Generalized Duality, Hamiltonian Formalism and New Brackets

It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2014
1. Verfasser:	Duplij, S.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2014
Schriftenreihe:	Журнал математической физики, анализа, геометрии
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/106791
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Generalized Duality, Hamiltonian Formalism and New Brackets / S. Duplij // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 2. — С. 189-220. — Бібліогр.: 76 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-106791
record_format	dspace
spelling	irk-123456789-1067912016-10-06T03:02:17Z Generalized Duality, Hamiltonian Formalism and New Brackets Duplij, S. It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dynamics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed(envelope/general) solution of the multidimensional Clairaut equation. The equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given. Показано, что любая сингулярная лагранжева теория: 1) может быть сформулирована без привлечения связей с помощью Клеро-версии гамильтонового формализма; 2) приводит к специальному виду неабелевой калибровочной теории, которая подобна пуассоновой калибровочной теории; 3) может быть сформулирована как многовременная классическая динамика. Обобщение преобразования Лежандра на случай нулевого гессиана проведено с использованием смешанного (обертывающего/общего) решения многомерного уравнения Клеро. Уравнения движения записываются в гамильтоновой форме с помощью введения новых антисимметричных скобок. Отмечено, что любая классическая система с вырожденным лагранжианом эквивалентна многовременной классической динамике. В заключение приведено взаимоотношение представленного формализма и теории связей Дирака. 2014 Article Generalized Duality, Hamiltonian Formalism and New Brackets / S. Duplij // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 2. — С. 189-220. — Бібліогр.: 76 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106791 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dynamics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed(envelope/general) solution of the multidimensional Clairaut equation. The equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given.
format	Article
author	Duplij, S.
spellingShingle	Duplij, S. Generalized Duality, Hamiltonian Formalism and New Brackets Журнал математической физики, анализа, геометрии
author_facet	Duplij, S.
author_sort	Duplij, S.
title	Generalized Duality, Hamiltonian Formalism and New Brackets
title_short	Generalized Duality, Hamiltonian Formalism and New Brackets
title_full	Generalized Duality, Hamiltonian Formalism and New Brackets
title_fullStr	Generalized Duality, Hamiltonian Formalism and New Brackets
title_full_unstemmed	Generalized Duality, Hamiltonian Formalism and New Brackets
title_sort	generalized duality, hamiltonian formalism and new brackets
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/106791
citation_txt	Generalized Duality, Hamiltonian Formalism and New Brackets / S. Duplij // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 2. — С. 189-220. — Бібліогр.: 76 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT duplijs generalizeddualityhamiltonianformalismandnewbrackets
first_indexed	2025-07-07T19:00:44Z
last_indexed	2025-07-07T19:00:44Z
_version_	1837015889544740864
fulltext	Journal of Mathematical Physics, Analysis, Geometry 2014, vol. 10, No. 2, pp. 189–220 Generalized Duality, Hamiltonian Formalism and New Brackets S. Duplij Theory Group, Nuclear Physics Laboratory, V.N. Karazin Kharkiv National University 4 Svoboda Sq., Kharkiv 61022, Ukraine E-mail: sduplij@gmail.com, duplij@math.rutgers.edu URL: http://math.rutgers.edu/˜duplij Received February 28, 2013, revised July 16, 2013 It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dyna- mics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed (envelope/general) solution of the multidimensional Clairaut equation. The equations of motion are written in the Hamilton-like form by in- troducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given. Key words: Dirac constraints, nonabelian gauge theory, degenerate Lagrangian, Hessian, Legendre transform, multidimensional Clairaut equation, gauge freedom, Poisson bracket, many-time dynamics. Mathematics Subject Classification 2010: 37J05, 44A15, 49K20, 70H45. 1. Introduction Nowadays, many fundamental physical models are based on gauge field theories [73, 17]. On the classical level, they are described by a singular (degenerate) Lagrangian, which makes the passage to the Hamiltonian description, which is important for quanti- zation, highly nontrivial and complicated (see, e.g., [66, 59]). A common way to deal with singular theories is the Dirac approach [19] based on ex- tending the phase space and constraints. This treatment of constrained theories has been deeply reviewed, e.g., in lecture notes [74] and books [30, 38]. In spite of its general suc- cess, the Dirac approach has some problems [20, 57, 72] and is not directly applicable in some cases, e.g., for irregular constrained systems (with linearly dependent constraints) c© S. Duplij, 2014 S. Duplij [56, 6] or so-called “pathological examples” [50]. Therefore, it is worthwhile to recon- sider basic ideas of the Hamiltonian formalism in general from another point of view [24, 23]. In the standard approach to nonsingular theories [64, 4], the transition from La- grangian to Hamiltonian description is carried out by using the Legendre transform and then finding the Hamiltonian as an envelope solution of the corresponding Clairaut equa- tion [42, 41]. The main idea of our formulation is the following [25]: for singular theo- ries, instead of the Lagrange multiplier procedure developed by Dirac [19], we construct and solve the corresponding multidimensional Clairaut equation [41]. In this way, we state that the ordinary duality can be generalized to the Clairaut duality [25]. In this paper we use our previous works [24, 25] to construct a self-consistent analog of the canonical (Hamiltonian) formalism and present a general algorithm for describing a Lagrangian system (singular or not) as a set of first-order differential equations without introducing the Lagrange multipliers. From mathematical viewpoint, we extend to the singular dynamical systems the well-known construction of Hamiltonian as a solution of the Clairaut equation developed in [4] for unconstrained systems. To simplify mat- ters, we consider the systems with a finite number of degrees of freedom, use the local coordinates and the clear language of differential equations together with the Clairaut equation theory [42, 41]. Using the fact that for a singular Lagrangian system the Hessian matrix is degene- rate and therefore has the rank less than its size, we separate out the group of “physical” (or regular/non-degenerate) and “non-physical” (degenerate) dynamical variables such that the Hessian matrix of the former is non-degenerate. On the other hand, the Clairaut equation has two kinds of solutions: the general solution and the envelope one [41]. The key idea is to use the envelope solution for “physical” variables and the general solution for “non-physical” ones, and therefore the separation of variables is unavoidable. In this way we obtain a unique analog of the Hamiltonian (called the mixed Hamilton–Clairaut) function which (formally) coincides with the the Hamiltonian function derived by the ge- ometric approach [11, 67] and by the generalized Legendre transformation [13]. Then, using the mixed Hamilton–Clairaut function, we pass from the second-order Lagrange equations of motion to a set of the first-order Hamilton-like equations. The next impor- tant step is to exclude the so-called degenerate “momenta” and introduce the “physical” Hamilton–Clairaut function (which corresponds to the total Dirac Hamiltonian), which allows us to present the equations of motion as a system of differential equations for “physical” coordinates and momenta together with a system of linear equations for un- resolved (“non-physical”) velocities. Different kinds of solutions of this system of linear equations lead to the classification of singular systems which reminds the classifica- tion of constraints but does not coincide with it: the former does not contain analogs of higher constraints because there are no corresponding degenerate “momenta” at all. Some formulations without (primary) constraints were given in [32, 18, 51], and without any constraints but for special (regularizable) kind of Lagrangians, in [47, 46]. 