Eschrig H. Topology and Geometry for Physics

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order to distinguish this case from constraints in a totally different context con-

sidered afterwards.

The so far outlined theory presupposes that the Lagrange function L is strictly

convex in V. If as usual L has second derivatives with respect to the v

, this means

that for all q 2 M and for any base in T

(M) the symmetric m 9 m-matrix of those

second derivatives (the Hessian) has maximal rank:

rank



¼ m; det



[ 0: ð3:51Þ

This is not always the case. The theory considered now was pioneered by Dirac in

the 1950 and 1960 [8], essentially to the goal of canonical quantization of gauge

ﬁeld theories (which itself is beyond the scope of the present text). With the rise of

importance of Yang-Mills theories it was a very active subject of research in the

1980 and 1990 (see, e.g. [9]), and it holds unsolved problems till now.

For the sake of simplicity of notation the theory is usually presented for a ﬁnite

number of degrees of freedom (m in our text), although corresponding systems of

that type with a ﬁnite number of degrees of freedom look rather academic.

However, the results readily transfer to ﬁelds (with a continuum of degrees of

freedom), and all gauge ﬁelds are standard cases of this transfer.

Consider, in appropriate coordinates, a Lagrange function

Lðq; VÞ¼hf ðq;

VÞ; Viþ L



ðq; VÞ;

f ¼ðf

mþ1

; ...; f

Þ; V ¼ðV; VÞ¼ðv

; ...; v

; v

mþ1

; ...; v

Þ;

rank



m\m; det



[ 0:

ð3:52Þ

Deﬁne the action integral as

S½qðtÞ; VðtÞ; PðtÞ ¼

Lðq; VÞþhP; dq=dt  ViðÞdt ð3:53Þ

where the P

; i ¼ 1; ...; m, are Lagrange multipliers for the conditions dq= dt ¼ V

along the trajectories q(t) through M. The variation with ﬁxed end points

(q, V) yields the usual Lagrange equations of motion

 V ¼ 0;

 P



V¼dq=dt

¼ 0;





V¼dq=dt

¼ 0 ð3:54Þ

with a self-explanatory vector notation of the derivatives of L (remark after 3.40).

Now, for instance,

sup

 f

ðq; VÞv

0 for

¼ f

ðq; VÞ

þ1 else;



ð3:55Þ

3.7 Examples from Physics 87

(exercise), and

sup

hP; ViLð q; VÞ



) P 

¼ 0 )

V ¼Vðq; P; VÞ: ð3:56Þ

From the last line of (3.52) it follows that the middle relations of (3.56) can be

resolved for

V; and for P ¼ðP; PÞ the same notation as for V was used. Inserting

the last result into f yields

¼ f

ðq; Vðq; P; VÞÞ ¼ f



ðq; PÞ; i ¼ m þ1; ...; m: ð3:57Þ

If the

V would not drop out from the functions f



; then obviously the rank in

(3.52) would be larger than

m: Altogether (3.38) results in

Hðq; PÞ¼



ðq; PÞ for P ¼ f



ðq; PÞ;

þ1 else

(

ð3:58Þ

with



ðq; PÞ¼hP; Vðq; PÞi  L



ðq; Vðq; PÞÞ: ð3:59Þ

There are

m ¼ m m primary constraints

ð1Þ

ðq; PÞ¼p

 f



ðq; PÞ¼0; i ¼ m þ1; ...; m; ð3:60Þ

for the momenta P on the trajectories.

