General Variational Principle, General Noether Theorem, Their Classical and Quantum New Physics and Solution to Crisis Deducing All Physics Laws

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

General variational principle, general Noether theorem, their classical and quantum new physics and solution to crisis deducing all physics laws

C. Huang 1,2∗ and Yong-Chang Huang 3† 1Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA 94720, USA 2Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, W.Lafayette, IN 47907-2036, USA 3Institute of Theoretical Physics, Beijing University of Technology, Beijing 100124, China (Dated: October 13, 2020) This paper discovers that current variational principle and Noether theorem for different physics systems with (in)finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether’s proposing her famous theorem, which result in the crisis deducing relevant mathematical laws and all physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text)books. The real physics should be after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical pictures and physics for (in)finite freedom systems are discovered. This paper gives both general variational principle and general Noether theorem as well as their classical and quantum new physics, which would rewrite all relevant current different branches of science, as key tools of studying and processing them. Key words: variational principle, Noether theorem, mathematical physics, fundamental interaction, physics law, unification theory, classical and quantum new physics

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I.Introduction ple in mathematical optimization; (iii) Variational principle in mathematical extremum problems; (iv) Variational A variational principle in science is to enables a prob- principle in mathematical motion equations and invari- lem being solved by using the calculus of variations, ant quantities; (v) The finite element method; ...... In which optimizes the values of these quantities in the vari- physics [2, 10–14]: (i) Fermat’s principle in geometrical ational systems [1]. optics; (ii) Maupertuis’ principle in classical mechanics; Fundamental physics laws can be expressed by a vari- (iii) The principle of least action in mechanics, electro- ational principle, which can give Euler-Lagrange equa- magnetic theory and so on; (iv) The variational method tions and the corresponding convervation quantity [2, 3]. in quantum mechanics; (v) Gauss’s principle of least con- Noether generalized the variational principle to a now straint and variational principle of least curvature; (vi) called Noether theorem by finding the transformation Hilbert’s action principle in general relativity leading to symmetry properties of variational systems and giving Einstein field equations; (vii) Palatini variational princi- both Euler-Lagrange equations and the many conserva- ple; (viii) Variational principle in different field theories; tion quantities depending on the corresponding many ...... In astronmy and astrophysics [15], in chemistry symmetries [4, 5]. [16, 17], even in engineering [9] and so on. Current variational principle and Noether theorem Various variational principles and their application- have been extensively used in different branches of sci- s are very well investigated, e.g., see [18–23]. Role of ence and have become key tools of studying and process- Noether’s Theorem at the Deconfined Quantum Criti- ing the different branches, for examples: cal Point is studied [24], Noether’s theorems and con- In mathematics [1, 6–9]: (i) The extremum method for served currents in gauge theories in the presence of fixed solving boundary-value problems; (ii) Variational princi- fields are explored [25], furthermore, Noether’s theorem and conserved quantities for the crystal- and ligand-field Hamiltonians invariant under continuous rotational symmetry are investigated [26]. ∗Electronic address: [email protected] †Electronic address: [email protected] In current variational principle and Noether theorem,