190 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets The “shortened” approach can play an important role for quantization of such com- plicated constrained systems as gauge field theories [43] and gravity [58]. To illustrate the power and simplicity of our method, we consider such examples, as the Cawley Lagrangian [12], which leads to difficulties in the Dirac approach, and the relativistic particle. The above Hamilton-like form of the equations of motion is described in terms of the newly defined antisymmetric brackets. Quantization of brackets can be done by means of the standard methods (see, e.g., [37]) without using Dirac quantization [19]. While analyzing the equations of motion corresponding to “unresolved” velocities, we arrive effectively at a kind of nonabelian gauge theory in the “degenerate” coordi- nate subspace which is similar to the Poisson gauge theory [28]. But in our case partial derivatives and Poisson brackets “live” in different subspaces. We also outline that the Clairaut-type formulation is equivalent to the many-time classical dynamics developed in [21, 48], if “nondynamical” (degenerate) coordinates are treated as additional “times”. Finally, in Appendix, after introduction of “non-dynamical” momenta corresponding La- grange multipliers and respective constraints, we show that the Clairaut-type formulation presented here corresponds to the Dirac approach [19]. 2. The Legendre–Fenchel and Legendre Transforms We start with a brief description of the standard Legendre–Fenchel and Legendre transforms for the theory with nondegenerate Lagrangian [5, 61]. Let L ( qA, vA ) , A = 1, . . . n, be a Lagrangian given by a function of 2n variables (n generalized coordinates qA and n velocities vA = q̇A = dqA/dt) on the configuration space TM , where M is a smooth manifold. We use the indices in the arguments to distinguish different kinds of coordinates (similarly to [69]). For the same reason, we use the summation signs with explicit ranges. Also, we consider the time-independent case for simplicity and conciseness, which will not influence on the main procedure. By the convex approach (see, e.g., [60, 5]), a Hamiltonian H ( qA, pA ) is a dual func- tion on the phase space T∗M (or convex conjugate [61]) to the Lagrangian (in the second set of variables pA) constructed by using the Legendre–Fenchel transform L LegFen 7−→ HFen defined by [27, 60], HFen ( qA, pA ) = sup vA G ( qA, vA, pA ) , (2.1) G ( qA, vA, pA ) = n∑ B=1 pBvB − L ( qA, vA ) . (2.2) Note that this definition is very general and it can be applied to nonconvex [2] and nondifferentiable [70] functions L ( qA, vA ) , which can lead to numerous extended ver- sions of Hamiltonian formalism (see, e.g., [15, 62, 40]). Also, a generalization of con- vex conjugacy can be achieved by substituting in (2.2) the form pAvA by any function Ψ ( pA, vA ) satisfying special conditions [33]. Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 191 S. Duplij In the standard mechanics [36], one usually restricts to convex, smooth and diffe- rentiable Lagrangians (see, e.g., [5, 65]). Then the coordinates qA(t) are treated as fixed (passive with respect to the Legendre transform) parameters, and the velocities vA(t) are assumed to be independent functions of time. According to our assumptions, the supremum (2.1) is attained by finding an ex- tremum point vA = vA extr of the (“pre-Hamiltonian”) function G ( qA, vA, pA ) which leads to the supremum condition pB = ∂L ( qA, vA ) ∂vB ∣∣∣∣∣ vA=vA extr . (2.3) It is commonly assumed (see, e.g., [5, 65, 36]) that the only way to get rid of de- pendence on the velocities vA in the r.h.s. of (2.1) is to resolve (2.3) with respect to velocities and find its solution given by a set of functions vB extr = V B ( qA, pA ) . (2.4) This can be done only in the class of nondegenerate Lagrangians L ( qA, vA ) = Lnondeg ( qA, vA ) (in the second set of variables vA), which is equivalent to the case det ∥∥∥∥∥ ∂2Lnondeg ( qA, vA ) ∂vB∂vC ∥∥∥∥∥ 6= 0. (2.5) Then, substituting vA extr to (2.1), we can obtain the standard Hamiltonian (see, e.g., [5, 36]), H ( qA, pA ) def = G ( qA, vA extr, pA ) = n∑ B=1 pBV B ( qA, pA )− Lnondeg ( qA, V A ( qA, pA )) . (2.6) The passage from the nondegenerate Lagrangian Lnondeg ( qA, vA ) to the Hamilto- nian H ( qA, pA ) is called the Legendre transform (of functions) which will be denoted by Lnondeg Leg7−→ H . By the geometric approach [68, 54, 1], the Legendre transform of the functions Lnondeg Leg7−→ H is tantamount to the Legendre transformation from the configuration space to the phase space Leg : TM → T∗M (or between submanifolds in the presence of constraints [55, 8, 29]). Nevertheless, here we will use local coordinates and the language of differential equations associated with function transforms, in particular the Clairaut equation theory [42, 41]. 192 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets 3. The Legendre–Clairaut Transform The connection between the Legendre transform, convexity and the Clairaut equa- tion has a long story [42, 64] (see also [4]). Here we present an alternative way by applying the supremum condition (2.3) and considering the related multidimensional Clairaut equation proposed in [24]. We differentiate (2.6) by the momenta pA and use the supremum condition (2.3) to get ∂H ( qA, pA ) ∂pB = V B ( qA, pA ) + n∑ C=1  pC − ∂L ( qA, vA ) ∂vC ∣∣∣∣∣ vC=V C(qA,pA)   ∂V C ( qA, pA ) ∂pB = V B ( qA, pA ) , (3.1) which can be called the dual supremum condition (indeed, this gives the first set of the Hamilton equations, see below). Relations (2.3), (2.6) and (3.1) represent a particular case of the Donkin theorem (see, e.g., [36]). Then we substitute (3.1) in (2.6) and obtain H ( qA, pA ) ≡ n∑ B=1 pB ∂H ( qA, pA ) ∂pB − Lnondeg ( qA, ∂H ( qA, pA ) ∂pC ) , (3.2) which contains no manifest dependence on velocities at all. It is important that for nonsingular Lagrangians, relation (3.2) is an identity, which follows from (2.3), (2.6) and (3.1) by our construction. This relation can also be obtained if the geometric approach from [10] is used. Now we make the main step: we consider equation (3.2) by itself (without referring to (2.3), (2.6) and (3.1)) as a definition of a new transform which is a solution of the nonlinear partial differential equation (the multidimensional Clairaut equation) [24, 25] HCl ( qA, λA ) = n∑ B=1 λB ∂HCl ( qA, λA ) ∂λB − L ( qA, ∂HCl ( qA, λA ) ∂λA ) (3.3) in the formal independent variables λA (initially not connected with pA defined by (2.3)) and L ( qA, vA ) , that is, any differentiable smooth function of 2n variables qA, vA, where the coordinates qA play the role of external parameters. It is very important that in (3.3) we do not demand that nondegeneracy condition (2.5) be imposed on L ( qA, vA ) . We call the transform defined by (3.3), L LegCl 7−→ HCl, a Clairaut duality transform (or the Legendre–Clairaut transform) and HCl ( qA, λA ) , a Hamilton–Clairaut function [24, 25]. Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 193 S. Duplij Note that relation (2.3), which is commonly treated as a definition of all dynamical momenta pA, in our approach is the supremum condition for some of the independent variables of the Clairaut duality transform λA. In the differential equation language, λA are independent mathematical variables having no connection with any physical dy- namics. Before solving the Clairaut equation (3.3) and applying supremum condition (2.3), which in our language is λA = pA = ∂L�∂vA, we must notice that the inde- pendent variables λA are not connected with the Lagrangian and therefore cannot be called momenta. The independent variables λA are used to find all possible solutions of the Clairaut Eq. (3.3) for nondegenerate and degenerate Lagrangians L ( qA, vA ) . Only those of λA which will be restricted by supremum condition (2.3) can be interpreted as momenta with the corresponding geometric description in terms of the cotangent space. The difference between the Legendre–Clairaut transform and the Legendre trans- form is crucial for degenerate Lagrangian theories [24]. Specifically, multidimensional Clairaut Eq. (3.3) has solutions even for degenerate Lagrangians L ( qA, vA ) = Ldeg ( qA, vA ) when the Hessian is