s. duplij, constraintless approach to singular theories, new brackets and multi time dynamics

Constraintless approach to singular theories,new brackets and multi-time dynamics

STEVEN DUPLIJ

Institute of Mathematics, University of Munster

http://wwwmath.uni-muenster.de/u/duplij

11 May 2015

1

Abstract

2

1. Description of singular Lagrangian theories by using a Clairaut-

type version of the Hamiltonian formalism.

2. Formulation of a some kind of a nonabelian gauge theory, such

that “nonabelianity” appears due to the Poisson bracket in the

physical phase space.

3. Partial Hamiltonian formalism.

4. Introducing a new (Lie) bracket.

5. Equivalence of a classical singular Lagrangian theory to the

multi-time classical dynamics.

Introduction

Most fundamental physical models are based on gauge field theories

WEINBERG [2000], DELIGNE ET AL. [1999]. On the classical level they

are described by degenerate Lagrangians, which makes the passage

to Hamiltonian description (necessary for quantization) highly

nontrivial and complicated DIRAC [1964], SUNDERMEYER [1982],

GITMAN AND TYUTIN [1986], REGGE AND TEITELBOIM [1976]. A

common way to deal with such theories is considering the Dirac

approach DIRAC [1964] based on extending the phase space and use

of constraints. In spite of its general success, the Dirac approach has

limited applicability and some inner problems PONS [2005].

3

We revisit basic ideas of the Hamiltonian formalism for degenerate

Lagrangians. First result DUPLIJ [2009a, 2011, 2014] is in

generalizing the Legendre transform to degenerate Lagrangians (with

zero Hessian matrix). For that, a mixed (general/envelope) solution of

the multidimensional Clairaut equation was introduced.

Then, a partial Hamiltonian formalism in the phase space having an

initially arbitrary number of momenta (which can be smaller than the

number of velocities) is described DUPLIJ [2015].

We apply the above ideas to construct a self-consistent analog of the

canonical (Hamiltonian) formalism and present an algorithm to

describe any degenerate Lagrangian system without introducing

constraints DIRAC [1964].

4

1 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, which still keeps the emphasis on some obvious details

to appear important below ARNOLD [1989], ROCKAFELLAR [1970].

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 = qA = dqA/dt) on the tangent bundleTM , whereM is a

smooth manifold (configuration spaceQn).

5


By the convex approach definition ( ROCKAFELLAR [1967], ARNOLD

[1989]), a HamiltonianH(qA, pA

)is a dual function on the phase

space T∗M (or convex conjugate ROCKAFELLAR [1970]) to the

Lagrangian (in the second set of variables pA) and is constructed by

means of the Legendre-Fenchel transform LLegFen

7−→ HFen defined by

FENCHEL [1949], ROCKAFELLAR [1967]

HFen(qA, pA

)= sup

vAG(qA, vA, pA

), (1.1)

G(qA, vA, pA

)= pAv

A − L(qA, vA

). (1.2)

It was applied to nonconvex ALART ET AL. [2006] and

nondifferentiable VALLEE ET AL. [2004] L, leading to extensions of

the Hamiltonian formalism ( CLARKE [1977], IOFFE [1997]).

6


In the standard mechanics GOLDSTEIN [1990] one usually restricts to

strictly convex, smooth and differentiable Lagrangians (see, e.g.,

ARNOLD [1989], SUDARSHAN AND MUKUNDA [1974]). Then the

coordinates qA are treated as fixed (passive with respect to the

transform) parameters, and the velocities vA are assumed

independent functions of time. The supremum (1.1) is attained by

finding an extremum point vA = vAextr of the (“pre-Hamiltonian”)

functionG(qA, vA, pA

), which leads to the supremum condition

pB =∂L(qA, vA

)

∂vB

∣∣∣∣∣vA=vAextr

. (1.3)

7


It is assumed ( ARNOLD [1989], SUDARSHAN AND MUKUNDA [1974],

GOLDSTEIN [1990]) that the only way to get rid of dependence on the

velocities vA in the r.h.s. of (1.1) is to resolve (1.3) and find its

solution given by a set of functions

vAextr = VA(qA, pA

). (1.4)

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 Hessian being non-zero

det

∥∥∥∥∥∂2Lnondeg

(qA, vA

)

∂vB∂vC

∥∥∥∥∥6= 0. (1.5)

8


Then substitute vAextr to (1.1) and obtain the standard Hamiltonian

(see, e.g., ARNOLD [1989], GOLDSTEIN [1990])

H(qA, pA

) def= G

(qA, vAextr, pA

)

= pBVB(qA, pA

)− Lnondeg

(qA, V A

(qA, pA

)). (1.6)

The passage from the nondegenerate Lagrangian Lnondeg(qA, vA

)to

the HamiltonianH(qA, pA

)is called the Legendre transform which

will be denoted by Lnondeg Leg7−→ H .

Geometric viewpoint TULCZYJEW [1989], MARMO ET AL. [1985],

ABRAHAM AND MARSDEN [1978], the Legendre transform being

applied to L(qA, vA

)is tantamount to the Legendre transformation

of the tangent to cotangent bundle Leg : TM → T∗M .

9

2 The Legendre-Clairaut transform


Another way to deal with the supremum condition

pB =∂L(qA, vA

)

∂vB(2.1)

is to consider the related multidimensional Clairaut equation DUPLIJ

[2009a, 2011]. The connection of the Legendre transform, convexity

and the Clairaut equation has a long story KAMKE [1928],

STERNBERG [1954] (see also ARNOLD [1988]).

10


We differentiate (1.6) by momenta pA and use the supremum

condition (2.1) to obtain

∂H(qA, pA

)

∂pB= V B

(qA, pA

)

+

pC −∂L(qA, vA

)

∂vC

∣∣∣∣∣vC=V C(qA,pA)

∂VC(qA, pA

)

∂pB= V B

(qA, pA

),

(2.2)

which can be called the dual supremum condition (indeed this gives

the first set of the Hamilton equations, see below). The relations (2.1),

(1.6) and (2.2) together represent a particular case of the Donkin

theorem (see e.g. GOLDSTEIN [1990]).

11


Then we substitute V B(qA, pA

)=∂H

(qA, pA

)

∂pBto (1.5) and

obtain

H(qA, pA

)= pB

∂H(qA, pA

)

∂pB− Lnondeg

(

qA,∂H

(qA, pA

)

∂pA

)

.

(2.3)

which contains no manifest dependence on velocities.

For nondegenerate Lagrangians the relation (2.3) is the identity which

follows from (2.1), (1.6) and (2.2) by our construction.

12


Main idea: to treat (2.4) by itself (without referring to (2.1), (1.6) and

(2.2)) as a definition of a new transform being a solution of the

following nonlinear partial differential equation (the multidimensional

Clairaut equation) DUPLIJ [2009a, 2011]

HCl(qA, λA

)= λB

∂HCl(qA, λA

)

∂λB−L

(

qA,∂HCl

(qA, λA

)

∂λA

)

,

(2.4)

where λA are parameters and L(qA, vA

)is any differentiable

smooth function of 2n variables (without nondegeneracy condition

(1.5)). Call the transform defined by (2.4) LLegCl

7−→ HCl a Clairaut

duality transform (or Legendre-Clairaut transform) andHCl(qA, λA

)

a Hamilton-Clairaut function DUPLIJ [2009a, 2011, 2014].

13


Note that pB =∂L(qA, vA

)

∂vBis commonly treated as a definition of

all dynamical momenta pA, but we should distinguish them from the

parameters of the Clairaut duality transform λA.