there are the needs in advance to assume existing some are [27–29] conditions which are equivalent to Euler-Lagrange equations and conservation quantities, and then deducing 0 0 σ Euler-Lagrange equations and conservation quantities, t = t (q, q,˙ q,¨ t, α) = t + ∆t = t + εστ , (2) which are related to a hidden logic cycle problem and are not both exact and natural. 0(r) 0(r) (r) (r) (r) σ (r) Furthermore, we find that all the investigations on qi = qi (q, q,˙ q,¨ t, α) = qi + ∆qi = qi + εσ(ξi ) , variational principle and Noether theorem for differen- (3) t physics systems have missed the key studies on the in which r = 0, 1, 2, α = (α1, α2, ..., αm) are independent double extremum processes related to the general ex- continuous variable parameters of Lie group G and tremum functional that is deduced via the least action 0(r) principle and should be key largely taken in deducing al- σ (r) ∂qi (q, q,˙ q,¨ t, α) (ξi ) = |α=0 , σ = 1, 2, ..., m; r = 0, 1, 2, l the physics laws, but the current variational principle ∂ασ and current Noether theorem have missed the general ex- (4) treme functionals and their minimum extremums for over a century since Noether’s proposing her theorem [4, 5], 0 which result in the crisis of no objectively deducing all the σ ∂t (q, q,˙ q,¨ t, α) τ = |α=0 , σ = 1, 2, ..., m. (5) physics laws. Using the studies on the double extremum ∂ασ processes related to the general extremum functionals in this paper, the crisis and the hidden logic cycle problem Eqs.(4) and (5) are the infinitesimal generating func- are solved, and the new physical pictures are discovered. tions under the operation of group G, εσ (σ = 1,2,. . . ,m) No losing generality, all physics laws always can be ex- are independent infinitesimal parameters corresponding pressed as some equations, these equations always can to α, one dot and two dots denote the first and second be viewed as some Euler-Lagrange equations, the Euler- order time derivatives respectively, the curve q(t) is pa- Lagrange equations always can be deduced by the gen- rameterized by time, and the path takes extremum cor- eral variational principle and/or Noether theorem [4, 5]. responding ∆A = 0. Especially, the four fundamental interaction theories in Doing as the well-known Refs. [2, 9, 13, 27, 29], we the universe, i.e., the strong, weak, electromagnetic and define granvitational interaction theories, are directly duduced by variational principle and Noether theorem. Therefore, σ 0 0 0 0 0 0 0 0 0 dΩ there always is the crisis deducing all the physics laws. L (q , q˙ , q¨ , t ) = L(q , q˙ , q¨ , t )+εσ , σ = 1, 2, ..., m. dt This paper wants to solve the crisis. (6) The arrangements of this paper are: Sect. 2 shows Putting Eq.(6) into Eq.(1), one has unification studies on variational principle and Noether 0 theorem for finite freedom systems; Sect. 3 investigates Z t2 σ Z t2 0 0 0 0 dΩ 0 crisis of deducing physics laws and its solution to the cri- ∆A = [L(q , q˙ , q¨ , t )+εσ ]dt − L(q, q,˙ q,¨ t)dt = 0. t0 dt t sis for finite freedom systems; Sect. 4 gives unification 1 1 studies on variational principle and Noether theorem for (7) infinite freedom systems; Sect. 5 studies crisis of deduc- Using the technique of deducing Euler-Lagrange equa- ing physics laws and its solution to the crisis for infinite tions to simplify Eq.(7) and neglecting second-order in- freedom systems; Sect. 6 shows discussions and applica- finitesimal quantities, we get tions; Sect. 7 gives summary and conclusions. Z t2 dΩ X ∂L d ∂L d2 ∂L II.Unification studies on variational principle ∆A = [ + [ − + ]δq + dt ∂q dt ∂q˙ dt2 ∂q¨ i and Noether theorem for finite freedom systems t1 i i i i The exact mathematical descriptions of the least action principle for a general case are: the variation of the d X ∂L ∂L d ∂L integral ( i.e., the action ) of the Lagrangian L during [ ( δq + δq˙ − δq ) + L∆(t)]]dt. (8) dt ∂q˙ i ∂q¨ i dt ∂q¨ i [t1, t2] about N generalized coordinates q = (q1, q2,...,qN ) i i i i is [13, 27] σ where Ω = εσΩ . Eq.(8) is simplified as

0 Z t2 Z t2 0 0 0 0 0 0 0 ∆A = A −A = L (q , q˙ , q¨ , t )dt − L(q, q,˙ q,¨ t)dt = 0. Z t2 ∂L d ∂L d2 ∂L 0 X t t1 ∆A = 0 = { [ − + ]δq + 1 ∂q dt ∂q˙ dt2 ∂q¨ i (1) t1 i i i i It is no losing the generality, because the results of the systems with higher derivatives of q are the similar but d X ∂L ∂L d ∂L more terms relevant to higher derivatives of q. [ ( δq + δq˙ − δq )+L∆(t)+Ω]}dt. (9) dt ∂q˙ i ∂q¨ i dt ∂q¨ i Among them, the general inﬁnitesimal transformations i i i i Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

For Eq.(9), about the degree of freedom, there are still three different cases: Z t2 X ∂L d ∂L d2 ∂L Z t2 d Case (i): When assuming [ − + ]δq dt = − [ ∂q dt ∂q˙ dt2 ∂q¨ i dt t1 i i i i t1 Z t2 d X ∂L ∂L d ∂L ∆A = [ ( δq + δq˙ − δq )+L∆(t)+Ω]dt, dt ∂q˙ i ∂q¨ i dt ∂q¨ i X ∂L ∂L d ∂L t1 i i i ( δq + δq˙ − δq ) + L∆(t) + Ω]dt. (16) i ∂q˙ i ∂q¨ i dt ∂q¨ i (10) i i i i using Eq.(10), one has Eq.(16) comes from the general systems’ taking extremum of the Lagrangian, but when the system has no ∂L d ∂L d2 ∂L Eq.(10) or Eq.(11), or no Eqs.(10) and (11), then the − + 2 = 0, (11) ∂qi dt ∂q˙i dt ∂qï systems cannot give Euler-Lagrange equations and the corresponding conservation quantities. Namely, this case where i = 1, 2,...,N, because δqi (i =1,2,. . . ,N) are lin- cannot give real physics laws, which is just the reason ear independent each other because qi are independent that current variational principle and current Noether coordinates. theorem have missed the case (iii) [27–29]. Using Eq.(10), we deduce conservation quantity Cases (i) and (ii) are necessary and sufficient conditions that just give real physics laws, and accordint to current variational principle and current Noether theorem [28, X ∂L ∂L d ∂L ( δq + δq˙ − δq ) + L∆(t) + Ω = const., 29], case (iii) at all cannot give real physics laws. ∂q˙ i ∂q¨ i dt ∂q¨ i i i i i Using Eq.(16) derived from the variational extremum, (12) we can exactly define a general extremum functional because