zero, det ∥∥∥∥∥ ∂2Ldeg ( qA, vA ) ∂vB∂vC ∥∥∥∥∥ = 0. (3.4) In this case, LegCl, the Legendre–Clairaut transform of functions (3.3), is another along with the Legendre–Fenchel transform LegFen counterpart to the ordinary Legen- dre transform (2.6) in the case of degenerate Lagrangians. The Clairaut equation (3.3) always has a solution which is independent of the properties of the Hessian as well as of solving the supremum condition (2.3) with respect to velocities. To find the solutions of (3.3), we differentiate it by λC to obtain n∑ B=1  λB − ∂L ( qA, vA ) ∂vB ∣∣∣∣∣ vB= ∂HCl(qA,λA) ∂λB   ∂2HCl ( qA, λA ) ∂λB∂λC = 0. (3.5) Now we apply the ordinary method of solving the Clairaut equation (see Appendix A). There are two possible solutions of (3.5), the first, in which square brackets vanish (enve- lope solution), and the second, in which double derivative in velocity vanishes (general solution). The l.h.s. of (3.5) is a sum over B and it is quite conceivable that one may vanish for some B and the other vanishes for other B. The physical reason of choos- ing a particular solution is given in Sec. 4. Thus we have two solutions of the Clairaut equation: 1) The envelope solution defined by the first multiplier in (3.5) being zero, that is, λB = pB = ∂L ( qA, vA ) ∂vB , (3.6) 194 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets which coincides with supremum condition (2.3), together with (3.1). In this way, we obtain the standard Hamiltonian (2.6), HCl env ( qA, λA ) \|λA=pA = H ( qA, pA ) . (3.7) Thus, in the nondegenerate case, the “envelope” Legendre–Clairaut transform LegCl env : L → HCl env coincides with the ordinary Legendre transform constructed here. 2) A general solution defined by ∂2HCl ( qA, λA ) ∂λB∂λC = 0, (3.8) which gives ∂HCl ( qA, λA ) ∂λB = cB , where cB are arbitrary smooth functions of qA, and the latter are considered in (3.3) as parameters (passive variables). Then the general solution takes the form HCl gen ( qA, λA, cA ) = n∑ B=1 λBcB − L ( qA, cA ) , (3.9) which corresponds to a “general” Legendre–Clairaut transform LegCl gen : L → HCl gen. Note that the general solution HCl gen ( qA, λA, cA ) is always linear in the variables λA and the latter are not actually the dynamical momenta pA, because we do not have the envelope solution condition (3.6), and therefore now there is no supremum condition (2.3). The variables cA are in fact unresolved velocities vA in the case of the general solution. Note that in the standard way, LegCl env can be also obtained by finding the envelope of the general solution [4], i.e., differentiating (3.9) by cA, ∂HCl gen ( qA, λA, cA ) ∂cB = λB − ∂L ( qA, cA ) ∂cB = 0, (3.10) which coincides with (3.6) and (2.3). This means that HCl gen ( qA, λA, cA ) \|cA=vA is the “pre-Hamiltonian” (2.2) needed to find the supremum in (2.1). Let us consider the classical example of a one-dimensional oscillator. E x a m p l e 3.1. Let L (x, v) = mv2/2 − kx2/2 (m, k are constants), then the corresponding Clairaut equation (3.3) for H = HCl (x, λ) is H = λH ′ λ − m 2 ( H ′ λ )2 + kx2 2 , (3.11) where the prime denotes partial differentiation with respect to λ. The general solution of (3.11) is HCl gen (x, λ, c) = λc− mc2 2 + kx2 2 , (3.12) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 195 S. Duplij where c is an arbitrary function (“unresolved velocity” v). The envelope solution (with λ = p) can be found from the condition ∂HCl ∂c = p−mc = 0 =⇒ cextr = p m , which gives HCl env (x, p) = p2 2m + kx2 2 (3.13) in the standard way. E x a m p l e 3.2. Let L (x, v) = x exp kv, then the corresponding Clairaut equation for H = HCl (x, λ) is H = λH ′ λ − x exp ( kH ′ λ ) . (3.14) The general solution is HCl gen (x, λ) = λc− x exp kc, (3.15) where c is a smooth function of x. The envelope solution (with λ = p) can be found by differentiating the general solution (3.15), ∂HCl ∂c = p− x exp kc = 0 =⇒ cextr = 1 k ln p x , which leads to HCl env (x, p) = p k ln p x − p. (3.16) 4. The Mixed Legendre–Clairaut Transform Now consider a singular Lagrangian L ( qA, vA ) = Ldeg ( qA, vA ) for which the Hessian is zero (3.4). This means that the rank of the Hessian matrix WAB = ∂2L(qA,vA) ∂vB∂vC is r < n, and we suppose that r is constant. We rearrange the indices of WAB in such a way that a nonsingular minor of rank r appears in the upper left corner [31]. Represent the index A as follows: if A = 1, . . . , r, we replace A with i (the “regular” index), and if A = r + 1, . . . , n, we replace A with α (the “degenerate” index). Obviously, detWij 6= 0, and rankWij = r. Thus any set of the variables labelled by a single index splits as a disjoint union of two subsets. We call these subsets regular (having Latin indices) and degenerate (having Greek indices). The standard Legendre transform Leg is not applicable in the degenerate case be- cause condition (2.5) is not valid [11, 67]. Therefore the supremum condition (2.3) cannot be resolved with respect to degenerate A, but it can be resolved only for regular 196 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets A because det Wij 6= 0. On the contrary, the Clairaut duality transform given by (3.3) is independent in spite of whether the Hessian is zero or not [24]. Thus we state the main idea of the formalism we present here: the ordinary duality can be generalized to the Clairaut duality, i.e., the standard Legendre transform Leg, given by (2.6), can be gen- eralized to the singular Lagrangian theory using the Legendre-Clairaut transform LegCl given by the multidimensional Clairaut equation (3.3). To find its solutions, we differentiate (3.3) by λA and split the sum (3.5) in B as follows: r∑ i=1 [ λi − ∂L ( qA, vA ) ∂vi ] ∂2HCl ( qA, λA ) ∂λi∂λC + n∑ α=r+1 [ λα − ∂L ( qA, vA ) ∂vα ] ∂2HCl ( qA, λA ) ∂λα∂λC = 0. (4.1) As det Wij 6= 0, we suggest to replace (4.1) by the conditions λi = pi = ∂L ( qA, vA ) ∂vi , i = 1, . . . , r, (4.2) ∂2HCl ( qA, λA ) ∂λα∂λC = 0, α = r + 1, . . . n. (4.3) In this way we obtain a mixed envelope/general solution of the Clairaut equation (cf. [24]). We resolve (4.2) by the regular velocities vi = V i ( qA, pi, c α ) and write down a solution of (4.3) as ∂HCl ( qA, λA ) ∂λα = cα, (4.4) where cα are arbitrary variables corresponding to the unresolved velocities vα. Finally we obtain a mixed Hamilton–Clairaut function HCl mix ( qA, pi, λα, vα ) = r∑ i=1 piV i ( qA, pi, v α ) + n∑ β=r+1 λβvβ − L ( qA, V i ( qA, pi, v α ) , vα ) , (4.5) which is the desired “mixed” Legendre–Clairaut transform of the functions L LegCl mix7−→ HCl mix written in coordinates. Note that (4.5) was obtained formally as a mixed general/envelope solution of the Clairaut equation for the sought-for Hamilton–Clairaut function without any reference to the dynamics (this connection will be considered in the next section). Nevertheless, HCl mix coincides with the corresponding functions derived from the “slow and careful Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 197 S. Duplij Legendre transformation” [69] and the “generalized Legendre transformation” [13], as well as from the implicit partial differential equation on the cotangent bundle [52, 16] in the local coordinates [75] and in the general geometric approach [53]. E x a m p l e 4.3. Let L (x, y, vx, vy) = myv2 x/2 + kxvy, then the corresponding Clairaut equation for H = HCl (x, y, λx, λy) is H = λxH ′ λx + λyH ′ λy − my 2 ( H ′ λx )2 − kxH ′ λy . (4.6) The general solution of (4.6) is HCl gen (x, y, λx, λy, cx, cy) = λxcx + λycy − myc2 x 2 − kxcy, where cx, cy are arbitrary functions of the passive variables x, y. Then we differentiate ∂HCl gen ∂cx = px −mycx = 0, =⇒ cextr x = px my , ∂HCl gen ∂cy = λy − kx. Finally, we solve the first equation with respect to cx and treat cy 7−→ vy as an “unre- solved velocity ” . This way we obtain the mixed Hamiltonian–Clairaut function HCl mix (x, y, px, λy, vy) = p2 x 2my + vy (λy − kx) . (4.7) This result can be compared with that obtained in the geometric approach based on the reduction of the Hamiltonian Morse family in [69]. 