In our approach before solving Clairaut equation and applying the

supremum condition (2.1) the parameters λA are not related to the

Lagrangian.

The corresponding Legendre-Clairaut transformation is the map

LegCl : TM → T∗ClM, (2.5)

where the space T∗ClM in local coordinates is just(qA, λA

), and we

call T∗ClM a Clairaut phase space. Note T∗ClM is not a cotangent

bundle, but its submanifold can be the cotangent bundle T∗M .

14

3 Construction of the Hamilton-Clairaut function


The difference between the Legendre-Clairaut transform and the

Legendre transform is crucial for degenerate Lagrangian theories

DUPLIJ [2011, 2014].

The multidimensional Clairaut equation (2.4) has solutions even for

degenerate Lagrangians L(qA, vA

)= Ldeg

(qA, vA

)when the

Hessian is zero

det

∥∥∥∥∥∂2Ldeg

(qA, vA

)

∂vB∂vC

∥∥∥∥∥= 0. (3.1)

15


So the Legendre-Clairaut transform (2.4) LegCl is another (along with

LegFen) counterpart to the ordinary Legendre transform (1.6) in the

case of degenerate Lagrangians. To make this manifest and to find

solutions of the Clairaut equation (2.4), we differentiate it by λA to

obtain

λB −

∂L(qA, vA

)

∂vB

∣∣∣∣∣vA=

∂HCl(qA,λA)∂λA

∙∂2HCl

(qA, λA

)

∂λB∂λC= 0.

(3.2)

Take into account two possibilities depending on which multiplier in

(3.2) is zero:

16


1) The envelope solution defined by the first multiplier in (3.2) is zero

λB = pB =∂L(qA, vA

)

∂vB, (3.3)

which coincides with the supremum condition (2.1) together with (2.2).

So we obtain the standard Hamiltonian (1.6)

HClenv

(qA, λA

)|λA=pA = H

(qA, pA

). (3.4)

Thus, in the nondegenerate case the “envelope” Legendre-Clairaut

transform LegClenv coincides with the ordinary Legendre transform by

our early construction DUPLIJ [2009a, 2011].

17


2) A general solution is defined by

∂2HCl(qA, λA

)

∂λA∂λB= 0 (3.5)

which gives∂HCl

(qA, λA

)

∂λB= cB , here cA are smooth functions of

qA, and cA are considered as parameters (passive variables). Then

HClgen

(qA, λA, c

A)= λBc

B − L(qA, cA

), (3.6)

corresponding to a “general” Legendre-Clairaut transform LegClgen.

The general solutionHClgen

(qA, λA, c

A)

is always linear in the

variables λA, which do not coincide with the dynamical momenta pA

i.e. λA 6= pA because we do not have the envelope solution

18


condition (3.3) and therefore now there is no supremum condition

(2.1) pB =∂L(qA, vA

)

∂vB.

The variables cA are in fact unresolved velocities vA in case of the

general solution.

In the standard way LegClenv can be obtained by finding the envelope

of the general solution ARNOLD [1988], i.e. differentiating (3.6) by cA

∂HClgen

(qA, λA, c

A)

∂cB= λB −

∂L(qA, cA

)

∂cB= 0 (3.7)

which coincides with (3.3) and (2.1). ThusHClgen is actually the

“pre-Hamiltonian” (1.2)

HClgen

(qA, λA, c

A)|cA=vA,λA=pA = G

(qA, vA, pA

)=

pAvA − L

(qA, vA

).

19

4 The mixed Legendre-Clairaut transform


Consider a degenerate Lagrangian theory

L(qA, vA

)= Ldeg

(qA, vA

)for which the Hessian is zero (3.1).

The rank of Hessian matrixWAB =∂2L(qA,vA)∂vB∂vC

is r < n, and we

suppose that r is constant. We rearrange indices ofWAB in such a

way that a nonsingular minor of rank r appears in the upper left

corner GITMAN AND TYUTIN [1990]. Represent the index A as

follows: if A = 1, . . . , r, we replace A with i (the “regular” Latin

index), and, if A = r + 1, . . . , n we replace A with α (the

“degenerate” Greek index).

Obviously, detWij 6= 0, and rankWij = r.

20


The standard Legendre transform Leg is not applicable in the

degenerate case because the condition

det

∥∥∥∥∥∂2Lnondeg

(qA, vA

)

∂vB∂vC

∥∥∥∥∥= detWAB 6= 0. (4.1)

is not valid CARINENA [1990], TULCZYJEW [1977]. Therefore the

supremum condition pB =∂L(qA, vA

)

∂vBcannot be resolved under

degenerate A = α, but it can be resolved under regular A = i only,

because detWij 6= 0.

On the contrary, the Clairaut duality transform given by (2.4) is

independent of the Hessian being zero or not DUPLIJ [2009a, 2011].

21


Main idea:

The ordinary duality can be generalized to the Clairaut duality .

The standard Legendre transform Leg as

H(qA, pA

)= pBV

B(qA, pA

)− L

(qA, V A

(qA, pA

))(4.2)

can be generalized to the degenerate Lagrangian theory using the

Legendre-Clairaut transform LegCl given by the multidimensional

Clairaut equation (2.4)

H(qA, λA

)= λB

∂H(qA, λA

)

∂λB−L

(

qA,∂H

(qA, λA

)

∂λA

)

(4.3)

with λA = pA at least for some A.

22


We differentiate (2.4) by λA and split the sum (3.2) in B as follows[

λi −∂L(qA, vA

)

∂vi

]

∙∂2HCl

(qA, λA

)

∂λi∂λC

+

[

λα −∂L(qA, vA

)

∂vα

]

∙∂2HCl

(qA, λA

)

∂λα∂λC= 0. (4.4)

As detWij 6= 0, we suggest to replace (4.4) by the conditions

λi = pi =∂L(qA, vA

)

∂vi, (4.5)

∂2HCl(qA, λA

)

∂λα∂λC= 0. (4.6)

23


In this way we obtain a mixed envelope/general solution of the Clairaut

equation DUPLIJ [2009a, 2011, 2014]. After resolving (4.5) under

regular velocities vi = V i(qA, pi, c

α)

and writing down a solution of

(4.6) as∂HCl

(qA, λA

)

∂λα= cα (where cα being actually unresolved

velocities vα) we obtain a mixed Hamilton-Clairaut function

HClmix

(qA, pi, λα, v

α)= piV

i(qA, pi, v

α)

+ λαvα − L

(qA, V i

(qA, pi, v

α), vα), (4.7)

which is the “mixed” Legendre-Clairaut transform LLegCl

mix7−→ HClmix . a

aNote that (4.7) coincides with the “slow and careful Legendre transformation” of

TULCZYJEW AND URBANSKI [1999] and with the “generalized Legendre

transformation” of CENDRA ET AL. [1998].

24


We obtain in a standard way the system of equations which gives a

Hamiltonian-Clairaut description of a degenerate Lagrangian theory

∂HClmix∂pi

=dqi

dt, (4.8)

∂HClmix∂λα

=dqα

dt, (4.9)

∂HClmix∂qi

= −dpi

dt+[λβ − Bβ

(qA, pi

)] ∂vβ

∂qi, (4.10)

∂HClmix∂qα

=dBα

(qA, pi

)

dt+[λβ − Bβ

(qA, pi

)] ∂vβ

∂qα. (4.11)

This system has two disadvantages: 1) It contains the “nondynamical”

momenta λα; 2) It has derivatives of unresolved velocities vα.