(r) (r) (r+1) Z t2 2 Z t2 ∆qi = δqi (t) + qi (t)∆t, r = 0, 1, 2, (13) X ∂L d ∂L d ∂L d F = [ − + 2 ]δqidt = − [ t ∂qi dt ∂q˙i dt ∂q¨i t dt Eq.(12) can be rewritten as 1 i 1

X ∂L ∂L d ∂L X ∂L ∂L ( δqi + δq˙i − δqi) + L∆(t) + Ω]dt. (17) ( (∆qi − q˙i∆t) + (∆q ˙i − q¨i∆t) ∂q˙ ∂q¨ dt ∂q¨ ∂q˙ ∂q¨ i i i i i i i d ∂L The new general equal equation functional F between − (∆qi − q˙i∆t)) + L∆(t) + Ω = const. (14) dt ∂q¨i the functional F1 of deducing Euler-Lagrange equation- s having merged like terms relevant to Euler-Lagrange Eq.(14) is the result of variational principle. equations and the functional F2 of deducing the gener- Putting Eqs.(2) and (3) into Eq.(14), we deduce m al conservation quantities having merged like terms rel- conservation quantities of the systems evant to the general conservation quantities is deduced

. by satisfying variational principle, i.e., F = F1 = −F2, X ∂L σ σ ∂L σ σ ( (ξ − q˙ τ ) + (ξ − q¨ τ ) namely, F1 + F2 = 0, which just shows the variational ∂q˙ i i ∂q¨ i i i i i extreme value, but these still cannot give real physics, d ∂L see the studies below, these are very important classical − (ξσ − q˙ τ σ) + Lτ σ + Ωσ = constσ. (15) dt ∂q¨ i i and quantum new physics processes of general physics i systems, because this Lagrangian is a general classical or where we have used that εσ(σ = 1, 2, . . . , m) are inde- quantum Lagrangian. pendent infinitesimal parameters. Namely, eq.(15) is the When the absolute value of the general extremum func- Noetther theorem’s result. tional F is taken as zero, because the minimum absolute We can see that both variational principle and Noether value of any functional is zero, i.e., a general extremum ( theorem all give the same Euler-Lagrange equations (11), because the general extremum functional F may gener- but they give the convervation quantities are very differ- ally take a lot of different values, e.g., arbitrary positive ent, i.e., Eq.(14) and Eq.(15) respectively. and/or negative values ), then we generally have Case (ii): When assuming that there exists Eq.(11), then putting Eq.(11) into Eq.(9), one has Eq.(10). In the Z t2 X ∂L d ∂L d2 ∂L following, there are the almost same discussions below [ − + ]δq dt = 0 ∂q dt ∂q˙ dt2 ∂q¨ i Eq.(11) in Case (i). t1 i i i i III.Crisis of deducing physics laws and its solution to the crisis for finite freedom systems Z t2 X d X ∂L ∂L Case (iii): Using Eq.(9) and the rule of merging like = − [ ( δq + δq˙ dt ∂q˙ i ∂q¨ i terms, we exactly have a general functional expression t1 i i i i Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