5. Hamiltonian Formulation of Singular Lagrangian Systems Let us use the mixed Hamilton–Clairaut function HCl mix ( qA, pi, λα, vα ) (4.5) to de- scribe a singular Lagrangian theory in terms of the system of ordinary first-order differ- ential equations. In our formulation we split a set of the standard Lagrange equations of motion d dt ∂L ( qA, vA ) ∂vB = ∂L ( qA, vA ) ∂qB (5.1) into two subsets according to the index B being either regular (B = i = 1, . . . , r) or degenerate (B = α = r +1, . . . n). We use the designation of “physical” momenta (4.2) in the regular subset only such that the Lagrange equations become dpi dt = ∂L ( qA, vA ) ∂qi , (5.2) dBα ( qA, pi ) dt = ∂L ( qA, vA ) ∂qα ∣∣∣∣∣ vi=V i(qA,pi,vα) , (5.3) 198 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets where Bα ( qA, pi ) def = ∂L ( qA, vA ) ∂vα ∣∣∣∣∣ vi=V i(qA,pi,vα) (5.4) are given functions determining the dynamics of the singular Lagrangian system in the “degenerate” sector. The functions Bα ( qA, pi ) are independent of the unresolved ve- locities vα since the rankWAB = r. One should also take into account that now dqi dt = V i ( qA, pi, v α ) , dqα dt = vα. (5.5) Note that before imposing the Lagrange Eqs. (5.2), when solving the Clairaut Eq. (3.3), the arguments of L ( qA, vA ) were treated as independent variables. A passage to an analog of the Hamiltonian formalism can be done by the standard procedure: consider the full differential of both sides of (4.5) and use supremum con- dition (4.2) which gives (note that in the previous sections the Lagrange equations of motion (5.1) were not used) ∂HCl mix ∂pi = V i ( qA, pi, v α ) , ∂HCl mix ∂λα = vα, ∂HCl mix ∂qi = − ∂L ( qA, vA ) ∂qi ∣∣∣∣∣ vi=V i(qA,pi,vα) + n∑ β=r+1 [ λβ −Bβ ( qA, pi )] ∂vβ ∂qi , ∂HCl mix ∂qα = − ∂L ( qA, vA ) ∂qα ∣∣∣∣∣ vi=V i(qA,pi,vα) + n∑ β=r+1 [ λβ −Bβ ( qA, pi )] ∂vβ ∂qα . Applying of (5.2) yields the system of equations which gives a Hamiltonian–Clairaut description of a singular Lagrangian system ∂HCl mix ∂pi = dqi dt , (5.6) ∂HCl mix ∂λα = dqα dt , (5.7) ∂HCl mix ∂qi = −dpi dt + n∑ β=r+1 [ λβ −Bβ ( qA, pi )] ∂vβ ∂qi , (5.8) ∂HCl mix ∂qα = dBα ( qA, pi ) dt + n∑ β=r+1 [ λβ −Bβ ( qA, pi )] ∂vβ ∂qα . (5.9) The system (5.6)–(5.9) has two disadvantages: the first, it contains the “nondynami- cal momenta” λα; the second, it has derivatives of unresolved velocities vα. We observe Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 199 S. Duplij that we can get rid of these difficulties if we reformulate (5.6)–(5.9) by introducing a “physical” Hamiltonian Hphys ( qA, pi ) = HCl mix ( qA, pi, λα, vα )− n∑ β=r+1 [ λβ −Bβ ( qA, pi )] vβ (5.10) which does not depend on the variables λα (“nondynamical momenta”) at all by the construction ∂Hphys ∂λα = 0 (5.11) (cf. (4.4) and (4.5)). Then the “physical” Hamiltonian (5.10) can be written in the form Hphys ( qA, pi ) = r∑ i=1 piV i ( qA, pi, v α ) + n∑ α=r+1 Bα ( qA, pi ) vα − L ( qA, V i ( qA, pi, v α ) , vα ) . (5.12) Using (4.2), we can show that the r.h.s. of (5.12) does not depend on λα, and for the degenerate velocities vα one has ∂Hphys ∂vα = 0, (5.13) which justifies the term “physical”. Therefore, the time evolution of the singular La- grangian system (5.1) is determined by (n− r + 1) functions Hphys ≡ Hphys ( qA, pi ) and Bα ≡ Bα ( qA, pi ) . Writing ( qA, pi ) = ( qα\|qi, pi ) ∈ Rn−r × Sp (r, r) ≡ Mphys, where Rn−r is a real space of the dimension (n− r), and Sp (r, r) is a symplectic space of the dimension (r, r), we observe that Hphys : Rn−r × Sp (r, r) → R and Bα : Rn−r × Sp (r, r) → Rn−r. Then we use (5.6)–(5.9) to deduce the main result of our Clairaut-type formulation that the sought-for system of ordinary first-order differential equations (the Hamilton– Clairaut system) which describes any singular Lagrangian classical system (satisfying the second-order Lagrange Eqs. (5.1)), has the form dqi dt = { qi,Hphys } phys − n∑ β=r+1 { qi, Bβ } phys dqβ dt , i = 1, . . . r, (5.14) dpi dt = {pi,Hphys}phys − n∑ β=r+1 {pi, Bβ}phys dqβ dt , i = 1, . . . r, (5.15) n∑ β=r+1 [ ∂Bβ ∂qα − ∂Bα ∂qβ + {Bα, Bβ}phys ] dqβ dt = ∂Hphys ∂qα + {Bα,Hphys}phys , α = r + 1, . . . , n, (5.16) 200 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets where {X,Y }phys = n−r∑ i=1 ( ∂X ∂qi ∂Y ∂pi − ∂Y ∂qi ∂X ∂pi ) (5.17) is the “physical” Poisson bracket (in regular variables qi, pi) for the functions X and Y on Mphys. The system (5.14)–(5.16) is equivalent to the Lagrange equations of motion (5.1) by the construction. Thus, the Clairaut-type formulation (5.14)–(5.16) is valid for any Lagrangian theory without additional conditions, as opposite to other approaches (see, e.g., [57, 72]). E x a m p l e 5.4. (Cawley [12]) Let L = ẋẏ + zy2/2, then the equations of motion are ẍ = yz, ÿ = 0, y2 = 0. (5.18) Because the Hessian has rank 2, and the velocity ż does not enter into the Lagrangian, the only degenerate velocity is ż (α = z), the regular momenta are px = ẏ, py = ẋ (i = x, y). Thus, we have Hphys = pxpy − 1 2 zy2, Bz = 0. The equations of motion (5.14)–(5.15) are ṗx = 0, ṗy = yz, (5.19) and condition (5.16) gives ∂Hphys ∂z = −1 2 y2 = 0. (5.20) Observe that (5.19) and (5.20) coincide with the initial Lagrange equations of motion (5.18). Since the number of equations r + r + n − r = n + r coincides with the number of the sought-for variables nqi = r, npi = r, nqα = n − r, we deduce that there are no constraints in (5.14)–(5.16) at all. In particular, the system (5.16) has (n− r) equations, which exactly coincides with the number of the sought-for “unresolved” velocities vα = dqα dt . Therefore, (5.16) is a standard system of linear algebraic equations with respect to vα, but not constraints (when there are more sought-for variables than equations). E x a m p l e 5.5. ([7, 71]) Let us consider a classical particle on R3 with the regular Lagrangian ( ẋ2 + ẏ2 + ż2 ) /2 subject to the nonholonomic constraint ż = yẋ. To apply the Clairaut equation method, we introduce an extra coordinate u. Then this system is equivalent to the singular Lagrangian system on R4 described by L = ẋ2 + ẏ2 + ż2 2 + u (ż − yẋ) . (5.21) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 201 S. Duplij The Lagrange equations of motion are straightforward (cf. [7]) ẍ− u̇y − ẏu = 0, ÿ + uẋ = 0, z̈ + u̇ = 0, ż − yẋ = 0. (5.22) The Hessian of (5.21) is zero, and thus the system is singular. The rank of the Hessian matrix diag (1, 1, 1, 0) being 3, we have 3 regular and 1 degenerate variables. First, we should find the “physical” Hamiltonian using the Clairaut equation formalism and then pass from the second-order Eqs. (5.22) to the first-order equations similar to (5.14)– (5.16). Let us consider multidimensional Clairaut Eq. (3.3) for the Hamilton–Clairaut function H ≡ HCl (x, y, z, u, λx, λy, λz, λu), H = λxH ′ λx + λyH ′ λy + λzH ′ λz + λuH ′ λu − 1 2 ( H ′ λx )2 − 1 2 ( H ′ λy )2 − 1 2 ( H ′ λz )2 − uH ′ λz + yuH ′ λx . (5.23) The general solution of (5.23) is Hgen = λxcx + λycy + λzcz + λucu − c2 x + c2 y + c2 z 2 − ucz + yucx, (5.24) where initially cx, cy, cz, cu are arbitrary functions of the passive (with respect to the Clairaut Eq. (5.23)) variables x, y, z, u. To find supremum conditions (3.10), we write the derivatives ∂Hgen ∂cx = λx − cx + yu = 0, (5.25) ∂Hgen ∂cy = λy − cy = 0, (5.26) ∂Hgen ∂cz = λz − cz − u = 0, (5.27) ∂Hgen ∂cu = λu. (5.28) Observe that only 3 first conditions here can be resolved with respect to ci (i = x, y, z), and therefore these λi correspond to the “physical” momenta (4.2), that is, λi = pi = ∂L�∂vi (i = x, y, z). Thus, the extremum values of ci are cextr x = px + yu, cextr y = py, cextr z = pz − u, (5.29) while cu becomes the “unresolved” velocity cu = vu. In this way, inserting (5.29) into (5.24), for the mixed Hamilton–Clairaut function (4.5) we have HCl mix = p2 x + p2 y + p2 z 2 + λuvu + u (ypx − pz) + u2 1 + y2 2 . (5.30) 202 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets Now we calculate function (5.4) and the “physical” Hamiltonian (5.10), Bu = ∂L ∂u̇ = 0, (5.31) Hphys = p2 x + p2 y + p2 z 2 + u (ypx − pz) + u2 1 + y2 2 (5.32) not depending on λu and vu. Using (5.14)–(5.16), we obtain the Hamilton–Clairaut system ẋ = px + yu, ẏ = py, ż = pz − u, (5.33) ṗx = 0, ṗy = −u (px + yu) , ṗz = 0, (5.34) ypx − pz + u ( 1 + y2 ) = 0, (5.35) which coincides with the system of Lagrange equations of motion (5.22) by the con- struction. It is remarkable that the “degenerate”variable u is determined by the algebraic equation (5.35), u = pz − ypx 1 + y2 , (5.36) and therefore the singular system (5.21) has no “gauge” degrees of freedom. In general, if a dynamical system is nonsingular, it has no “degenerate” variables at all because the rank r of the Hessian is full (r = n). The distinguishing property of any singular system (r < n) is clear and simple in our Clairaut-type approach: it contains an additional system of the linear algebraic equations (5.16) for the “unresolved” velocities vα (not constraints), which can be analyzed and solved by the standard linear algebra methods. Indeed, the linear algebraic system (5.16) gives a full classification of singular Lagrangian theories presented in the next section. E x a m p l e 5.6. The classical relativistic particle is described by L = −mR, R = √ ẋ2 0 − ∑ i=x,y,z ẋ2 i , (5.37) where a dot denotes a derivative with respect to the proper time. Because the rank of the Hessian is 3, we will treat the velocities ẋi as regular variables and the velocity ẋ0 as a degenerate variable. Then for the regular canonical momenta we have pi = ∂L�∂ẋi = mẋi�R, which can be resolved with respect to the regular velocities, ẋi = ẋ0 pi E , E = √ m2 + ∑ i=x,y,z p2 i . (5.38) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 203 S. Duplij Using (5.4) and (5.12), we obtain Hphys = 0, Bx0 = ∂L ∂ẋ0 = −m ẋ0 R = −E. (5.39) The “physical sense” of (−Bx0) is just the energy (5.38), while the “physical” Hamilto- nian is zero. Equations of motion (5.14)–(5.15) are ẋi = ẋ0 pi E , ṗi = ∂Bx0 ∂xi ẋ0 = 0, which coincide with the Lagrange equations following from (5.37). Note that the veloc- ity ẋ0 is arbitrary here, and therefore we have one “gauge” degree of freedom. 