25


To get rid of them, we introduce a “physical” Hamiltonian

Hphys(qA, pi

)= HClmix

(qA, pi, λα, v

α)−[λβ − Bβ

(qA, pi

)]vβ.

(4.12)

We can rewrite the “physical” Hamiltonian (4.12) using (4.7) as

Hphys(qA, pi

)= piV

i(qA, pi, v

α)

+Bα(qA, pi

)vα − L

(qA, V i

(qA, pi, v

α), vα), (4.13)

Using (4.5), we see that the r.h.s. of (4.12) indeed doesn’t depend on

“nondynamical” momenta λα and degenerate velocities vα, such that

∂Hphys

∂vα= 0. (4.14)

26


Finally, we obtain from (4.8)–(4.11) the sought-for system of (first

order differential) equations (the Hamilton-Clairaut system of

equations) which describes any degenerate Lagrangian theory

dqi

dt={qi, Hphys

(qA, pi

)}reg+{qi, Bβ

(qA, pi

)}reg

dqβ

dt, i = 1, . . . r

(4.15)

dpi

dt={pi, Hphys

(qA, pi

)}phys+{pi, Bβ

(qA, pi

)}phys

dqβ

dt, i = 1, . . . r

(4.16)

Fαβ(qA, pi

) dqβ

dt= DαHphys

(qA, pi

), α = r + 1, . . . , n.

(4.17)

27


Here {X,Y }phys =∂X∂qi∂Y∂pi− ∂Y∂qi∂X∂pi

is the regular (physical)

Poisson bracket (in regular variables).

A “qα-long (covariant) derivative”

DαX =∂X

∂qα+{Bα(qA, pi

), X}phys, (4.18)

whereX = X(qA, pi

)is a smooth scalar function.

A “qα-field strength” of the “qα-gauge fields”

Fαβ(qA, pi

)=∂Bβ

(qA, pi

)

∂qα−∂Bα

(qA, pi

)

∂qβ

+{Bα(qA, pi

), Bβ

(qA, pi

)}phys. (4.19)

28

5 Nonabelian gauge theory interpretation


Let us consider a “qα-long derivative”

DαX =∂X

∂qα+ {Bα, X}phys , (5.1)

whereX = X(qA, pi

)is a smooth scalar function. We also

interpret in (4.17) a “qα-field strength” Fαβ ≡ Fαβ(qA, pi

)of the

“qα-gauge fields” Bα defined by

Fαβ =∂Bβ

∂qα−∂Bα

∂qβ+ {Bα, Bβ}phys . (5.2)

29


The system of equations (4.17) for unresolved velocities is

n∑

β=r+1

Fαβdqβ

dt= DαHphys, α = r + 1, . . . , n. (5.3)

The “qα-field strength” Fαβ is nonabelian due to the presence of the

“physical” Poisson bracket in r.h.s. of (5.2). It is important to observe

that, in distinct of the ordinary Yang-Mills theory, the partial derivatives

of Bα in (5.2) are defined in the qα-subspace Rn−r, while the

“nonabelianity” (third term) is due to the Poisson bracket in another

symplectic subspace Sp (r, r).

The “qα-long derivative” satisfies the Leibniz rule

Dα {Bβ, Bγ}phys = {DαBβ, Bγ}phys + {Bβ, DαBγ}phys

30


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 . (5.4)

It follows from (5.4)

DαDβFαβ = 0. (5.5)

31


Let us introduce the Bα-transformation

δBαX = {Bα, X}phys , (5.6)

which satisfies

(δBαδBβ − δBβδBα

)Bγ = δ{Bα,Bβ}

phys

Bγ, (5.7)

δBαFβγ(qA, pi

)= (DγDβ −DβDγ)Bα, (5.8)

δBα {Bβ, Bγ}phys = {δBαBβ, Bγ}phys + {Bβ, δBαBγ}phys .

(5.9)

This means that the “qα-long derivative”Dα (4.18) is in fact a

“qα-covariant derivative” with respect to the Bα-transformation (5.6).

32


Observe that “Dα transforms as fields” (5.6), which proves that it is

really covariant (note the cyclic permutations in both sides)

δBαDβBγ + δBγDαBβ + δBβDγBα

= {Bα, DβBγ}phys + {Bγ, DαBβ}phys + {Bβ, DγBα}phys .

(5.10)

The “qα-Maxwell” equations of motion for the “qα-field strength” are

DαFαβ = Jβ, (5.11)

DαFβγ +DγFαβ +DβFγα = 0, (5.12)

where Jα ≡ Jα(qA, pi

)is a “qα-current” which is a function of the

initial Lagrangian (1.2) and its derivatives up to third order.

33


Due to (5.5) the “qα-current” Jα is conserved

DαJα = 0. (5.13)

Thus, a singular Lagrangian system leads effectively to some special

kind of the nonabelian gauge theory in the direct product space

Mphys = Rn−r × Sp (r, r). Here the “nonabelianity” (third term in

(5.2)) 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.,

BUEKEN [1996]).

34


The analogous Poisson type of “nonabelianity” (5.2) appears in the

N →∞ limit of Yang-Mills theory, and it is called the “Poisson gauge

theory” FLORATOS ET AL. [1989]. In the SU (∞) Yang-Mills theory

the group indices become surface coordinates FAIRLIE ET AL. [1990],

and it is connected with the Schild string ZACHOS [1990]. The related

algebra generalizations are called the continuum graded Lie algebras

SAVELIEV AND VERSHIK [1990]. Here, because of the direct product

structure of the spaceMphys, the similar construction appears in

(5.2), while the “long derivative” (4.18), “gauge transformations” (5.6)

and the analog of the Maxwell equations (5.11)–(5.12) differ from the

“Poisson gauge theory” FLORATOS ET AL. [1989].

Some applications of the presented formalism for two-color QCD was

recently done in WALKER AND DUPLIJ [2015] (Phys. Rev. D).

35

6 Partial Hamiltonian formalism


6.1 Preliminaries. The standard Hamiltonian formalism

The standard principle of least action LANCZOS [1962] considers the

functional S =∫ tt0Ldt′, where a differentiable function

L = L(t, qA, qA

)is a Lagrangian, and the functional (on extremals)

S = S(t, qA

)is the action of a dynamical system as a function of

the upper limit of t (a fixed lower limit t0). Without loss of generality,

δqA (t0) = 0, while the upper denotes the trajectory δqA.

For the variation of δS after integration by parts

δS =

∫t

t0

(∂L

∂qA−d

dt′

(∂L

∂qA

))δqA(t′)dt′+∂L

∂qAδqA+

(L−

∂L

∂qAqA)δt, (6.1)

36


Then, from δS = 0, we obtain the equations of motion

(Euler-Lagrange equations) LANDAU AND LIFSHITZ [1969]

∂L

∂qA−d

dt

(∂L

∂qA

)

= 0 A = 1, . . . , n. (6.2)

The second and third terms in (6.1) define the total differential of the

action (on extremals) as a function of (n+ 1) variables: the

coordinates and the upper limit of integration in (6.1)

dS =∂L

∂qAdqA +

(

L−∂L

∂qAqA)

dt. (6.3)

Thus, from the definition of the action (6.1) and (6.3), it follows thatdS

dt= L,

∂S

∂qA=∂L

∂qA,∂S

∂t= L−

∂L

∂qAqA. In the standard

Hamiltonian formalism LANDAU AND LIFSHITZ [1969], to each

37


coordinate qA one can assign the canonically conjugate momentum

pA by the formula

pA =∂L

∂qA, A = 1, . . . , n. (6.4)

If the system of equations (6.4) is solvable for all velocities qA, we can

define the Hamiltonian by the Legendre transform LANDAU AND

LIFSHITZ [1969]

H = pAqA − L, (6.5)

which defines the mapping TQ∗n → TQn ARNOLD [1989].

