) so that Eq. (1)�s right handside 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 usefulfor 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 inEq. (1) (Djukic, 1989; Li, 1987), the general expression of variational principle of differential style is ob-

tained as follows

DGðqðtÞ; _qqðtÞ; tÞ � GðqðtÞ; _qqðtÞ; tÞ ¼Xi

ðGqiDqi þ G _qqiD _qqiÞ þ GtDt ð2Þ

where

DqðrÞi ¼ dqðrÞi ðt0Þ þ qðrþ1Þi Dt ð3Þ

Therefore we can have

DG ¼Xi

oGoqi

�� d

dtoGo _qqi

�dqiðt0Þ þ

d

dt

Xi

oGo _qqi

dqiðt0Þ þdGdt

Dt ð4Þ

Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572 569

When Dt is infinitesimal constant quantity, it follows that

Xi

oGoqi

�� d

dtoGo _qqi

�dqiðt0Þ þ

d

dt

Xi

oGo _qqi

dqiðt0Þ(

þ GDt

)¼ 0 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

Xi

oGo _qqi

dqiðt0Þ(

þ GDt

)¼ const: ð6Þ

On the other hand, when integrating Eq. (5) during ½t1; t2�, and taking condition (i): qijt¼t1¼ dqijt¼t2

¼ 0; ortaking condition (ii): f� � �gðt1Þ ¼ f� � �gðt2Þ, we have

Xi

oGoqi

�� d

dtoGo _qqi

�dqiðt0Þ ¼ 0 ð7Þ

When G is taken as Lagrangian L of the holonomic system or non-holonomic system that can be trans-

formed into holonomic system (Mei, 1987), G can be generally written as

G ¼ T0ð _qqÞ � V0ðqÞ þXl

Llðq; _qq; tÞ ð8Þ

Then it follows that

Xi

(� oV0

oqi� d

dtoT0o _qqi

þXl

oLl

oqi

�� d

dtoLl

o _qqi

�)dqiðt0Þ ¼ 0 ð9Þ

Defining

mi€qqi ¼d

dtoT0o _qqi

; Fiðq; _qq; €qq; tÞ þ Riðq; _qq; €qq; tÞ ¼ � oV0oqi

þXl

oLl

oqi

�� d

dtoLl

o _qqi

�ð10Þ

where Fi and Ri are general force and constraint force of the ith components of system respectively.

Ther�eefore we have

Xi

½Fiðq; _qq; €qq; tÞ þ Riðq; _qq; €qq; tÞ � mi€qqi�dqiðt0Þ ¼ 0 ð11Þ

Using arbitration of dqi, we acquire the general equation with constraint force

Fiðq; _qq; €qq; tÞ þ Riðq; _qq; €qq; tÞ ¼ mi€qqi ð12Þ

As considering ideal constraint X

i

Riðq; _qq; €qq; tÞdqi ¼ 0 ð13Þ

it follows that

Xi

½Fiðq; _qq; €qq; tÞ � mi€qqi�dqiðt0Þ ¼ 0 ð14Þ

570 Y.C. Huang / Mechanics Research Communications 30 (2003) 567–572

And we have

dqiðt0Þ ¼ dqiðtÞ þ d _qqiðtÞDt þ � � � þ dqðrÞi ðtÞDtr

r!þ � � � ð15Þ

When q0ðrÞi ðtÞ ¼ qðrÞi ðtÞ ðr 2 z; r 6¼ 0Þ, we obtain the formula of D�Alembert–Lagrange principle (Rosenberg,1977)

X

i

½Fiðq; _qq; €qq; tÞ � mi€qqi�dqiðtÞ ¼ 0 ð16Þ

As acceleration is zero, taking qi as r* i, Fi as F*

i, we get the formula of virtual work principle

Xi

F*

i ðq; _qq; €qq; tÞ � dr*i ¼ 0 ð17Þ

When q0ðrÞi ðtÞ ¼ qðrÞi ðtÞ ðr 6¼ 1Þ, we obtain the expression of Jourdian principle (Pars, 1965)

Xi

½Fiðq; _qq; €qq; tÞ � mi€qqi�d _qqiðtÞ ¼ 0 ð18Þ

Under the conditions q0ðrÞðtÞ ¼ qðrÞi ðtÞ ðr 6¼ 2Þ, we acquire the expression of Gauss principle (Rosenberg,1977)

X

i

½Fiðq; _qq; €qq; tÞ � mi€qqi�d€qqiðtÞ ¼ 0 ð19Þ

Generally, when q0ðrÞðtÞ ¼ qðrÞi ðtÞ ðr 2 z; r 6¼ 8kÞ, we deduce the general expression of general D�Alembert–Lagrange principle

X

i

½Fiðq; _qq; €qq; tÞ � mi€qqi�dqðkÞi ðtÞ ¼ 0 ð20Þ

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 thedifferent variational principle. In the past, these principles are obtained by postulating Eq. (12) existence,

multiplying Eq. (12) with dqðrÞi ðtÞ and summing subscript i. Only D�Alembert–Lagrange principle and Gaussprinciple 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

Xi

oGo _qqi

dqi

�þ GDt

ðt1Þ ¼

Xi

oGo _qqi

dqi

�þ GDt

ðt2Þ ¼

Xi

oGo _qqi

dqi

�þ GDt

ðtÞ ¼ const: ð21Þ

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. Whenthe 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) 567–572 571

DqðrÞi ¼ dqðrÞi þ qðrþ1Þi Dt ¼ dqðrÞi þ qðrþ1Þi ersr ¼ erðnr

i ÞðrÞ; r ¼ 0; 1 ð22Þ

where

sr ¼ ot0ðqðtÞ; _qqðtÞ; t; aÞoar

a¼0

; ðr ¼ 1; 2; . . . ;mÞ ð23Þ

ðnri Þ

ðrÞ ¼ oq0ðrÞi ðq; _qq; t; aÞoar

a¼0

; ðr ¼ 1; 2; . . . ;m; r ¼ 0; 1Þ ð24Þ

in which ar ðr ¼ 1; 2; . . . ;mÞ are m linearly independent infinitesimal continuous transformation parametersof Lie group Dm, Eqs. (23) and (24) are infinitesimal generated functions under the transformation ofspacetime coordinates about Lie group Dm, er ðr ¼ 1; 2; . . . ;mÞ are infinitesimal parameters correspondingto ar. Using Eqs. (5) and (21)–(24) and er�s arbitration, we obtain Noether theorem (Djukic, 1989; Li, 1987),and the Noether conservation charges of the all differential variational principles are

Xi

oGo _qqi

ððnri Þ

ðrÞ�

� qðrþ1Þi srÞ þ Gsr

ðtÞ ¼ const: ðr ¼ 1; 2; . . . ;mÞ ð25Þ

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

dt ¼ 0, i.e., the system takes extreme value about time t, the system is simplified, the results of theabove 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 D�Alembert–Lagrange, virtual work, Jourdian, Gauss and general D�Alembert–Lagrangeprinciples 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 con-servation 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.