6. Nonabelian Gauge Theory Interpretation We observe that (5.16) can be written in a more compact form using the gauge theory notation. Let us introduce a “qα-long derivative” DαX = ∂X ∂qα + {Bα, X}phys , (6.1) where X = X ( qA, pi ) is a smooth scalar function on Mphys. We also notice that a multiplier in (5.16) to be called a “qα-field strength” Fαβ ≡ Fαβ ( qA, pi ) of the “qα- gauge fields” Bα on Mphys defined by Fαβ = ∂Bβ ∂qα − ∂Bα ∂qβ + {Bα, Bβ}phys . (6.2) Then the linear system of equations (5.16) for unresolved velocities can be written in a compact form n∑ β=r+1 Fαβ dqβ dt = DαHphys, α = r + 1, . . . , n. (6.3) The “qα-field strength” Fαβ is nonabelian due to the presence of the “physical” Poisson bracket in r.h.s. of (6.2). It is important to observe that in distinct of the ordinary Yang– Mills theory, the partial derivatives of Bα in (6.2) are defined in the qα-subspace Rn−r, while the “noncompactivity” (the third term) is due to the Poisson bracket (5.17) in another symplectic subspace Sp (r, r). Note that the “qα-long derivative” satisfies the Leibniz rule, Dα {Bβ, Bγ}phys = {DαBβ, Bγ}phys + {Bβ, DαBγ}phys , 204 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets which is valid while acting on “qα-gauge fields” Bα. The commutator of the “qα-long derivatives” is now equal to the Poisson bracket with the “qα-field strength” (DαDβ −DβDα)X = {Fαβ , X}phys . (6.4) It follows from (6.4) that DαDβFαβ = 0. (6.5) Let us introduce the Bα-transformation δBαX = {Bα, X}phys (6.6) which satisfies ( δBαδBβ − δBβ δBα ) Bγ = δ{Bα,Bβ}phys Bγ , (6.7) δBαFβγ ( qA, pi ) = (DγDβ −DβDγ) Bα, (6.8) δBα {Bβ, Bγ}phys = {δBαBβ, Bγ}phys + {Bβ, δBαBγ}phys . (6.9) This means that the “qα-long derivative” Dα (6.1) is in fact a “qα-covariant derivative” with respect to the Bα-transformation (6.6). Indeed, observe that Dα transforms as fields (6.6), which proves that it is really covariant (note the cyclic permutations in both sides) δBαDβBγ + δBγDαBβ + δBβ DγBα = {Bα, DβBγ}phys + {Bγ , DαBβ}phys + {Bβ, DγBα}phys . (6.10) The “qα-Maxwell” equations of motion for the “qα-field strength” are DαFαβ = Jβ, (6.11) DαFβγ + DγFαβ + DβFγα = 0, (6.12) where Jα ≡ Jα ( qA, pi ) is a “qα-current” in Mphys which is a function of the initial Lagrangian (2.2) and its derivatives up to the third order. Due to (6.5), the “qα-current” Jα is conserved DαJα = 0. (6.13) Thus, a singular Lagrangian system leads effectively to a special kind of the nonabelian gauge theory in the direct product space Mphys = Rn−r × Sp (r, r). Here the “non- commutativity” (the third term in (6.2)) appears not due to a Lie algebra (as in the Yang- Mills theory), but “classically”, due to the Poisson bracket in the symplectic subspace Sp (r, r). The corresponding manifold can be interpreted locally as a special kind of the degenerate Poisson manifold (see, e.g., [9]). The analogous Poisson type of “nonabelianity” (6.2) appears in the N →∞ limit of the Yang-Mills theory, and it is called the “Poisson gauge theory” [28]. In the SU (∞) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 205 S. Duplij Yang–Mills theory the group indices become the surface coordinates [26], which is con- nected with the Schild string [76]. The related algebra generalizations are called the continuum graded Lie algebras [63] (see also [45]). Here, because of the direct prod- uct structure of the space Mphys, the similar construction appears in (6.2) (in another initial context), while the “long derivative” (6.1), the “gauge transformations” (6.6) and the analog of the Maxwell equations (6.11)–(6.12) differ from the “Poisson gauge the- ory” [28]. 7. Classification, Gauge Freedom and New Brackets Next we can classify singular Lagrangian theories as follows: 1. Gaugeless theory. The rank of the skew-symmetric matrix Fαβ is “full”, i.e., the rankFαβ = n − r is constant, and therefore the matrix Fαβ is invertible, and all the (degenerate) velocities vα can be found from the system of the linear equations (5.16) (and (6.3)) in a purely algebraic way. 2. Gauge theory. The skew-symmetric matrix Fαβ is singular. If the rankFαβ = rF < n − r, then a singular Lagrangian theory has n − r − rF gauge degrees of freedom. We can take them arbitrary, which corresponds to the presence of some symmetries in the theory. Note that the rank rF is even due to the skew-symmetry of Fαβ . In the first case (gaugeless theory) one can resolve (6.3) as follows: vβ = n∑ α=r+1 F̄ βαDαHphys, (7.1) where F̄αβis the inverse matrix to Fαβ , i.e., n∑ β=r+1 FαβF̄ βγ = n∑ β=r+1 F̄ γβFβα = δγ α. (7.2) Substitute (7.1) in (5.14)–(5.15) to present the system of equations for a gaugeless de- generate Lagrangian theory in the Hamilton-like form dqi dt = { qi,Hphys } nongauge , (7.3) dpi dt = {pi,Hphys}nongauge , (7.4) where the “nongauge” bracket is defined by {X, Y }nongauge = {X,Y }phys − n∑ α=r+1 n∑ β=r+1 DαX · F̄αβ ·DβY. (7.5) 206 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets Then the time evolution of any function of the dynamical variables X = X ( qA, pi ) is also determined by the bracket (7.5) as follows: dX dt = {X,Hphys}nongauge . (7.6) The meaning of the new nongauge bracket (7.5) (which appears naturally in the Clairaut-type formulation [25]) is the same as the meaning of the ordinary Poisson bracket in the unconstrained Hamiltonian dynamics: it governs the dynamics by the set of first-order differential equations in the Hamilton-like form (7.3)–(7.4) and is re- sponsible for the time evolution of any dynamical variable (7.6). Also, the second term in the new bracket (7.5) has a complicated coordinate dependence and is analogous to that of the Dirac bracket [19]. In the extended phase space both brackets coincide (see Appendix B). On the other hand, the appearance of the second term in (7.5) can be treated as a deformation of the Poisson bracket, which can lead to another kind of the generalized symplectic geometry [39]. In the second case (gauge theory), with the singular matrix Fαβ of rank rF , we rearrange its rows and columns to obtain a nonsingular rF × rF submatrix in the left upper corner. Thus, the first rF equations of the system of linear (under also rearranged vβ) Eqs. (6.3) are independent. Then we express the indices α and β as the pairs α = (α1, α2) and β = (β1, β2), where α1 and β1 denote the first rF rows and columns, while α2 and β2 denote the rest of n− r − rF rows and columns. Correspondingly, we decompose the system (6.3), r+rF∑ β1=r+1 Fα1β1v β1 + n∑ β2=r+rF +1 Fα1β2v β2 = Dα1Hphys, (7.7) r+rF∑ β1=r+1 Fα2β1v β1 + n∑ β2=r+rF +1 Fα2β2v β2 = Dα2Hphys. (7.8) The matrix Fα1β1 being nonsingular by the construction, we can find the first rF velocities vβ1 = r+rF∑ α1=r+1 F̄ β1α1Dα1Hphys − r+rF∑ α1=r+1 F̄ β1α1Fα1β2v β2 , (7.9) where F̄ β1α1 is the inverse of the nonsingular rF × rF submatrix Fα1β1satisfying (7.2). Then, since the rankFαβ = rF , the last n − r − rF equations (7.8) are the linear Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 207 S. Duplij combinations of the first rF independent equations (7.7), which gives Fα2β1 = r+rF∑ α1=r+1 λα1 α2 Fα1β1 , (7.10) Fα2β2 = r+rF∑ α1=r+1 λα1 α2 Fα1β2 , (7.11) Dα2Hphys = r+rF∑ α1=r+1 λα1 α2 Dα1Hphys, (7.12) where