The r.h.s. of (6.5) is expressed in terms of the momenta, so that

H = H(t, qA, pA

)is a function on the phase space (or the

cotangent bundle TQ∗n of rank n and the dimension of the total space

38


2n), that is independent of 2n canonical coordinates(qA, pA

). In the

standard canonical formalism, each coordinate qA has its conjugate

momentum pA by the formula (6.4), we call it a full Hamiltonian

formalism (in the full phase space TQ∗n). Then the differential of the

action (6.3) can also be written as

dS = pAdqA −Hdt. (6.6)

∂S

∂qA= pA,

∂S

∂t= −H

(t, qA, pA

), (6.7)

The Hamilton-Jacobi differential equation

∂S

∂t+H

(

t, qA,∂S

∂qA

)

= 0, (6.8)

which determines the dynamics in terms ofH(t, qA, pA

).

39


Variation of the action S =∫ (pAdq

A −Hdt)

while considering the

coordinates and momenta as independent variables, gives the

Hamilton’s equations in the differential form LANDAU AND LIFSHITZ

[1969] to

dqA =∂H

∂pAdt, dpA = −

∂H

∂qAdt (6.9)

for the full Hamiltonian formalism.

40


In terms of the (full) Poisson bracket

{A,B}full =∂A

∂qA∂B

∂pA−∂B

∂qA∂A

∂pA(6.10)

(6.9) can be written in standard form LANDAU AND LIFSHITZ [1969]

dqA ={qA, H

}fulldt, dpA = {pA, H}full dt. (6.11)

It is clear that the two formulations of the principle of the least action,

that is (6.1) and (6.1), are completely equivalent. Now we will show in

this simple language, how to describe the same dynamics using fewer

generalized momenta than the number of generalized coordinates.

41


6.2 Hamiltonian formalism with reduced number of momenta

The transition from a full to a partial Hamiltonian formalism and a

multi-time dynamics can be analyzed using the following well-known

classical analogy LANCZOS [1962], but in its reverse form. Recall, the

extended phase space with the following additional position and

momentum

qn+1 = t, (6.12)

pn+1 = −H. (6.13)

42


Then the action (6.1) takes the symmetrical form and contains only

the first term LANCZOS [1962] (see, also ARNOWITT ET AL. [1962])

S =

∫ n+1∑

A=1

pAdqA. (6.14)

Here we proceed in the opposite way, and ask: can we, on the

contrary, not increase, but reduce the number of momenta that

describe the dynamical system? Can we formulate a version of

so-called partial Hamiltonian formalism, which would be equivalent (at

the classical level) to the Lagrangian formalism?

The answer is: Yes.

43


We define a partial Hamiltonian formalism DUPLIJ [2013, 2015], so

that the conjugate momentum is associated not to every qA by the

formula (6.4), but only for the first np < n generalized coordinates,

which are called canonical and denoted by qi, i = 1, . . . np. The

resulting tangent bundle TQ∗np is defined by 2np (reduced) canonical

coordinates (qi, pi). The rest of the generalized coordinates are

called noncanonical qα, α = np + 1, . . . n, and they form a

configuration subspaceQn−np , which corresponds to the tangent

bundle TQn−np (a subscript indicates the corresponding dimension

of the total space). Thus, the dynamical system is now given on the

direct product (of manifolds) TQ∗np × TQn−np .

44


For reduced generalized momenta we have

pi =∂L

∂qi, i = 1, . . . , np. (6.15)

A partial Hamiltonian, similar to (B.5), is defined by a partial Legendre

transform

H0 = piqi +∂L

∂qαqα − L, (6.16)

which defines a mapping of TQ∗np × TQn−np → TQn. In (6.16) the

canonical generalized velocities qi are expressed in terms of the

reduced canonical momenta pi by (6.15). For the action differential

(6.6) we can write

dS = pidqi +∂L

∂qαdqα −H0dt. (6.17)

45


We define

Hα = −∂L

∂qα, α = np + 1, . . . n (6.18)

dS = pidqi −Hαdq

α −H0dt. (6.19)

Without the second term in (6.19) the partial Hamiltonian (6.16) is the Routh function in terms of which the Lagrange equations of motion

can be reformulated LANDAU AND LIFSHITZ [1969].

In the partial Hamiltonian formalism the dynamics is completely

determined by the set of (n− np + 1) HamiltoniansH0,Hα,

α = np + 1, . . . n.

46

and call them additional Hamiltonians (note H α = −B α )


It follows from (6.19) that the partial derivatives of S = S (t, qi, qα)

are ∂S

∂qi= pi,

∂S

∂qα= −Hα

(t, qi, pi, q

α, qα),∂S

∂t= −H0

(t, qi, pi, q

α, qα), which

yields the system (n− np + 1) of Hamilton-Jacobi equations

∂S

∂t+H0

(

t, qi,∂S

∂qi, qα, qα

)

= 0, (6.20)

∂S

∂qα+Hα

(

t, qi,∂S

∂qi, qα, qα

)

= 0. (6.21)

Now, on the direct product of TQ∗np × TQn−np , the action is

S =

∫ (pidq

i −Hαdqα −H0dt

). (6.22)

Variation of (6.22) should be made independently on 2np-reduced

coordinates qi, pi and (n− np) noncanonical coordinates qα.

47


The equations of motion for the partial Hamiltonian formalism can be

derived from δS = 0. We introduce the reduced Poisson bracket for

two functions A and B in the reduced phase space TQ∗np

{A,B} =∂A

∂qi∂B

∂pi−∂B

∂qi∂A

∂pi. (6.23)

48


The equations of motion on TQ∗np × TQn−np in the differential form

dqi ={qi, H0

}dt+

{qi, Hβ

}dqβ, (6.24)

dpi = {pi, H0} dt+ {pi, Hβ} dqβ, (6.25)

∂Hα

∂qβdqβ +

d

dt

(∂H0

∂qα+∂Hβ

∂qαqβ)

dt =

(∂Hβ

∂qα−∂Hα

∂qβ+ {Hβ, Hα}

)

dqβ

+

(∂H0

∂qα−∂Hα

∂t+ {H0, Hα}

)

dt. (6.26)

On TQ∗np the first-order equations (6.24)–(6.25). On (noncanonical)

subspace TQn−np the equation (6.26) is still of second order.

49


The equations of motions of the partial Hamiltonian formalism are

qi ={qi, H0

}+{qi, Hβ

}qβ, (6.27)

pi = {pi, H0}+ {pi, Hβ} qβ, (6.28)

∂Hα

∂qβqβ +

d

dt

(∂H0

∂qα+∂Hβ

∂qαqβ)

=

(∂Hβ

∂qα−∂Hα

∂qβ+ {Hβ, Hα}

)

qβ

+

(∂H0

∂qα−∂Hα

∂t+ {H0, Hα}

)

. (6.29)

The number of reduced momenta np is a free parameter together with

the number (n− np) of the additional HamiltoniansHα

0 ≤ np ≤ n. (6.30)

50


In other words, the dynamics is independent of the dimension of the

reduced phase space. In this case, the boundary values of np

correspond to the standard Lagrangian and Hamiltonian formalisms,

respectively, so that we have the three cases:

1. np = 0— the Lagrangian formalism on TQn (we have only the

last equation (6.29) without the Poisson brackets), and

α = 1, . . . , n;

2. 0 < np < n— the partial Hamiltonian formalism for

TQ∗np × TQn−np (all of the equations are considered);

3. np = n— the standard Hamiltonian formalism on TQ∗n (the first

two equations (6.27 )–(6.28) without the second terms, coinciding

with the standard Hamiltonian equations (6.9), i = 1, . . . , n.