The trajectories are now obtained from the extremum of the action integral

½qðtÞ; PðtÞ; kðtÞ ¼

ðhP; dq=dtiH



ðq; PÞhU

ð1Þ

ðq; PÞ; kiÞdt; ð3:61Þ

where k

; i ¼ m þ 1; ...; m; are the Lagrange multipliers for the constraints. This

may be abbreviated as

hP; dq=dtiH

ð1Þ

ðq; P; kÞ



dt;

ð1Þ

ðq; P; kÞ¼H



ðq; PÞþhU

ð1Þ

ðq; PÞ; ki:

ð3:62Þ

Again with a notation q ¼ð

q; qÞ; the variation yields

ð1Þ

;

¼ k



;

¼

ð1Þ

;

P ¼ f



: ð3:63Þ

88 3 Manifolds

The Lagrange multipliers turn out to be the velocities on the trajectories which

remained unresolved in (3.56). It is seen that the time derivative along a trajectory

of any function A(q, P) is obtained from the Poisson bracket,

Aðq; PÞ :

¼fH

ð1Þ

; Ag

ð1Þ

¼0

;

ð1Þ





ð1Þ

;



ð1Þ

¼0

ð3:64Þ

where the constraints must be taken after the calculation of the brackets.

Note that the constraints (3.60) form a Pfafﬁan equation system deﬁning an

dimensional distribution in T(M). Any trajectory being solution of (3.54) must be

kept in this distribution at all times, whence the time derivatives of the constraints

should vanish,

ð1Þ

¼fH

ð1Þ

; U

ð1Þ

¼0

¼fH



; U

ð1Þ

¼0

þfU

ð1Þ

; U

ð1Þ

¼0

; k

¼ 0:

ð3:65Þ

Here, U

(1)

and fH



; U

ð1Þ

g are understood as cotangent vector ﬁelds on M while

ð1Þ

; U

ð1Þ

g is a 2-form on M (a q-dependent antisymmetric m  m-matrix in local

coordinates q

Let, as a matrix in local coordinates q

rank fU

ð1Þ

; U

ð1Þ

¼0



¼ q

; 0 q

m: ð3:66Þ

As the rank of an antisymmetric matrix, q

is even. Then, there exist m q

non-

zero vector ﬁelds U

(k)

so that

ð1Þ

; U

ð1Þ

¼0

; U

ðkÞ

¼ 0; U

ðkÞ

¼ðU

ðkÞ

; ...; U

ðkÞ

Þ; k ¼ q

þ 1; ...; m:

ð3:67Þ

Hence, Eq. 3.65 determines the

m linearly independent vector functions

k(q, P, t) modulo U

(k)

; a number of m q

linear combinations remain undeter-

mined. Instead, (3.65) comprise the conditions



; U

ð1Þ

¼0

; U

ðkÞ



¼ 0; ð3:68Þ

some of which may be identities, some may not be independent of the constraints

(1)

, but a number m

of them may form new constraints U

(2,1)

. Inserting the q

determined expressions k in terms of q, P (as well as of the suppressed variable t)

and the remaining undetermined combinations k

into (3.65) yields a combination

of q

independent expressions denoted as W

ð1Þ

ðq; P; k

Þ,ofm primary constraints

(1)

(q, P) and of m

new constraints U

(2,1)

(q, P). Since {H

(1)

, U

(1)

} is linear in

3.7 Examples from Physics 89

the constraints, (3.65) has a structure, with U = (U

) further on written as a

column,

ð1Þ

; U

ð1Þ

¼0

¼ 0 , K

ð1Þ

ðq; PÞ

ð1Þ

ðq; P; k

ð1Þ

ðq; PÞ

ð2;1Þ

ðq; PÞ

¼ 0; ð3:69Þ

where K

(1)

is an (m ðq

þ m þ m

ÞÞ-matrix function divided into three blocks

with rank K

ð1Þ

¼ q

; K

ð1Þ

¼ 1

; (unit matrix) and rank K

ð1Þ

ð2;1Þ

¼ m

an obvious block matrix notation. At the same step, introducing the determined

expressions for the k into the Hamiltonian (3.62) results in a new Hamiltonian

ð1Þ

ðq; P; k

Þ for which (3.65) transforms into fH

ð1Þ

; U

ð1Þ

¼0

¼ 0: Repeating

this process with H

ð1Þ

ðq; P; k

Þ and with the new constraints results in a second step

ð1Þ

; U

ð2;1Þ

ð1Þ

¼0

ð2;1Þ

¼0

¼ 0 , K

ð2Þ

ðq; PÞ

ð2;1Þ

ðq; P; k

ð1Þ

ðq; PÞ

ð2;1Þ

ðq; PÞ

ð2;2Þ

ðq; PÞ

¼ 0: ð3:70Þ

After a ﬁnite number of l steps there appear no new independent constraints in the

l ? 1st step although there may still remain unresolved multipliers k. (The number

of independent constraints cannot exceed the dimension of T

(M).) All constraints

ð2; jÞ

are called secondary constraints.