d ∂L tremum functional F in this paper, the crisis and the − δqi) + L∆(t) + Ω]dt = 0 (18) dt ∂qï hidden logic cycle are not only solved, but also the new mathematical and physical double extremum processes thus the first line of Eq.(18) is equivalent to case (ii), and their new mathematical and physical pictures are and the sum of the second and third lines of Eq.(18) are discovered. Therefore, the general variantional principle equivalent to case (i), all these can give physics laws. and the general Noether theorem for finite freedom sys- Namely, the general extremum functional F takes the tems are given, which solve the crisis and the hidden logic minimum absolute value, i.e., zero, all the physics laws cycle. can be deduced, otherwise, all the physics laws cannot be IV.Unification studies on variational principle deduced. That is, Eq.(17) is deduced from the variational and Noether theorem for infinite freedom systems extremum, Eq.(18) is further taking the absolute extreme value zero, i.e., the minimum absolute extremum, of the general extremum functional F . Therefore, we, for the For general field variables X(x) = {Ψ(x), ϕ(x), first time, discover that it is the double extreme values ωµ(x), gµν (x), ..., } , the exact mathematical description- (i.e., the extreme functional F ’s extremum) that result s of the least action principle for a general case are: in that all the physics laws can be deduced, otherwise, the variation of the action about N field components all the physics laws cannot be deduced. These are very X = (X1,X2,...,XN ) is important classical and quantum new physics processes 0 Z x2 of general physics systems. 0 0 0 0 0 0 0 0 0 0 0 0 04 Therefore, the systems first choose extreme value (i.e., ∆A = A − A = L (X (x ), ∂αX (x ), ∂α∂βX (x ), x )dx x0 via Eq.(1)) of the Lagrangian, and then we naturally d- 1 Z x2 4 educe Eq.(9), there are needs as usual in advance to as- − L((X(x), ∂αX(x), ∂α∂βX(x), x)dx = 0. sume existing case (i) or (ii), because which are equiva- x1 lent to Euler-Lagrange equations and conservation quan- (19) tities, and then deducing Euler-Lagrange equations and conservation quantities, which are related to a hidden in which the general infinitesimal transformations are [28, logic cycle and are not both exact and natural. 29] Actually, there naturally exists the general extremum functional F so that we can choose the absolute extreme 0µ µ µ µ µσ value zero of the general extremum functional F , then x = x +∆x = x +εσ(x)τ (x, X(x), ∂αX(x), ∂α∂βX(x)), case (i) or (ii) can be naturally deduced ( e.g., see the s- (20) tudies below Eq.(18) ). Making these natural deductions 0α 0 a aσ reflects the systems’ intrinsical properties, namely, the in- X (x ) = X (x)+εσ(x)ξ (x, X(x), ∂µX(x), ∂µ∂ν X(x)) trinsical mathematical and physical double extreme value (21) 0α 0 α α procceses. Otherwise, the systems cannot get real physi- where X (x ) = X (x) + ∆X (x), ω = (ω1, ω2, ..., ωm) cal laws. These results are not only supplementary devel- are independent continuous variable parameters of Lie opments of the current variational principle and current group G and Noether theorem but also classsical and quantum new physics corresponding to classical and quantum physics systems, because this Lagrangian is a general Lagrangian. ∂xµ(x, X(x), ∂ X(x), ∂ ∂ X(x), ω) τ µσ = µ µ ν | , (22) For all times, both the Lagrangian and the action con- ωσ =0 ∂ωσ tain the systems’ dynamics, and the real appearance case is that the path taken by the systems during [t1, t2] takes extreme value corresponding ∆A = 0, which means that α aσ ∂X (x, X(x), ∂αX(x), ∂α∂βX(x), ω) the systems can not only choose but also make the least ξ = |ωσ =0 , extremum choice and further choose the minimum abso- ∂ωσ (23) lute extremum of the general extremum functional F . where τ µσ and ξaσ(σ = 1, 2, ..., m) are infinitesimal trans- We discover that, up to now, all the investigations on formation functions. variational principle and Noether theorem for different Eqs.(22) and (23) are the infinitesimal generating func- physics systems have missed the key studies on the dou- tion under the operation of group G, ε (σ = 1,2,. . . ,m) ble extremum processes related to the general extremum σ are independent infinitesimal parameters corresponding functional F that is deduced via the least action princi- to ω. ple and should be key largely taken in deducing all the physics laws, but the current variational principle and Without loss of generality, we define current Noether theorem have neglected the general extreme function F and F ’s minimum extremum, which 0 0 0 0 0 0 0 0 0 0 0 0 0 0 results in the crisis and the hidden logic cycle of no ob- L (X (x ), ∂αX (x ), ∂α∂βX (x ), x ) = L(X (x ), ∂αX (x ), 0 0 0 0 0 σ jectively deducing all physics laws. Using the studies on ∂α∂βX (x ), x ) + εσ∂µΩ (X(x), ∂αX(x), ∂α∂βX(x), x), the double extremum processes related to the general ex- (24) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

σ where σ = 1, 2, ..., m. Putting Eq.(24) into Eq.(19), one in which Ω = εσΩ (σ = 1, 2, ..., m) is one order in- has ﬁnitesimal quantity. Eq.(27) cannot directly give Euler-

0 Lagrange equations, and there are some additional de- Z x2 0 0 0 0 0 0 0 0 0 0 σ grees of freedom. ∆A = [L(X (x ), ∂αX (x ), ∂α∂βX (x ), x )+εσ∂µΩ (X( 0 For Eq.(27)), about the degree of freedom, there are x1 still three cases: Case (I): When assuming x), ∂ X(x), ∂ ∂ X(x), x) − L(X(x0), ∂0 X(x0), ∂0 ∂0 X(x0), x0) α α β α α β Z Z x2 ∂L ∂L a 0 0 0 0 0 0 0 04 ∂µ[( a − ∂ν a )δX + L(X(x ), ∂αX(x ), ∂α∂βX(x ), x )]dx − L(X(x), M 4 ∂X ,µ ∂X ,µν x1 0 Z x2 4 0 0 0 0 0 ∂αX(x), ∂α∂βX(x), x)dx = [δ(L(X (x ), ∂αX (x ), ∂L x0 a µ µ 4 1 + a δX ,ν + L∆x + Ω ]}d x = 0, (28) ∂X ,µν