51


We can show that in the case 1) we obtain the standard Lagrange

equations for the noncanonical variables qα. We derive from (4.9)

d

dt

∂

∂qα(H0 +Hβ q

β)=∂

∂qα(H0 +Hβ q

β). (6.31)

The formula to determine the partial Hamiltonian (6.16) without

variables qi, pi is

H0 = −Hαqα − L. (6.32)

Hence,H0 +Hβ qβ = −L, we obtain the standard Lagrange

equations in the noncanonical sector

d

dt

∂

∂qαL =

∂

∂qαL. (6.33)

52


A non-trivial dynamics in the noncanonical sector is determined by the

presence of terms with second derivatives. Consider a special case of

the partial Hamiltonian formalism, when these terms (with second

derivatives) vanish and call it nondynamical in the noncanonical sector

(because, there are no noncanonical accelerations qα). This requires

∂H0

∂qβ= 0,

∂Hα

∂qβ= 0, α, β = np + 1, . . . , n. (6.34)

Then (6.29) will have only the right side, which can be written as(∂Hβ

∂qα−∂Hα

∂qβ+ {Hβ, Hα}

)

qβ = −

(∂H0

∂qα−∂Hα

∂t+ {H0, Hα}

)

,

(6.35)

which is a system of linear algebraic equations for the noncanonical

velocities qα with the given HamiltoniansH0,Hα.

53

7 Singular theories in partial Hamiltonian formalism


We express the conditions (6.34) in terms of the Lagrangian to obtain

∂H0

∂qβ= 0,

∂Hα

∂qβ= 0 =⇒

∂2L

∂qα∂qβ= 0, α, β = np+1, . . . , n.

(7.1)

Therefore, the rank r of the Hessian matrix

WAB =

∥∥∥∥∂2L

∂qA∂qB

∥∥∥∥ , A,B = 1, . . . , n (7.2)

is not only less than the dimension n of the configuration space, but

less than or equal to the number of momenta (due to (7.1))

r ≤ np. (7.3)

54


The definition of “extra ” (np − r) momenta results in a (np − r)

relations, just as in the Dirac theory of constraints DIRAC [1964].

1) In the standard Hamiltonian formalism there are exactly (n− r)

constraints, because the dimension n of the configuration space

(np = n) and rank of the Hessian r are fixed by the problem.

2) In the partial Hamiltonian formalism the number np of momenta is

a free parameter that can be chosen so that the constraints do not

appear i.e. (np − r) = 0. Thus, to describe singular theories without

using constraints it is natural to equate the number of momenta with

the rank of the Hessian

np = r. (7.4)

55


Main idea. The singular dynamics (theory with degenerate

Lagrangian) can be formulated in a way that constraints will not occur.

In the Hessian matrixWAB (7.2) we rename the indices so that the

non-singular minor rank r is in the upper left hand corner, denote with

the Latin alphabet i, j the first r indices and with Greek letters α, β

the remaining (n− r) indices. The equations of motion are

qi ={qi, H0

}+{qi, Hβ

}qβ, (7.5)

pi = {pi, H0}+ {pi, Hβ} qβ, (7.6)

Fαβ qβ = Gα, (7.7)

where nowH0 = Hphys and { } becomes { }phys.

56


Here the index values are connected with the rank of the Hessian by

i = 1, . . . , r, α, β = r + 1, . . . , n, and

Fαβ =∂Hα

∂qβ−∂Hβ

∂qα+ {Hα, Hβ} , (7.8)

Gα = DαH0 =∂H0

∂qα−∂Hα

∂t+ {H0, Hα} . (7.9)

The system of equations (7.5 )–(7.9) coincides with the equations

derived in the approach to singular theories (4.15 )–(4.17), which uses

the mixed solutions of Clairaut’s equation DUPLIJ [2009b, 2011]

(except for the term with the partial time derivative ofHα in (7.9)).

57

8 Classification, gauge freedom and new brackets


Next we classify degenerate Lagrangian theories as follows. There

are two possibilities determined by the rank of the matrix Fαβ :

1. Nongauge theory, when rankFαβ = rF = n− r is full, so that

the matrix Fαβ is invertible. Then

qα = F αβGβ, (8.1)

where F αβ is the inverse matrix of Fαβ , defined by the equation

F αβFβγ = FγβFβα = δαγ .

2. Gauge theory, the rank of the Fαβ is incomplete, that is,

rF < n− r, and the matrix Fαβ is noninvertible.

58


In this case, we can find from (7.7) only rF noncanonical

velocities, while the rest (n− r − rF ) of the velocities remain

arbitrary gauge parameters that correspond to the symmetries of

a singular dynamical system. In the particular case rF = 0 (or

zero matrix Fαβ) from (7.7) we obtain

Gα = 0, (8.2)

and all the noncanonical velocities correspond to (n− r) gauge

parameters of the theory.

59


In the first case all the noncanonical velocities can be excluded and

the Hamilton-like equations for nongauge singular systems

qi ={qi, H0

}nongauge

, (8.3)

pi = {pi, H0}nongauge , (8.4)

where a new (nongauge) bracket for A,B is

{A,B}nongauge = {A,B}+DαA ∙ Fαβ ∙DβB, (8.5)

andDα is defined in (7.9). The nongauge bracket (8.5) uniquely

determines the evolution of any dynamical quantity A in time

dA

dt=∂A

∂t+ {A,H0}nongauge . (8.6)

60


The nongauge bracket (8.5) has all the properties of the Poisson

bracket: it is antisymmetric and satisfies the Jacobi identity.

In the second case (gauge theory), only some of the noncanonical

velocities qα can be eliminated, the number of which is equal to the

rank rF of the matrix Fαβ , and the rest (n− r − rF ) of the

velocities are still arbitrary and can serve as gauge parameters. Then

in the system (7.7), only the first rF equations are independent. We

present (“noncanonical”) indices α, β = r + 1, . . . , n as pairs

(α1, α2), (β1, β2), where α1, β1 = r + 1, . . . , rF number the first

rF independent rows of the matrix Fαβ and correspond to the

nonsingular minor of Fα1β1 , the remaining (n− r − rF ) rows will be

dependent on the first ones, and α2, β2 = rF + 1, . . . , n.

61


Then the system (7.7) can be written as

Fα1β1 qβ1 + Fα1β2 q

β2 = Gα1 , (8.7)

Fα2β1 qβ1 + Fα2β2 q

β2 = Gα2 . (8.8)

Since Fα1β1 is non-singular by construction, we can express

qα1 = F α1β1Gβ1 − Fα1β1Fβ1α2 q

α2 , (8.9)

where F α1β1 is rF × rF -matrix which is inverse to Fα1β1 .