This process of ‘breeding constraints’ results in a number l ¼

m þm

þþ

; unique for a given Lagrange function, of constraints U, independent in the

sense that the rank of the Jacobi matrix

rank

DðUÞ

Dðq; PÞ



U¼0



¼ l: ð3:71Þ

Let (superscript t meaning the transposed, U

being a row)

m ¼ l  rank fU; U

U¼0



: ð3:72Þ

These ranks are independent of a linear functional transformation with a ðl  lÞ-

matrix V,

ðq; PÞ¼Vðq; PÞUðq; PÞ; det Vj

U¼0

6¼ 0: ð3:73Þ

There is such a transformation that

VU ¼



; v ¼ðv

; ...; v

; u ¼ðu

mþ1

; ...; u

; ð3:74Þ

with

90 3 Manifolds

fv; Ug¼OðUÞ; det f u; ug

U¼0



6¼ 0; ð3:75Þ

where O(U) means order of U; OðUÞj

U¼0

¼ 0. The constraints ðv; uÞ¼0 are

equivalent to U = 0. The v are called ﬁrst-class constraints, and the u are called

second-class constraints.

Consider ﬁrst the simpler case if there are only second-class constraints.

Compared to (3.60) we subtract

m from all subscripts i of the constraints and of all

superscripts of the k in what follows. In the actually considered case, in any local

coordinate system, the matrix {U, U

} has maximal rank in the neighborhood of

U = 0, so that its inverse {U, U

}

-1

exists,

fU; U

1

fU; U

g¼1

¼fU; U

gfU; U

1

; ð3:76Þ

where 1

means the (l 9 l) unit matrix. This means that all k may be determined

from (3.65)tobe

¼

l¼1

ðfU; U

1

; H



gþOðUÞ; k ¼ 1; ...; m; ð3:77Þ

while

l¼1

ðfU; U

1

; H



g¼OðUÞ; k ¼ m þ1; ...; l: ð3:78Þ

In the equations of motion (3.64), terms of higher than ﬁrst order in the U may

be added to the Hamiltonian before the Poisson brackets are calculated, since

after their calculation now all U are put to zero. One hence may deﬁne a

Hamiltonian

¼ H



 U

fU; U

1

; H



g; ð3:79Þ

and obtain canonical equations of motion

Aðq; PÞ :

¼fH

; Ag

U¼0

: ð3:80Þ

Alternatively, one may deﬁne Dirac’s brackets

instead of the Poisson brackets

fA; Bg

¼fA; BgfA; U

gfU; U

1

fU; Bgð3:81Þ

and have

Do not confuse them with Dirac’s notation of Hilbert vectors by bras and kets.

3.7 Examples from Physics 91

¼fH



; Ag

U¼0

: ð3:82Þ

The brackets f; g

have algebraic properties (3.42) like the Poisson brackets.

As the rank of the antisymmetric matrix fu; ug of (3.75) must be even, the

number of second-class constraint is always even. It can be shown [9] that a

canonical transformation from (q, P) to new canonical variables (g, /) exists so

that the g and the / separately consist of pairs of canonically conjugate variables

and the / form the constraints in the new coordinates. Then, the equations of

motion take the form

¼fH

; gg; / ¼ 0; H

¼ Hj

/¼0

¼ HðgÞ: ð3:83Þ

That means, the presence of only second-class constraints simply reduces the

system to an ordinary Hamilton system on a submanifold of M deﬁned by the

distribution of the 1-forms /.