0 0 0 0 0 σ using Eq.(28), one has ∂α∂βX (x ), x ) + εσ∂µΩ (X(x), ∂αX(x), ∂α∂βX(x), x) + Z x2 DL µ 04 L + µ ∆x ]dx − L(X(x), ∂αX(x), ∂α∂βX(x), ∂L ∂L ∂L Dx x 1 a − ∂µ a + ∂µ∂ν a = 0 (29) ∂X ∂X ,µ ∂X ,µν

a Z x2 ∂L ∂L because δX (a = 1, 2,...,N) are independent each oth- x)dx4 = [( δXa + δ∂ Xa+ ∂Xa ∂X,a ν er. Eq.(29) are just the usual Euler-Lagrange equations. x1 ν Using Eq.(28), we deduce a general continuous equation Z x2 ∂L a 4 σ DL µ a δ∂ν ∂ρX )dx + [εσ∂µΩ + L + µ ∆x ]( ∂X, x Dx νρ 1 µ ∂L ∂L a ∂µJ = 0 = ∂µ[( a − ∂ν a )δX + ∂X ,µ ∂X ,µν

β Z x2 ∂L a µ µ ∂∆x 4 4 a δX ,ν +L∆x + Ω ] (30) 1 + β )dx − L(X(x), ∂αX(x), ∂α∂βX(x), x)dx ∂X ,µν ∂x x1 (25) and its general conservation current where DL/Dxµ is whole derivative for the whole La- grangian. Using the technique of deducing Euler- µ ∂L ∂L a ∂L a µ µ Lagrange equations to make Eq.(25) into order and ne- J = ( a −∂ν a )δX + a δX ,ν +L∆x +Ω ∂X ,µ ∂X ,µν ∂X ,µν glecting two order infinitesimal quantities, we get (31) Z Because σ ∂L ∆A = {εσ∂µΩ (X(x), ∂αX(x), ∂α∂βX(x), x)+[ a M 4 ∂X a a a ν a a a ν δX = ∆X − X ,ν ∆x ; δX,β = ∆X,β −X,βν ∆x , (32) Eqs. (30) and (31) can be rewritten as ∂L ∂L a ∂L − ∂µ a + ∂µ∂ν a ]δX + ∂µ[( a ∂X ,µ ∂X ,µν ∂X ,µ ∂L ∂L 0 ∂L a ∂L a µ 4 µ a a ν ∂µJ = ∂µ[( − ∂ν )(∆X − X ,ν0 ∆x )+ − ∂ν a )δX + a δX ,ν + L∆x ]}d x (26) a a ∂X ,µν ∂X ,µν ∂X ,µ ∂X ,µν ∂L a a ν0 µ µ 0 Omitting the higher order infinitesimal quantities, a (∆X,ν −X ,νν ∆x ) + L∆x + Ω ] = 0 ∂X ,µν Eq.(26) is simplified as (33)

Z ∂L ∂L ∂L and a general conservation current ∆A = 0 = {[ a − ∂µ a + ∂µ∂ν a M 4 ∂X ∂X ,µ ∂X ,µν µ ∂L ∂L a a ν0 J = ( − ∂ )(∆X − X , 0 ∆x )+ ∂Xa, ν ∂Xa, ν ∂L ∂L µ µν ]δXa + ∂ [( − ∂ )δXa µ a ν a ∂L a a ν0 µ µ ∂X ,µ ∂X ,µν 0 a (∆X,ν −X ,νν ∆x ) + L∆x + Ω (34) ∂X ,µν ∂L a µ µ 4 + a δX ,ν + L∆x + Ω ]}d x (27) ∂X ,µν Namely, eq.(34) is the variational principle’s result. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