Since rankFα1β1 = rF , other blocks can be expressed via Fα1β1

Fα2β1 = λα1α2Fα1β1 , (8.10)

Fα2β2 = λα1α2Fα1β2 = λ

α1α2λγ1β2Fα1γ1 , (8.11)

Gα2 = λα1α2Gα1 , (8.12)

62


where λα1α2 = λα1α2(qi, pi, q

α) are rF × (n− r − rF ) smooth

functions. Since the matrix Fαβ is given, we can determine the

function λα1α2 by rF × (n− r − rF ) equations (8.10)

λα1α2 = Fα2β1Fα1β1 . (8.13)

Because (n− r − rF ) velocities qα2 are arbitrary, we can put them

equal to zero

qα2 = 0, α2 = rF + 1, . . . , n, (8.14)

which can be considered as a gauge condition. Then from (4.15), it

follows that

qα1 = F α1β1Gβ1 , α1 = r + 1, . . . , rF . (8.15)

63


By analogy with (8.5), we introduce a new (gauge) bracket

{A,B}gauge = {A,B}+Dα1A ∙ Fα1β1 ∙Dβ1B. (8.16)

The equations of motion can be written in the Hamiltonian-like form

qi ={qi, H0

}gauge

, (8.17)

pi = {pi, H0}gauge . (8.18)

The evolution of a physical quantity A in time, as (8.6), is determined

by the gauge bracket (8.16)

dA

dt=∂A

∂t+ {A,H0}gauge . (8.19)

64


In the particular case, when rF = 0, we have

Fαβ = 0, (8.20)

and hence all additional Hamiltonians vanishHα = 0, then it is seen

from the definition (6.18) that the Lagrangian does not depend on the

noncanonical velocities qα. Therefore, taking into account (8.20) and

(7.7) we find that the partial HamiltonianH0 does not depend on the

noncanonical generalized coordinates qα

∂H0

∂qα= 0, (8.21)

ifH0 is manifestly independent of time. In this case, the gauge

bracket coincides with the Poisson bracket, since the second term in

(8.16) vanishes.

65

9 Singular theory as multi-time dynamics


The conditions (6.34) ∂H0∂qβ

= 0,∂Hα

∂qβ= 0 mean that all the Hamiltonians

in the partial Hamiltonian formalism do not depend on the velocities

qα, that is,H0 = H0 (t, qi, pi, q

α),Hα = Hα (t, qi, pi, q

α),

i = 1, . . . , np, α = np + 1, . . . , n. Thus, the dynamical problem is

defined on the manifold TQ∗np ×Qn−np , so that qα actually play the

role of real parameters, similar to the time. In this case the

nondynamicalQn−np is isomorphic to the real space Rn−np .

66


Therefore, we can interpret (n− np) canonical generalized

coordinates qα as (n− np) “extra” (to t) times, andHα as (n− np)corresponding Hamiltonians. Indeed, we introduce DUPLIJ [2015]

τμ = t, Hμ = H0, μ = 0, (9.1)

τμ = qμ+np , Hμ = Hμ+np , μ = 1, . . . , (n− np) , (9.2)

where Hμ = Hμ (τμ, qi, pi) are Hamiltonians of the multi-time

dynamics with nμ = n− np + 1 times τμ. Note that τμ can be

called generalized times, because they are not real times (for the real

multi-time physics see DORLING [1970], DVALI ET AL. [2000] and

review BARS AND TERNING [2010]), in the same way as generalized

coordinates have nothing to do with space-time coordinates LANDAU

AND LIFSHITZ [1969].

67


The differential of the action S = S (τμ, qi) in the multi-time dynamics

is LONGHI ET AL. [1989] (see also ARNOWITT ET AL. [1962])

dS = pidqi − Hμdτ

μ, i = 1, . . . , np, μ = 0, . . . , (n− np)

(9.3)∂S

∂qi= pi,

∂S

∂τμ= −Hμ. (9.4)

The system of (n− np + 1) Hamilton-Jacobi equations is

∂S

∂τμ+ Hμ

(

τμ, qi,∂S

∂qi

)

= 0, μ = 0, . . . , (n− np) . (9.5)

68


We note that from (6.1) and (9.4) it follows the relation between Hμ.

Differentiating the Hamilton-Jacobi equations (6.1) and further

antisymmetrization gives the integrability condition

∂2S

∂τ ν∂τμ−∂2S

∂τμ∂τ ν=∂Hμ∂τ ν−∂Hν∂τμ+ {Hμ,Hν} = 0. (9.6)

We set δS = 0, where S =∫(pidq

i − Hμdτμ), and obtain

δS =

∫δpi

(

dqi −∂Hμ∂pidτμ)

+

∫δqi(

−dpi −∂Hμ∂qidτμ)

.

(9.7)

69


Therefore, the equations of motion for the multi-time dynamics can be

written in differential form LONGHI ET AL. [1989]

dqi ={qi,Hμ

}dτμ, (9.8)

dpi = {pi,Hμ} dτμ, (9.9)

which coincide with (6.24)–(6.25) by construction. The integrability

conditions (9.6) can be also written in differential forma

(∂Hμ∂τ ν−∂Hν∂τμ+ {Hμ,Hν}

)

dτ ν = 0, μ, ν = 0, . . . , (n− np)

(9.10)

which coincide with the equations (6.35), written in differential form.aAs in the standard case, ARNOLD [1989], the equations (9.8 )–(9.10) are the

conditions for closure of a differential 1-form (9.3).

70


We have shown that the nondynamical sector in the noncanonical

version of the partial Hamiltonian formalism without any conditions on

the number of moments np can be formulated as the multi-time

dynamics with the number of generalized times (and corresponding

Hamiltonians Hμ) nμ = n− np + 1 . In this formulation the number

of generalized times nμ is not fixed, and 1 ≤ nμ ≤ n+ 1, and

nμ + np = n+ 1, (9.11)

which can be called a times-momenta rule DUPLIJ [2015].

71


In the particular case of singular theories (with degenerate

Lagrangians), the number of momenta np is fixed by the condition

(7.4) np = r. From (9.11) we obtain

nμ + r = n+ 1, (9.12)

which can be called a times-rank rule DUPLIJ [2015].

Thus, a singular theory can be effectively described as the multi-time

dynamics, such that (6.34) ∂H0∂qβ

= 0,∂Hα

∂qβ= 0 and (9.12) are fulfilled.

72

10 Origin of constraints in singular theories


Introduction of additional dynamical variables must necessarily give

rise to additional relationships between them. Consider the “extra”

(n− r) momenta pα (since we have a complete description of the

dynamics without them), which correspond to noncanonical

generalized velocities qα for the standard definition DIRAC [1964]

pα =∂L

∂qα, α = r + 1, . . . , n. (10.1)

So (10.1), together with the definition of the partial generalized

canonical momenta (6.15) coincide with “full” momenta (6.4)

pA =∂L

∂qA, A = 1, . . . , n.

73


From the definition of additional Hamiltonians (6.18)Hα, we get

(n− r) relations

Φα = pα +Hα = 0, α = r + 1, . . . , n, (10.2)

which are called the (primary) constraints DIRAC [1964] (in a resolved

form). These relations (10.2) are similar to the standard procedure of

extension of the phase space (6.13). One can enter any number

n(add)p of “extra” momenta 0 ≤ n(add)p ≤ n− r, then the theory will

have the same number n(add)p of (primary) constraints. We have

1) In the partial Hamiltonian formalism n(add)p = 0.

2) In the itermediate case 0 ≤ n(add)p ≤ n− r (a specific task).

2) In the Dirac theory, n(add)p = n− r.

74


If we have the “extra” momenta pα, as in Dirac approach, the

transition to the Hamiltonian by the standard formula

Htotal = piqi + pαq

α − L, (10.3)

cannot be done directly, therefore,Htotal is also not a “true”

Hamiltonian, asH0 in (6.16). It is impossible to express the

noncanonical velocities qα through the “extra” momenta pα and then

apply the standard Legendre transformation. But it is possible to

transformHtotal (10.3) in such a way that one can use the method of

undetermined coefficients DIRAC [1964]. The total Hamiltonian can

be written as

Htotal = H0 + qαΦα, (10.4)

where qα play the role of undetermined coefficients.