λα1 α2 = λα1 α2 ( qA, pi ) are some rF×(n− r − rF ) smooth functions. Using relation (7.10) and invertibility of Fα1β1 , we eliminate the functions λα1 α2 by λα1 α2 = r+rF∑ α1=r+1 r+rF∑ β1=r+1 Fα2β1F̄ β1α1 . (7.13) This indicates that the gauge theory is fully determined by the first rF rows of the (rearranged) matrix Fαβ and the first rF (rearranged) derivatives Dα1Hphys only. Next, we can make the unresolved n− r − rF velocities vanish vβ2 = 0 (7.14) by some “gauge fixing” condition. Then (7.9) becomes vβ1 = r+rF∑ α1=r+1 F̄ β1α1Dα1Hphys. (7.15) By analogy with (7.3)–(7.4), in the gauge case we can also write the system of equations for a singular Lagrangian theory in the Hamilton-like form. Now we introduce another new (gauge) bracket {X,Y }gauge = {X, Y }phys − r+rF∑ α1=r+1 r+rF∑ β1=r+1 Dα1X · F̄α1β1 ·Dβ1Y. (7.16) Then substituting (7.14)–(7.15) into (5.14)–(5.15) and using (7.16), we obtain dqi dt = { qi, Hphys } gauge , (7.17) dpi dt = {pi,Hphys}gauge . (7.18) 208 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets Thus the gauge bracket (7.16) governs the time evolution in the gauge case dX dt = {X,Hphys}gauge . (7.19) Note that the brackets (7.5) and (7.16) are antisymmetric and satisfy the Jacobi iden- tity. Therefore the standard quantization scheme is applicable here (see, e.g., [37]). The difference is that only the canonical (regular) dynamic variables ( qi, pi ) can be quan- tized, while the degenerate coordinates can be treated as some continuous parameters. It is worthwhile to consider the limit case, when rF = 0, i.e., Fαβ = 0 (7.20) identically, which can mean that Bα = 0, so the Lagrangian can be independent of the degenerate velocities vα. It follows from (5.16) that DαHphys = ∂Hphys ∂qα = 0, (7.21) which leads to the “independence” statement: the “physical” Hamiltonian Hphys does not depend on the degenerate coordinates qα iff the Lagrangian does not depend on the velocities vα. In the limit case, both brackets (7.5) and (7.16) coincide with the Poisson bracket in the reduced “physical” phase space { , }nongauge,gauge = { , }phys. E x a m p l e 7.7. (Christ–Lee model [14]). The Lagrangian of SU (2) Yang–Mills theory in 0 + 1 dimensions in our notation is L (xi, yα, vi) = 1 2 ∑ i=1,2,3  vi − ∑ j,α=1,2,3 εijαxjyα   2 − U ( x2 ) , (7.22) where i, α = 1, 2, 3, x2 = ∑ i x 2 i , vi = ẋi and εijk is the Levi–Civita symbol. Be- cause (7.22) is independent of the degenerate velocities ẏα, all Bα (5.4) = 0, and therefore Fαβ (6.2) = 0, we have the limit gauge case of the above classification. The corresponding Clairaut Eq. (3.3) for H = HCl (xi, yα, λi, λα) has the form H = ∑ i=1,2,3 λiH ′ λi + ∑ α=1,2,3 λαH ′ λα − 1 2 ∑ i=1,2,3  H ′ λi − ∑ j,α=1,2,3 εijαxjyα   2 +U ( x2 ) . (7.23) We show manifestly how to obtain the envelope solution for regular variables and the general solution for degenerate variables. The general solution is Hgen = ∑ i=1,2,3 λici + ∑ α=1,2,3 λαcα − 1 2 ∑ i=1,2,3  ci − ∑ j,α=1,2,3 εijαxjyα   2 + U ( x2 ) , (7.24) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 209 S. Duplij where ci, cα are arbitrary functions of coordinates. Recall that qA are passive variables under the Legendre transform. We differentiate (7.24) by ci, cα, ∂Hgen ∂ci = λi −  ci − ∑ j,α=1,2,3 εijαxjyα   , (7.25) ∂Hgen ∂cα = λα, (7.26) and observe that only (7.25) can be resolved with respect to ci, and therefore can lead to the envelope solution, while other cα cannot be resolved, and therefore we consider only the general solution of the Clairaut equation. So we can exclude half of the constants using (7.25) (with the substitution λi (4.2)→ pi) and get the mixed solution (4.5) to the Clairaut Eq. (7.23), HCl mix (xi, yα, pi, λα, cα) = 1 2 ∑ i=1,2,3 p2 i + ∑ i,j,α=1,2,3 εijαpixjyα+ ∑ α=1,2,3 λαcα+U ( x2 ) . (7.27) Using (5.10), we obtain the “physical” Hamiltonian Hphys (xi, yα, pi) = 1 2 ∑ i=1,2,3 p2 i + ∑ i,j,α=1,2,3 εijαpixjyα + U ( x2 ) . (7.28) On the other hand, the Hessian of (7.22) has the rank 3, and we choose xi, vi and yα to be regular and degenerate variables, respectively. The degenerate velocities vα = ẏα cannot be defined from (5.16) at all, they are arbitrary, and the first integrals (5.16), (7.21) of the system (5.14)–(5.15) become (also in accordance to the independence statement) ∂Hphys (xi, yα, pi) ∂yα = ∑ i,j=1,2,3 εijαpixj = 0. (7.29) The preservation in time (7.19) of (7.29) is fulfilled identically due to the antisymmetry properties of the Levi–Civita symbols. It is clear that only 2 equations from 3 of (7.29) are independent, so we choose p1x2 = p2x1, p1x3 = p3x1 and insert them into (7.28) to get H̃phys = 1 2 p2 1 x2 x2 1 + U ( x2 ) . (7.30) The transformation p̃ = p1 √ x2�x1, x̃ = √ x2 gives the well-known result [14, 34], H̃phys = 1 2 p̃2 + U (x̃) . (7.31) 210 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets 8. Singular Lagrangian Systems and Many-time Dynamics The many-time classical dynamics and its connection with constrained systems were studied in [48, 44] as a generalization of some relativistic two-particle models [22]. We consider this connection from a different viewpoint, that is, in the Clairaut-type approach [25]. Recall that the Hamiltonian–Clairaut dynamics (5.14)–(5.16) of a Lagrangian singular system (5.1) is governed by the “physical” Hamiltonian function Hphys and (n− r) “qα-gauge fields” Bα defined on the direct product space Rn−r × Sp (r, r). Let us treat the degenerate coordinates qα ∈ Rn−r as (n− r) additional “time” variables together with (n− r) corresponding “Hamiltonians”−Bα ( qα\|qi, pi ) , α = r+1, . . . , n (see (6.1)). Indeed, let us introduce (n− r + 1) generalized “times” tµ and the corre- sponding “many-time Hamiltonians” Hµ ( tµ\|qi, pi ) , µ = 0, . . . n− r defined by t0 = t, H0 ( tα\|qi, pi ) = Hphys ( qα\|qi, pi ) , µ = 0, (8.1) tµ = qµ, Hµ ( tµ\|qi, pi ) = −Br+µ ( qr+µ\|qi, pi ) , µ = 1, . . . , n− r. (8.2) Then Eqs. (5.14)–(5.15) can be presented in the differential form dqi = n−r∑ µ=0 { qi, Hµ } phys dtµ, (8.3) dpi = n−r∑ µ=0 {pi, Hµ}phys dtµ, (8.4) where { , }phys is defined in (5.17). The linear algebraic system of Eqs. (5.16) for the degenerate velocities then becomes n−r∑ µ=0 Gµνdtµ = 0, (8.5) where Gµν = ∂Hµ ∂tν − ∂Hν ∂tµ + {Hµ, Hν}phys . (8.6) It follows from (8.5) that the one-form ω = pidqi − Hµdtµ is closed, dω = 1 2 n−r∑ µ=0 n−r∑ ν=0 Gµνdtµ ∧ dtν = 0, (8.7) which agrees with the action principle for multi-time classical dynamics [21]. The corre- sponding set of the Hamilton–Jacobi equations for action S ( qα\|qi, pi ) 7−→ S ( tµ\|qi, pi ) is ∂S ∂tµ + Hµ ( tµ\|qi, ∂S ∂qi ) = 0. (8.8) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 211 S. Duplij Therefore, we have come to the conclusion that any singular Lagrangian theory (in the Clairaut-type formulation [24, 25]) is equivalent to the many-time classical dynamics [21, 48]: the equations of motion are (8.3)–(8.4), which coincide with (5.14)–(5.15), and the integrability condition is (8.5), which coincides with the system of the linear algebraic equations for unresolved velocities (5.16) by the construction. 9. Conclusions We have described the Hamilton-like evolution of singular Lagrangian systems by using n−r+1 functions Hphys ( qα\|qi, pi ) and Bα ( qα\|qi, pi ) on the direct product space Rn−r×Sp (r, r). To do this, we used the generalized Legendre–Clairaut transform, that is, we solved the corresponding multidimensional Clairaut equation without introducing the Lagrange multipliers. All variables are set as regular or degenerate according to the rank of the Hessian matrix of Lagrangian. We consider the reduced “physical” phase space formed by the regular coordinates qi and momenta pi only, while the degenerate coordinates qα play a role of parameters. There are two reasons why the degenerate mo- menta λα corresponding to qα need not be considered in the Clairaut-type formulation: 1) the mathematical reason: there is no possibility to find the degenerate velocities vα, as can be done for the regular velocities vi in (4.2), and “pre-Hamiltonian” (2.2) has no extremum in degenerate directions; 2) the physical reason: momentum is a measure of motion; however, there is no dynamics in “degenerate” directions and hence no reason to introduce the corresponding “physical” momenta at all. Note that some possibilities to avoid constraints were considered in a different con- text in [18, 58] and for special forms of the Lagrangian in [32]. The Hamilton-like form of the equations of motion (7.3)–(7.4) is achieved by intro- ducing new brackets (7.5) and (7.16) which are responsible for time evolution. They are antisymmetric and satisfy the Jacobi identity. Therefore we can quantize the brackets using the standard methods [37], but only for the regular variables, while the degenerate variables can be considered as some continuous parameters. In the “nonphysical” coordinate subspace, we formulate some kind of nonabelian gauge theory such that “nonabelianity” appears due to the Poisson bracket in the physical phase space (6.2). This makes it similar to the Poisson gauge theory [28], but do not coincide with the latter. Finally, we show that, in general, a singular Lagrangian system in the Clairaut-type formulation [24, 25] is equivalent to the many-time classical dynamics. 