Using Eqs.(20) and (21), we achieve m continuos equa- Eq.(39) comes from the general systems’ taking ex- tions and their conservative currents tremum of the Lagrangian, but when the systems have no Eq.(28) or Eq.(29), or no Eqs.(28) and (29), then the systems cannot give Euler-Lagrange equations and the µσ ∂L ∂L aσ a ν0σ 0 corresponding conservation quantities. Namely, this case ∂µJ = ∂µ[( a − ∂ν a )(ξ − X ,ν τ )+ ∂X ,µ ∂X ,µν cannot give real physics laws, which is just the reason ∂L aσ a ν0σ µσ µσ that current variational principle and current Noether 0 a (ξ,ν −X ,νν τ ) + Lτ + Ω ] = 0 ∂X ,µν theorem have missed the case (III) [28, 29]. (35) Cases (I) and (II) are necessary and sufficient conditions that just give real physics laws, and accordint to current variational principle and current Noether theorem [28, 29], case (III) at all cannot give real physics µσ ∂L ∂L aσ a ν0σ 0 J = ( a − ∂ν a )(ξ − X ,ν τ )+ laws. ∂X ,µ ∂X ,µν Using Eq.(39) derived from the variational extremum, ∂L aσ a ν0σ µσ µσ 0 we can exactly define a general extremum functional a (ξ,ν −X ,νν τ ) + Lτ + Ω (36) ∂X ,µν Z ∂L ∂L ∂L G = [ − ∂ + ∂ ∂ ]δXad4x where we have used that ε (σ = 1, 2, . . . , m) are inde- a µ a µ ν a σ M 4 ∂X ∂X ,µ ∂X ,µν pendent infinitesimal parameters. Namely, eq.(36) is the Z ∂L ∂L Noetther theorem’s result. a = − ∂µ[( a − ∂ν a )δX R 0 M 4 ∂X ,µ ∂X ,µν Using Eqs.(34) and (36) and M 3 ∂0J dV = R i i − M 2 J dSi → 0, ( Si → ∞,J → 0), we achieve conservation charges of variational principle and ∂L a µ µ 4 + a δX ,ν + L∆x + Ω ]d x (40) Noether theorem, respectively ∂X ,µν The new general equal equation functional G between Z ∂L ∂L a a ν0 0 the functional G1 of deducing Euler-Lagrange equation- Qvp = [( a − ∂ν a )(∆X − X ,ν ∆x )+ M 3 ∂X ,0 ∂X ,0ν s having merged like terms relevant to Euler-Lagrange equations and the functional G2 of deducing the general conservation quantities having merged like terms rel- ∂L a a ν0 0 0 0 evant to the general conservation quantities is deduced a (∆X,ν −X ,νν ∆x ) + L∆x + Ω ]dV, (37) ∂X ,0ν by satisfying variational principle, i.e., G = G1 = −G2, namely, G1 + G2 = 0, which just shows the variational Z extreme value, but these still cannot give real physics, σ ∂L ∂L aσ a ν0σ 0 see the studies below, these are very important classical QNt = [( a − ∂ν a )(ξ − X ,ν τ ) + M 3 ∂X ,0 ∂X ,0ν and quantum new physics processes of general physics ∂L aσ a ν0σ 0σ 0σ systems, because this Lagrangian is a general classical or 0 a (ξ,ν −X ,νν τ ) + Lτ + Ω dV,(38) ∂X ,0ν quantum Lagrangian. When the absolute value of the general extremum func- where σ = 1, 2, . . . , m. We can see that both variational tion G is taken as zero, because the minimum absolute principle and Noether theorem all give the same Euler- value of any function is zero, i.e., a general extremum ( Lagrange equations (29), but they give the convervation because the general extremum functional G may gener- currents (charges) are very different, i.e., Eq.(34) and ally take a lot of different values, e.g., arbitrary positive Eq.(36) (Eq.(37) and Eq.(38)) respectively. and/or negative values ), then we generally have Case (II): When assuming that there are Eq.(29), then Z putting Eq.(29) into Eq.(27), one has Eq.(28). In the fol- ∂L ∂L ∂L a 4 [ a − ∂µ a + ∂µ∂ν a ]δX d x = 0 lowing, there the almost same discussions below Eq.(29) M 4 ∂X ∂X ,µ ∂X ,µν in Case (I). Z ∂L ∂L a V.Crisis of deducing physics laws and its solu- = − ∂µ[( a − ∂ν a )δX 4 ∂X , ∂X , tion to the crisis for infinite freedom systems M µ µν Case (III): Using Eq.(27), we generally have ∂L Z ∂L ∂L ∂L Z + δXa, + L∆xµ + Ωµ]d4x = 0 (41) a 4 ∂Xa, ν {[ a −∂µ a +∂µ∂ν a ]δX d x = − ∂µ[( µν M 4 ∂X ∂X ,µ ∂X ,µν M 4 thus the first line of Eq.(41) is equivalent to case (II), and the sum of the second and third lines of Eq.(41) ∂L ∂L a ∂L a µ µ 4 are equivalent to case (I), these all can give physics laws. a −∂ν a )δX + a δX ,ν +L∆x +Ω ]}d x ∂X ,µ ∂X ,µν ∂X ,µν Namely, the general extremum function G takes the min- (39) imum absolute value, i.e., zero, all the physics laws can Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