75


The equations of motion can be written in a Hamiltonian-like form in

terms of the total HamiltonianHtotal and the full Poisson bracket

dqA ={qA, Htotal

}fulldt, (10.5)

dpA = {pA, Htotal}full dt (10.6)

with the (n− r) conditions (10.2) Φα = pα +Hα = 0.

However, the equations (10.5)–(10.6) and (10.2) are not sufficient to

solve the problem: to find the equations for the undetermined

coefficients qα in (10.4). These equations can be derived from some

additional principles, such as conservation relations (10.2) in time

DIRAC [1964]dΦαdt= 0. (10.7)

76


The time dependence of any physical quantity A is now determined

by the total Hamiltonian and the full Poisson bracket

dA

dt=∂A

∂t+ {A,Htotal}full . (10.8)

If the constraints do not depend explicitly on time, then from (10.8)

and (10.4) we obtain

{Φα, Htotal}full = {Φα, H0}full + {Φα,Φβ}full qβ = 0, (10.9)

which is a system of equations for the undetermined coefficients qα.

Note that (10.9) coincides with (7.7) Fαβ qβ = Gα = DαH0, since

Fαβ = {Φα,Φβ}full , (10.10)

DαH0 = {Φα, H0}full . (10.11)

77


However, in contrast to the reduced description (without the (n− r)

“extra” momenta pα), where (7.7) is a system of (n− r) linear

equations in (n− r) unknowns qα, the extended system (10.9) can

lead to additional constraints (of higher stages), which significantly

complicates the analysis of dynamics DIRAC [1964], SUNDERMEYER

[1982].

After introducing (n− r) additional (“extra”) momenta pα, it follows

from (10.10)–(10.11), that the new brackets (gauge (8.16) and

nongauge (8.5)) transform into the corresponding Dirac brackets.

Our classification on gauge and nongauge theories corresponds to

the first- and second-class constraints DIRAC [1964], and Fαβ = 0 -

to Abelian constraints GOGILIDZE ET AL. [1996], LORAN [2005].

78


Conclusions

A “shortened” Hamiltonian-like formulation of classical singular

theories is given by means of

1) The generalized Legendre-Clairaut transform.

2) The partial Hamiltonian formalism.

Also, we found

3) A kind of Poisson-like gauge theory.

4) New brackets (which can lead to consistent quantization).

5) Equivalence of singular theories to multi-time dynamics.

79


We consider the restricted phase space formed by the r (=rank of

Hessian) regular momenta pi only. There are two reasons why

additional momenta pα are not worthwhile to be considered:

1) the mathematical reason: there is no possibility to resolve the

degenerate velocities vα as can be done for the regular velocities vi ;

2) the physical reason: momentum is a “measure of movement”, but in

“degenerate” directions there is no dynamics, hence — no reason to

introduce the corresponding momenta at all.

Thus there is no notion of a constraint (as in DIRAC [1964]) as

restriction on “nondynamical” momenta pα, because in our

“shortened” approach pα are not introduced at all — thus nothing to

constrain.

80


The Hamiltonian formulation of singular theories is done by

introducing new brackets (gauge and nongauge), which have all the

properties of the Poisson brackets (antisymmetry, Jacobi identity and

their appearance in the equations of motion and evolution of the

system in time).

The quantization of singular systems under the proposed “shortened”

approach can be carried out in a standard way, while not all 2n

variables of the extended phase space will be quantized, but only 2r

variables of the (initially) reduced phase space. The remaining

(noncanonical) variables can be treated as continuous parameters.

81


In the “nonphysical” coordinate subspace, we can formulate a some

kind of a nonabelian gauge theory, such that “nonabelianity” appears

due to the Poisson bracket in the physical phase space.

Finally, we show that any degenerate Lagrangian theory in the

“shortened” formulation DUPLIJ [2014, 2015] is equivalent to the

multi-time classical dynamics.

82


Acknowledgements

The author would like to express his deep thanks to V. P. Akulov,

A. V. Antonyuk, H. Arodz, Yu. A. Berezhnoj, V. Berezovoj, Yu. Bespalov,

Yu. Bolotin, I. H. Brevik, B. Broda, B. Burgstaller, M. Dabrowski, S. Deser,

V. K. Dubovoy, M. Dudek, P. Etingof, V. Gershun, M. Gerstenhaber,

G. A. Goldin, J. Grabowski, U. Gunter, M. Hewitt, R. Jackiw, B. Jancewicz,

H. Jones, I. Kanatchikov, A. T. Kotvytskiy, M. Krivoruchenko, G. C. Kurinnoj,

M. Lapidus, J. Lukierski, P. Mahnke, N. Merenkov, A. A. Migdal, B. V. Novikov,

A. Nurmagambetov, L. A. Pastur, P. Orland, S. A. Ovsienko, M. Pavlov,

S. V. Peletminskij, D. Polyakov, B. Shapiro, Yu. Shtanov, V. Shtelen,

S. D. Sinel’shchikov, W. Siegel, J. Stasheff, V. Soroka, K. S. Stelle, M. Tonin,

W. M. Tulczyjew, R. Umble, P. Urbanski, A. Vainstein, A. Voronov, M. Walker,

A. A. Yantzevich, C. Zachos, A. A. Zheltukhin, M. Znojil and B. Zwiebach.

83

A Multidimensional Clairaut equation


The multidimensional Clairaut equation for a function y = y(xi) of nvariables xi is IZUMIYA [1994], ALEKSEEVSKIJ ET AL. [1991]

y =n∑

j=1

xjy′xj− f(y′xi), (A.1)

where prime denotes a partial differentiation by subscript and f is asmooth function of n arguments. To find and classify solutions of(A.1), we need to find first derivatives y′xi in some way, and thensubstitute them back to (A.1).

Note that here we changedHCl → y,L→ f , λA → xi.

84


We differentiate the Clairaut equation (A.1) by xj and obtain nequations

n∑

i=1

y′′xixj(xi − f′y′xi) = 0. (A.2)

The classification follows from ways of vanishing the multipliers in(A.2). Here, for our physical applications, it is sufficient to supposethat ranks of Hessians of y and f are equal

rank y′′xixj = rank f′′y′xiy′xj= r. (A.3)

This means that in each equation from (A.2) whether first or secondmultiplier is zero, but it is not necessary to vanish both of them. Thefirst multiplier can be set to zero without any additional assumptions.

85


So we have

1) The general solution. It is defined by the condition

y′′xixj = 0. (A.4)

After one integration we find y′xi = ci and substitution them to (A.1)and 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 thiswill give a solution, if we can resolve them under y′xi . It can be

possible, if the rank of Hessians off is full, i.e. r = n. In this case weobtain

86


2) The envelope solution. It is defined by

xi = f′y′xi. (A.6)

We resolve (A.6) under derivatives as y′xi = Ci (xj) and get

yenv =n∑

i=1

xiCi (xj)− f(Ci (xj)), (A.7)

where Ci (xj) are n smooth functions of n arguments.

In the intermediate case, we can use the envelope solution (A.7) forfirst s variables, while the general solution (A.5) for other n− svariables, and obtain

87


3) The s-mixed solution, as follows

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 Hessians f is not full and a nonsingular minor of therank r is in upper left corner, then we can resolve first r relations (A.6)only, and so s ≤ r. In our physical applications we use the limitedcase s = r.