212 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets A. Multidimensional Clairaut Equation The multidimensional Clairaut equation for a function y = y(xi) of n variables xi is [41, 3] y = n∑ j=1 xjy ′ xj − f(y′xi ), (A.1) where prime denotes a partial differentiation by subscript and f is a smooth function of n arguments. To find and classify the solutions of (A.1), we have to find first derivatives y′xi in some way and then substitute them back into (A.1). We differentiate (A.1) by xj and obtain n equations n∑ i=1 y′′xixj (xi − f ′y′xi ) = 0. (A.2) The classification follows from the ways the factors in (A.2) can be set to zero. Here, for our physical applications, it is sufficient to suppose that the ranks of Hessians of y and f are rank y′′xixj = rank f ′′y′xi y′xj = r. (A.3) This means that in each equation from (A.2) either the first or the second multiplier is zero, but it is not necessary to vanish both of them. The first multiplier can be set to zero without any additional assumptions. So we have 1) The general solution. It is defined by the condition y′′xixj = 0. (A.4) After one integration we can find y′xi = ci and substitute them into (A.1) to obtain ygen = n∑ j=1 xjcj − f(ci), (A.5) where ci are n constants. All second multipliers in (A.2) can be zero for i = 1, . . . , n, but this will give a solution if they can be resolved for y′xi . It is possible if the rank of Hessians f is full, i.e., r = n. In this case we obtain 2) The envelope solution. It is defined by xi = f ′y′xi . (A.6) We resolve (A.6) for the derivatives as y′xi = Ci (xj) and get yenv = n∑ i=1 xiCi (xj)− f(Ci (xj)), (A.7) Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 213 S. Duplij where Ci (xj) are n smooth functions of n arguments. In the intermediate case, we can use the envelope solution (A.7) for the initial s variables and the general solution (A.5) for the rest of n− s variables to obtain 3) The s-mixed solution y (s) mix = s∑ j=1 xjCj (xj) + n∑ j=s+1 xjcj − f(C1 (xj) , . . . , Cs (xj) , cs+1, . . . , cn). (A.8) If the rank r of the Hessians f is not full and a nonsingular minor of the rank r is in the upper left corner, then we can resolve the first r relations (A.6) only, and thus s ≤ r. In our physical applications we use the limited case s = r. E x a m p l e A.1. Let f (zi) = z2 1 + z2 2 + z3. Then the Clairaut equation for y = y (x1, x2,x3) has the form y = x1y ′ x1 + x2y ′ x2 + x3y ′ x3 − ( y′x1 )2 − ( y′x2 )2 − y′x3 , (A.9) and we have n = 3 and r = 2. The general solution can be found from (A.4) by integrating one time and using (A.5), ygen = c1 (x1 − c1) + c2 (x2 − c2) + c3 (x3 − 1) , (A.10) where ci are constants. Since r = 2, we can resolve only 2 relations from (A.6) by y′x1 = x1 2 , y′x2 = x2 2 . So there is no envelope solution (for all variables), but we have several mixed solutions corresponding to s = 1, 2: y (1) mix =    x2 1 4 + c2 (x2 − c2) + c3 (x3 − 1) , c1 (x1 − c1) + x2 2 4 + c3 (x3 − 1) , (A.11) y (2) mix = x2 1 4 + x2 2 4 + c3 (x3 − 1) . (A.12) The case f (zi) = z2 1 + z2 2 can be obtained from the above formulas by putting x3 = c3 = 0 and y (2) mix becomes the envelope solution yenv = x2 1 4 + x2 2 4 . B. Correspondence with the Dirac Approach The constraints appear due to additional variables introduced into the theory of ad- ditional dynamical variables (because the Hamilton-like form of the equations of mo- tion can be achieved without them in the presented approach), that is, momenta which correspond to the “degenerate” velocities. The relationship between the Clairaut-type 214 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets formulation and the Dirac approach can be clarified by treating the variables λα in the general solution of the Clairaut equation as the “physical” degenerate momenta pα using for them the same expression in terms of the Lagrangian as in (4.2), λα = pα = ∂L ( qA, vA ) ∂vα . (B.1) Then we obtain the primary Dirac constraints (in the resolved form and our notation (5.4)) Φα ( qA, pA ) = pα −Bα = 0, α = r + 1, . . . n, (B.2) which are defined now on the full phase space T∗M . Using (5.10) and (5.12), we can arrive at the complete Hamiltonian of the first-order formulation [31] (corresponding to the total Dirac Hamiltonian [19]), HT ( qA, pA, vα ) = HCl mix ( qA, pi, λα, vα )∣∣∣ λα=pα = Hphys ( qA, pi ) + n∑ α=r+1 vαΦα ( qA, pA ) , (B.3) which is equal to the mixed Hamilton–Clairaut function (4.5) with the substitution (B.1) and (B.2) being used. Then the Hamilton–Clairaut system of Eqs. (5.14)–(5.15) coin- cides with the Hamilton system in the first-order formulation [31], q̇A = { qA,HT } full , ṗA = {pA,HT }full , Φα = 0, (B.4) and (5.16) gives the second stage equations of the Dirac approach {Φα,HT }full = {Φα,Hphys}full + n∑ β=r+1 {Φα,Φβ}full v β = 0, (B.5) where {X, Y }full = n∑ A=1 ( ∂X ∂qA ∂Y ∂pA − ∂Y ∂qA ∂X ∂pA ) (B.6) is the (full) Poisson bracket on the whole phase space T∗M . Notice that Fαβ = {Φα,Φβ}full , (B.7) DαHphys = {Φα, Hphys}full . (B.8) It is important that the introduced new brackets (7.5) and (7.16) become the Dirac bracket [19]. Moreover, our cases 2) and 1) of Section 7 work as counterparts of the first and the second class constraints in the Dirac classification [19], respectively. The limit Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 215 S. Duplij case with zero “qα-field strength” Fαβ = 0 (7.20) (see (B.7)) corresponds to the Abelian constraints [35, 49]. Acknowledgements. The author is grateful to G. A. Goldin and J. Lebowitz for kind hospitality at the Rutgers University, where this work was finalized, and to the Fulbright Scholar Program for financial support, he would also like to express deep thankfulness to Jim Stasheff for careful reading the final version and making many corrections and important remarks. The author would like to thank a number of colleagues and friends for fruitful discussions. References [1] R. Abraham and J.E. Marsden, Foundations of Mechanics. Benjamin–Cummings, Reading, 1978. [2] P. Alart, O. Maisonneuve, and R. T. Rockafellar, Nonsmooth Mechanics and Analysis: Theoretical and Numerical Advances. Springer–Verlag, Berlin, 2006. [3] D.V. Alekseevskij, A.M. Vinogradov, and V.V. Lychagin, Geometry I: Basic Ideas and Con- cepts of Differential Geometry. Springer, New York, 1991. [4] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer–Verlag, New York, 1988. [5] V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer, Berlin, 1989. [6] D. Baleanu, Fractional Hamiltonian Analysis of Irregular Systems. — Signal Proc. 86 (2006), 2632–2636. [7] L. Bates and J. Śniatycki, Nonholonomic reduction. — Rep. Math. Phys. 32 (1993), 99– 115. [8] C. Battle, J. Gomis, J.M. Pons, and N. Roman-Roy, On the Legendre Transformation for Singular Lagrangians and Related Topics. — J. Phys. A: Math. Gen. 20 (1987), 5113–5123. [9] P. Bueken, Multi-Hamiltonian Formulation for a Class of Degenerate Completely Inte- grable Systems. — J. Math. Phys. 37 (1996), 2851–2862. [10] G. Caratù, G. Marmo, A. Simoni, B. Vitale, and F. Zaccaria, Lagrangian and Hamiltonian Eormalisms: an Analysis of Classical Mechanics on Tangent and Cotangent Bundles. — Nuovo Cim. B31 (1976), 152–172. [11] J.F. Carinena, Theory of Singular Lagrangians. — Fortsch. Physik 38 (1990), 641–679. [12] R. Cawley, Determination of the Hamiltonian in the Presence of Constraints. — Phys. Rev. Lett. 42 (1979), 413–416. [13] H. Cendra, D.D. Holm, M.J.V. Hoyle, and J.E. Marsden, The Maxwell–Vlasov Equations in Euler–Poincare form. — J. Math. Phys. 39 (1998), 3138–3157. [14] N.H. Christ and T.D. Lee, Operator Ordering and Feynman Rules in Gauge Theories. — Phys. Rev. D22 (1980), 939–958. 