be deduced. Otherwise, all the physics laws cannot be d- educed. That is, Eq.(39) is deduced from the variational X ∂L d ∂L d2 ∂L extremum, Eq.(41) is further taking the absolute extreme f = [ − + ]δq = ∂q dt ∂q˙ dt2 ∂q¨ i value zero, i.e., the minimum absolute extremum, of the i i i i general extremum functional G, therefore, we, for the first time, discover that it is the double extreme values (i.e., the extreme functional G’s extremum) that result d X ∂L ∂L d ∂L in that all the physics laws can be deduced, otherwise, − [ ( δqi + δq˙i − δqi)+L∆(t)+Ω]. (42) dt ∂q˙i ∂qï dt ∂qï all the physics laws cannot be deduced. These are very i important classical and quantum new physics processes where f can take any functional value and F = R t2 fdt. of general physics systems. t1 Therefore, the systems first choose extreme value (i.e., When the absolute value of the general extremum func- via Eq.(19)) of the Lagrangian, and then we naturally tional f is taken as zero, namely, taking the minimum deduce Eq.(27), there are needs as usual in advance to absolute extreme value of the general extremum func- assume existing case (I) or (II), because which are e- tional f, i.e., the general extremum functional f’s ex- quivalent to Euler-Lagrange equations and conservation tremum, that is, the double extremum process, Eq.(42) quantities, and then deducing Euler-Lagrange equations can directly deduce Euler-Lagrange equations due to the and conservation quantities, which are related to a hid- linear independent properties of δqi and the general con- den logic cycle and are not both exact and natural. servation quantity due to having taken the second line of Actually, there naturally exists the general extremum Eq.(42) as zero. functional G so that we can choose the absolute extreme Using Eq.(40) derived from the variational extremum, value zero of the general extremum functional G, then we deduce a general extremum functional expression for case (I) or (II) can be naturally deduced ( e.g., see the infinite freedom systems studies below Eq.(41) ). Making these natural deductions reflects the systems’ intrinsical properties, namely, ∂L ∂L ∂L a the intrinsical mathematical and physical double extreme g = [ − ∂µ + ∂µ∂ν ]δX = ∂Xa ∂Xa, ∂Xa, value procceses. Otherwise, the systems cannot get real µ µν physical laws. These results are not only supplementary developments of the current variational principle and cur- ∂L ∂L a ∂L a µ µ rent Noether theorem for infinite freedom systems, but −∂µ[( a −∂ν a )δX + a δX ,ν +L∆x +Ω ] ∂X ,µ ∂X ,µν ∂X ,µν also classsical and quantum new physics corresponding (43) to classical and quantum physics systems, because this R 4 where g can take any functional value and G = M 4 gd x. Lagrangian is a general Lagrangian. When the absolute value of the general extremum func- We discover that, up to now, all the investigations on tional g is taken as zero, namely, taking the minimum ab- variational principle and Noether theorem for different solute extreme value of the general extremum functional physics systems and infinite freedom systems have missed g, i.e., the general extremum functional g’s extremum, the key studies on the double extremum processes relat- that is, the double extremum process, Eq.(43) can di- ed to the general extremum functional G that both is rectly deduce Euler-Lagrange equations due to the linear deduced via the least action principle and should be key independent properties of δXa and the general conser- largely taken in deducing all the physics laws, but the vation current due to having taken the second line of current variational principle and current Noether theo- Eq.(43) as zero. rem for infinite freedom systems have missed the gen- Therefore, this paper discovers that the processes no eral extreme functional G and G’s minimum extremum, choosing the minimum absolute extremum zero of the which results in the crisis and the hidden logic cycle of no general extremum functional F (G) statisfying the varia- objectively deducing all physics laws. Using the studies tional extremum principle are still the virtual processes, on the double extremum processes related to the general because all current refererenes, e.g., refs.[4, 5],[2, 10–14], extremum functional G in this paper, the crisis and the think of cases (iii) and (III) satisfying the variational hidden logic cycle are not only solved, but also the new extreme value cannot derive out Euler-Lagrange equa- mathematical and physical double extremum processes tions and their correspoonding conservation quantities. and their new mathematical and physical pictures are Thus for choosing the processes of minimam absolute ex- discovered. Therefore, general variantional principle and tremum zero of the general extremum functional F (G), general Noether theorem for infinite freedom systems are the processes of the physics systems are just real physic- given, which solve the crisis and the hidden logic cycle. s processes and can give Euler-Lagrange eqautions and VI. Discussions and applications their corresponding conservation quantities. Especially, cases (i) and (ii) ((I) and (II)) are the two special taken Using Eq.(17) derived from the variational extremum, value cases and are included in case (iii) ((III)) as special we have a general extremum functional expression for cases, and there is the hidden logic cycle between case finite freedom systems (i) (assuming to exist Eq.(10) of deducing conservation Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