Example A.1. Let f (zi) = z21 + z

22 + z3, then the Clairaut equation

for y = y (x1, x2,x3) is

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

88


from (A.4) by one integration 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) byy′x1 = x1�2, y

′x2= x2�2 . So there is no the envelope solution (for

all variables), but we have several mixed solutions corresponding tos = 1, 2 :

y(1)mix =

x214+ c2 (x2 − c2) + c3 (x3 − 1) ,

c1 (x1 − c1) +x224+ c3 (x3 − 1) ,

(A.11)

y(2)mix =

x214+x224+ c3 (x3 − 1) . (A.12)

89


The case f (zi) = z21 + z

22 can be obtained from the above formulas

by putting x3 = c3 = 0, while y(2)mix becomes the envelope solution

yenv =x214+x224

.

90

B Examples

B Examples

Example B.1. Let L (x, v) = mv2/2− kx2/2, then the

corresponding Clairaut equation forH = HCl (x, λ) is

H = λH ′λ −m

2(H ′λ)

2+kx2

2, (B.1)

whereH ′λ denotes partial differentiation in λ. The general solution is

HClgen (x, λ, c) = λc−

mc2

2+kx2

2,

where c = c (x) is an arbitrary function (“unresolved velocity” v). Theenvelope solution (with λ = p) can be found from the condition

∂HCl

∂c= p−mc = 0 =⇒ cextr =

p

m,

91

B Examples

which gives

HClenv (x, p) =

p2

2m+kx2

2(B.2)

in the standard way.

Example B.2. Let L (x, v) = x exp v, then the corresponding

Clairaut equation forH = HCl (x, λ) is

H = λH ′λ − x exp (H′λ) . (B.3)

The general solution is

HClgen (x, λ) = λc (x)− x exp c (x) ,

The envelope solution (with λ = p) can be found from the condition

∂HCl

∂c= p− x exp c = 0 =⇒ cextr = ln

p

x,

92

B Examples

which leads to

HClenv (x, p) = p ln

p

x− p. (B.4)

Example B.3. Let L (x, y, vx, vy) = myv2x/2 + kxvy, then the

corresponding Clairaut equation forH = HCl (x, y, λx, λy) is

H = λxH′λx+ λyH

′λy−my

2

(H ′λx

)2− kxH ′λy , (B.5)

The general solution is

HClgen (x, y, λx, λy, cx, cy) = λxcx + λycy −

myc2x2− kxcy,

where cx, cy are arbitrary functions of the passive variables x, y.

93

B Examples

Then we differentiate

∂HClgen

∂cx= px −mycx = 0, =⇒ c

extrx =

px

my,

∂HClgen

∂cy= λy − kx.

Finally, resolve only the first equation and set cy 7−→ vy an“unresolved velocity” to obtain the mixed Legendre-Clairaut transform(cf. TULCZYJEW AND URBANSKI [1999], Examples 5,17)

HClmix (x, y, px, λy, vy) =

p2x2my

+ vy (λy − kx) . (B.6)

94

B Examples

Example B.4 (Cawley CAWLEY [1979]). Let L = xy + zy2/2, thenthe equations of motion are

x = yz, y = 0, y2 = 0. (B.7)

Because the Hessian has rank 2 and the velocity z does not enter intothe Lagrangian, the only degenerate velocity is z. The regularmomenta are px = y, py = x, and we have

Hphys = pxpy −1

2zy2, hy = 0.

The equations of motion (4.15)–(4.16) are

px = 0, py = yz, (B.8)

95

B Examples

and the condition (4.17) gives

DzHphys =∂Hphys

∂z= −1

2y2 = 0. (B.9)

Observe that (B.8) and (B.9) coincide with initial Lagrange equationsof motion (B.7).

Example B.5. Classical relativistic particle is described by

L = −mR, R =√x20 −

∑

i=x,y,z

x2i , (B.10)

where a dot denotes derivative with respect to the proper time.Because the rank of Hessian is 3, we consider velocities xi as regularvariables and the velocity x0 as degenerate variable. Then for theregular canonical momenta we have pi = mxi�R which can be

96

B Examples

resolved under the regular velocities as

xi = x0pi

E, E =

√m2 +

∑

i=x,y,z

p2i . (B.11)

Using (?? ) and (4.13) we obtain for Hamiltonians

Hphys = 0, hx0 = mx0

R= E, (B.12)

and so the “physical sense” of hx0 is, that it is indeed the energy(B.11). Equations of motion (4.15)–(4.16) are

xi = x0pi

E, pi = −

∂hx0∂xix0 = 0,

which coincide with the Lagrange equations following from (B.10)directly, the velocity x0 is arbitrary, and (4.17) becomes 0 = 0.

97

B Examples

Example B.6 (Christ-Lee model CHRIST AND LEE [1980]). TheLagrangian of SU (2) Yang-Mills theory in 0 + 1 dimensions is (inour notation)

L (xi, yα, vi) =1

2(vi − εijαxjyα)

2 − U(x2), (B.13)

where i, α = 1, 2, 3, x2 =∑i x2i , vi = xi and εijk is the

Levi-Civita symbol.

98

B Examples

The corresponding Clairaut equation (2.4) forH = HCl (xi, yα, λi, λα) has the form

H = λiH′λi+λαH

′λα−1

2

(H ′λi − εijαxjyα

)2+U

(x2). (B.14)

Its general solution is

Hgen = λici + λαcα −1

2(ci − εijαxjyα)

2 + U(x2), (B.15)

where ci, cα are functions of coordinates only (arbitrary constants).We differentiate (B.15) by them to get

∂Hgen

∂ci= λi − (ci − εijαxjyα) , (B.16)

∂Hgen

∂cα= λα, (B.17)

99

B Examples

and observe that only (B.16) can lead to the envelope solution. So we

can exclude the half of constants using (B.16) (with λi(6.15)→ pi) and

get the mixed solution (4.7) to the Clairaut equation (B.14) as

HClmix (xi, yα, pi, λα, cα) =1

2p2i + εijαpixjyα + λαcα + U

(x2).

(B.18)Using (4.12), we obtain the “physical” Hamiltonian

Hphys (xi, yα, pi) =1

2p2i + εijαpixjyα + U

(x2). (B.19)

From the other side, the Hessian of (B.13) has the rank 3, and wechoose xi, vi and yα to be regular and degenerate variablesrespectively.

100

B Examples

Because (B.13) independent of degenerate velocities yα, all

Bα(qA, pi

) (??)= 0, and therefore Fαβ

(qA, pi

) (7.8)= 0, we have the

limit gauge case of the above classification. So the degeneratevelocities vα = yα cannot be defined from (4.17) at all, they arearbitrary, and the first integrals (4.17), of the system (4.15)–(4.16)become (also in accordance to the independence statement)

∂Hphys (xi, yα, pi)

∂yα= εijαpixj = 0, (B.20)

and the preservation in time (B.20) is fulfilled identically due toproperties of the Levi-Civita symbols.

101

B Examples

It is clear, that only 2 equations from 3 ones of (B.20) are independent,we choose p1x2 = p2x1, p1x3 = p3x1 and insert in (B.19) to get

Hphys =1

2p21x2

x21+ U

(x2). (B.21)

The transformation p = p1√x2�x1, x =

√x2 gives the well-known

result CHRIST AND LEE [1980], GOGILIDZE ET AL. [1998]

Hphys =1

2p2 + U (x) . (B.22)

102

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