216 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets [15] F.H. Clarke, Extremal Arcs and Extended Hamiltonian Systems. — Trans. Amer. Math. Soc. 231 (1977), 349–367. [16] P. Dazord, Mécanique Hamiltonienne en Presence de Constraintes. — Illinois J. Math. 38 (1994), 148–1175. [17] Quantum Fields and Strings: A Cource for Mathematicians, P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison, and E. Witten (Eds.), Vol. 1, 2. AMS, Providence, 1999. [18] A.A. Deriglazov, Analysis of Constrained Theories Without Use of Primary Constraints. — Phys. Lett. B626 (2005), 243–248. [19] P.A.M. Dirac, Lectures on Quantum Mechanics. Yeshiva University, New York, 1964. [20] R.G. Di Stefano, A Modification of Dirac’s Method of Hamiltonian Analysis. — Phys. Rev. D27 (1983), 1752–1765. [21] D. Dominici, G. Longhi, J. Gomis, and J.M. Pons, Hamilton–Jacobi Theory for Constrained Systems. — J. Math. Phys. 25 (1984), 2439–2452. [22] P. Droz-Vincent, Is Interaction Possible without Heredity? — Lett. Nuovo Cim. 1 (1969), 839–841. [23] S. Duplij, Some Abstract Properties of Semigroups Appearing in Superconformal Theories. — Semigroup Forum 54 (1997), 253–260 (hep-th/9505179). [24] S. Duplij, Analysis of Constraint Systems Using the Clairaut Equation. Proceedings of 5th Mathematical Physics Meeting: Summer School in Modern Mathematical Physics, 6–17 July 2008, B. Dragovich and Z. Rakic (Eds.), Institute of Physics, Belgrade, 2009, 217– 225. [25] S. Duplij, A New Hamiltonian Formalism for Singular Lagrangian Theories. — J. Kharkov National Univ., ser. Nuclei, Particles and Fields 969 (2011), 34–39. [26] D.B. Fairlie, P. Fletcher, and C.K. Zachos, Infinite-Dimensional Algebras and a Trigono- metric Basis for the Classical Lie Algebras. — J. Math. Phys. 31 (1990), 1088–1094. [27] W. Fenchel, On Conjugate Convex Functions. — Canad. J. Math. 1 (1949), 73–77. [28] E. Floratos, J. Iliopoulos, and G. Tiktopoulos, A Note on SU(∞) Classical Yang–Mills Theories. — Phys. Lett. 217 (1989), 285–288. [29] G. Giachetta, L. Mangiarotti, and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, Singapore, 1997. [30] D.M. Gitman and I.V. Tyutin, Canonical Quantization of Constrained Fields. Nauka, Moskow, 1986. [31] D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints. Springer–Verlag, Berlin, 1990. [32] D.M. Gitman and I.V. Tyutin, Hamiltonization of Theories with Degenerate Coordinates. — Nucl. Phys. B630 (2002), 509–527. Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 217 S. Duplij [33] R. Goebel and R.T. Rockafellar, Generalized Conjugacy in Hamilton–Jacobi Theory for Fully Convex Lagrangians. — J. Convex Analysis 9 (2002), 463–474. [34] S.A. Gogilidze, A.M. Khvedelidze, D.M. Mladenov, and H.-P. Pavel, Hamiltonian Reduction of SU(2) Dirac–Yang–Mills Mechanics. — Phys. Rev. D57 (1998), 7488–7500. [35] S.A. Gogilidze, A.M. Khvedelidze, and V.N. Pervushin, On Abelianization of First Class Constraints. — J. Math. Phys. 37 (1996), 1760–1771. [36] H. Goldstein, Classical Mechanics. Addison–Wesley, Reading, 1990. [37] W. Greiner and J. Reinhardt, Field Quantization. Springer, Berlin, 1996. [38] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, Princeton, 1994. [39] N. Hitchin, Generalized Calabi–Yau manifolds. — Quart. J. Math. Oxford Ser. 54 (2003), 281–308. [40] A. Ioffe, Euler–Lagrange and Hamiltonian Formalisms in Dynamic Optimization. — Trans. Amer. Math. Soc. 349 (1997), 2871–2900. [41] S. Izumiya, Systems of Clairaut Type. — Colloq. Math. 66 (1994), 219–226. [42] E. Kamke, Üeber Clairautsche Differentialgleichung. — Math. Zeit. 27 (1928), 623–639. [43] A.M. Khvedelidze, On the Hamiltonian Formulation of Gauge Theories in Terms of Physi- cal Variables. — J. Math. Sci. 119 (2004), 513–555. [44] A. Komar, Constraint Formalism of Classical Mechanics. — Phys. Rev. D18 (1978), 1881– 1886. [45] A. Konechny and A. Schwarz, Introduction to Matrix Theory and Noncommutative Geom- etry. — Phys. Rep. 360 (2002), 353–465. [46] O. Krupkova, Hamiltonian Field Theory. — J. Geom. Phys. 43 (2002), 93–132. [47] O. Krupkova and D. Smetanova, Legendre Transformation for Regularizable Lagrangians in Field Theory. — Lett. Math. Phys. 58 (2001), 189–204. [48] G. Longhi, L. Lusanna, and J.M. Pons, On the Many-time Formulation of Classical Particle Dynamics. — J. Math. Phys. 30 (1989), 1893–1912. [49] F. Loran, Non-Abelianizable First Class Constraints. — Commun. Math. Phys. 254 (2005), 167–178. [50] L. Lusanna, The Second Noether Theorem as the Basis of the Theory of Singular La- grangians and Hamiltonian Constraints. — Riv. Nuovo Cim. 14 (1991), 1–75. [51] V.K. Maltsev, On Canonical Formulation of Field Theories with Singular Lagrangians. — JETP Lett. 27 (1978), 473–475. [52] C.M. Marle, Géométrie des Systèmes Méchaniques à Liaisons Actives. Symplectic Ge- ometry and Mathematical Physics, P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman (Eds.), Birkhäuser, Boston, 1991, 260–287. 218 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 Generalized Duality, Hamiltonian Formalism and New Brackets [53] G. Marmo, G. Mendella, and W. Tulczyjew, Constrained Hamiltonian Systems as Implicit Differential Equations. — J. Phys. A30 (1997), 277–293. [54] G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems: a Differential Geometric Approach to Symmetry and Reduction. J. Wiley, Chichester, 1985. [55] M.R. Menzo and W.M. Tulczyjew, Infinitesimal Symplectic Relations and Deneralized Hamiltonian Dynamics. — Ann. Inst. Henri Poincaré A4 (1978), 349–367. [56] O. Mišković and J. Zanelli, Dynamical Structure of Irregular Constrained Systems. — J. Math. Phys. 44 (2003), 3876–3887. [57] J.M. Pons, On Dirac’s Incomplete Analysis of Gauge Transformations. — Stud. Hist. Philos. Mod. Phys. 36 (2005), 491–518. [58] A. Randono, Canonical Lagrangian Dynamics and General Relativity. — Class. Quant. Grav. 25 (2008), 205017. [59] T. Regge and C. Teitelboim, Constrained Hamiltonian Systems. Academia Nazionale dei Lincei, Rome, 1976. [60] R.T. Rockafellar, Conjugates and Legendre Transforms of Convex Functions. — Canad. J. Math. 19 (1967), 200–205. [61] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, 1970. [62] R.T. Rockafellar, Equivalent Subgradient Versions of Hamiltonian and Euler–Lagrange Equations in Variational Analysis. — SIAM J. Control Opt. 34 (1996), 1300–1315. [63] M.V. Saveliev and A.M. Vershik, New Examples of Continuum Graded Lie Algebras. — Phys. Lett. A143 (1990), 121–128. [64] S. Sternberg, Legendre Transformation of Curves. — Proc. Amer. Math. Soc. 5 (1954), 942–945. [65] E.C. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective. Wiley, New York, 1974. [66] K. Sundermeyer, Constrained Dynamics. Springer–Verlag, Berlin, 1982. [67] W.M. Tulczyjew, The Legendre Transformation. — Ann. Inst. Henri Poincaré A27 (1977), 101–114. [68] W.M. Tulczyjew, Geometric Formulation of Physical Theories. Bibliopolis, Naples, 1989. [69] W.M. Tulczyjew and P. Urbański, A Slow and Careful Legendre Transformation for Singu- lar Lagrangians. — Acta Phys. Pol. B30 (1999), 2909–2977. [70] C. Vallee, M. Hjiaj, D. Fortune, and G. de Saxce, Generalized Legendre–Fenchel Transfor- mation. — Adv. Mech. Math. 6 (2004), 289–312. [71] A.J. van der Schaft, Implicit Hamiltonian Systems with Symmetry. — Rep. Math. Phys. 41 (1998), 203–221. [72] A.M. Wang and T.N. Ran, Corrected Forms of Physical Supplementary Conditions and Gauge Conditions in Singular Lagrangian Systems. — Phys. Rev. Lett. 73 (1994), 2011– 2014. Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 219 S. Duplij [73] S. Weinberg, The Quantum Theory of Fields. Vols. 1, 2, 3. Cambridge University Press, Cambridge, 1995–2000. [74] A. Wipf, Hamilton’s Formalism for Systems with Constraints. In: Canonical Gravity: From Classical to Quantum, J. Ehlers and H. Friedrich (Eds.), Springer, Heidelberg, 1994, 22–58. [75] H. Yoshimura and J.E. Marsden, Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems. In: Lagrangian and Hamiltonian Methods for Nonlinear Control, F. Bullo (Ed.), Springer, Berlin, 2007, 233–247. [76] C. Zachos, Hamiltonian Flows, SU(∞), SO(∞), USp(∞), and Strings, In: Differen- tial Geometric Methods in Theoretical Physics: Physics and Geometry, L.-L. Chau and W. Nahm (Eds.), NATO ASI, Plenum Press, New York, 1990, 423–430. 220 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2

Generalized Duality, Hamiltonian Formalism and New Brackets

Institution

Ähnliche Einträge