quantity, then putting Eq.(10) into Eq.(9), one can d- s and their corresponding conservation quantities, which educe Euler-Lagrange Eq.(11)) and case (ii) (assuming are the key new physics. to exist Euler-Lagrange Eq.(11), then putting Eq.(11) Especially, cases (i) and (ii) ((I) and (II)) are included into Eq.(9), one can deduce Eq.(10) of deducing conser- in case (iii) ((III)) as special cases of case (iii) ((III)), vation quantity), namely, cases (i) and (ii) are equiv- and there is the hidden logic cycle between case (i) and alent with each other, which means that one assumes case (ii) ((I) and (II)), namely, cases (i) and (ii) ((I) and Euler-Lagrange equations in case (ii), and then he final- (II)) are equivalent with each other, which means that ly deduces Euler-Lagrange equations in case (i) via the one assumes Euler-Lagrange equations, and then he fi- equivalent relation between cases (i) and (ii) in the w- nally deduces Euler-Lagrange equations via the equiva- hole processes, which is just the hidden logic cycle, so lent relation between cases (i) and (ii) ((I) and (II)) in does Eq.(10) of deducing conservation quantity (similar the whole processes, which is just the hidden logic cycle, for cases (I) and (II)). Especially, from this paper it can so does Eq.(10) of deducing conservation quantity. be seen that the current investigations about cases (i-iii) This paper corrects the current key mistakes that when ((I-III)) in all current references, e.g., refs.[4, 5],[2, 10– physics systems choose the variational extreme values, 14], are no the exact general investigations. the appearing processes of the physics systems are re- Therefore, this paper corrects the current key mistake al physics processes, otherwise, are virtual processes in concepts that when physics systems choose the variation- all current articles, reviews and (text)books. The real al extreme values, the appearing processes of the physics physics should be what after choosing the variational ex- systems are real physics processes, otherwise, are virtual treme values of physics systems, the general extremum processes in all current articles, reviews and (text)books, functional F (G) of the physics systems needs to further e.g., [4, 5],[2, 10–14]. The real physics should be what choose the minimum absolute extremum zero of the gen- after choosing the variational extreme values of physics eral extremum functional F (G), otherwise, the appear- systems, the general extremum functional F (G) of the ing processes of physics systems are still virtual processes physics systems needs to further choose the minimum ab- because the virtual process case (iii) (III) cannot deduce solute extremum zero of the general extremum functional Euler-Lagrange equations and their corresponding con- F (G), otherwise, the appearing processes of physics sys- servation quantities. tems are still virtual processes because the virtual process The systems first choose extreme value, and then must cases cannot deduce Euler-Lagrange equations and their choose the minimum absolute extremum of the general corresponding conservation quantities. extremum functional F (G), then cases (i) or (ii) (cas- All the investigations on functionals F and G in this es (I) or (II)) can be naturally deduced. Making these paper give the corresponding integral descriptions, using deductions shows the systems’ intrinsical properties of functional f and g we can give the corresponding dif- taking double extreme values, otherwise cannot get re- ferantial descriptions, the two descriptions are entirely al physical laws according exact deduction logic. These equivalent, thus we don’t repeat more here. results are not only supplementary developments of the VII. Summary and conclutions current variational principle and current Noether theo- People can deduce Euler-Lagrange equations and cor- rem for finite (infinite) freedom system, but also classsi- responding conservation quantities by utilizing the vari- cal and quantum new physics corresponding to classical ational principle and Noether theorem. But we discover and quantum physics systems, because our Lagrangian is the fact that the systems generally have extra intrinsical the most general. freedoms of choice. And if not assuming to exist Eq.(10) This paper discovers, up to now, all the studies on or (11) (Eq.(28) or (29)), then the Lagrange systems can- variational principle and Noether theorem for differen- not give true physical laws. Actually, Eqs.(10) and (11) t physics systems with finite (infinite) freedom systems (Eqs.(28) and (29)) are equivalent to Euler-Lagrange e- have neglected the key studies on the double extremum quations and conservation quantities, and then deduc- processes of the general extremum functional F (G) that ing Euler-Lagrange equations and conservation quanti- both is deduced by the least action principle and is key ties, which are related to a hidden logic cycle and are not largely taken in deducing all the physics laws, but these both exact and natural. have not been done, which result in the crisis of deducing This paper discovers that the processes no choosing relevant mathematical laws and all physics laws. Using the minimum absolute extremum zero of the general ex- the above studies on the double extremum processes of tremum functional F (G) statisfying the variational ex- the general extremum functional F (G) in this paper, i.e., tremum principle are still the virtual processes, because on the double extreme values, the crisis and the hidden all current refererenes think cases (iii) and (III) satisfying logic cycle are not only solved, but also the new math- the variational extreme value cannot derive out Euler- ematical and physical double extremum processes and Lagrange equations and their correspoonding conserva- their new mathematical pictures and physics are discov- tion current. For choosing the processes of minimam ab- ered. Therefore, general variantional principle and gen- solute extremum zero of the general extremum functional eral Noether theorem for (in)finite freedom systems are F (G), the processes of the physics systems are just real given in this paper, which solve both the crisis having physics processes and can give Euler-Lagrange eqaution- existed for over a century since Noether’s proposing her Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 October 2020 doi:10.20944/preprints202008.0334.v3

famous theorem and the hidden logic cycle. orem are the key firm bases in modern physics, and we Therefore, this paper gives general variational princi- just discover new right avenues of research in the estab- ple, general Noether theorem, their classical and quan- lished variational principle & Noether theorem for finite tum new physics and solution to crisis deducing al- (infinite) freedom systems, their applications and so on in l physics laws, opens a new area of research on variational modern sciences, and all the relevant current articles and principle and Noether theorem for finite (infinite) free- (text)books would be rewritten, supplied and updated. dom systems by choosing optima of the double extreme Acknowledgments: The work is supported by NSF values to explain origins of physics laws etc., and will sig- through grants PHY-0805948, DOE through grant DE- nificantly influence and rewrite the research of others in FG02- 91ER40681, National Natural Science Foundation relevant different branches of science, because the least of China (No. 11875081). action principle or variational principle and Noether the- References

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