First-class constraints cannot form in Dirac’s ‘breeding’ process, if there were

no primary ﬁrst-class constraints present: among ﬁrst-class constraints there are

necessarily primary ﬁrst-class constraints. In this case the existence of a canonical

transformation has been shown [9] to new canonical variables (g, Q, P, /)in

which the dynamics is described by the equations of motion

¼fH

; gg; P ¼ 0 ¼ /;

ð1Þ

¼ k

ð1Þ

;

ð2Þ

¼ AðgÞ; H

¼ Hj

ðP;/Þ¼0

;

ð3:84Þ

where P ¼ðP

ð1Þ

; P

ð2Þ

Þ; P

ð1Þ

are the primary and P

(2)

the secondary ﬁrst-class

constraints, and Q are the conjugate variables to ﬁrst-class constraints. A(g) are

ﬁxed functions appearing in Dirac’s procedure, and k

ð1Þ

¼ k

ð1Þ

ðg; Q; tÞ are the

remaining undetermined Lagrange multipliers which do not enter the other

equations of motion. Hence, for any initial condition the time evolution contains

arbitrary functions of the canonical variables and of time t, that is, an

m-dimen-

sional class of trajectories.

The only way to save causality, which is supported by experience, is to say

that each trajectory out of this class describes the same physical process. The

ð1Þ

merely constitute redundancy in the description, for whatsoever good rea-

sons. This redundancy is conventionally called gauge freedom. Since transitions

between the trajectories of a given class can be performed one after another and

the k

ð1Þ

may continuously be varied, those transitions form a continuous

transformation group the elements of which are continuously connected to the

identity transformation. This is the group of gauge transformations (see

Sect. 8.3).

There is a systematic actual way to arrive at the indicated special canonical

variables which is in general, however, quite technical (see [10]).

92 3 Manifolds

3.7.4 Classical Mechanics Under Velocity Constraints

We now consider sketchily a case which was already initiated towards the end of

nineteenth century, essentially by H.R. Hertz and S.A. Chaplygin, which devel-

oped into a huge branch of Mathematical Physics and of various disciplines of

engineering over the last century, and which is a very active ﬁeld of research even

today including also aspects of quantization [11].

We ﬁrst switch to a much broader understanding of the conﬁguration space

M. It may be any smooth manifold the points of which describe some mechanical

setting, for instance describing the group elements of translation and rotation

groups of rigid bodies. It may even describe the instantaneous state of some

sensor-actuator system which need not be mechanical.

Let L(q, V) be a regular Lagrange function on M,

Lðq; VÞ : det



¼ m ¼ dim M: ð3:85Þ

Let be given l \ m linearly independent smooth 1-forms f

(k)

(q)onM so that the

motion of the system is constrained by the conditions

ðkÞ

; VðtÞi ¼ 0 for all t; k ¼ 1; ...; l: ð3:86Þ

According to (3.34) the f

(k)

form the annihilator D

of an (m - l)-dimensional

distribution D of velocity vectors. Constraints nonlinear in V are also considered in

literature, but here we limit ourselves to linear constraints. The constraints cause

constraint forces during the motion which are not (and in general cannot be)

described by the Lagrange function (3.85) and which have to be added to the

equations of motion (understood as a cotangent vector equation as previously),



k¼1

ðkÞ

ðqÞk

ðkÞ

ðq; tÞ¼f k; ð3:87Þ

where the coefﬁcient functions k(q, t) are uniquely determined by the conditions

(3.86). (One has hf ; Vik ¼ 0, the constraint forces do not perform work along the

trajectories of motion; d’Alembert’s principle.) On transition to the Hamilton

function (3.38) corresponding to the Lagrange function (3.85) and adding the

constraint forces and the constraints to the canonical equations of motion one gets

;

¼

þ f k; f;



¼ 0: ð3:88Þ

The time derivative of the constraints must be zero. The time derivative of the last

equation of (3.88) along trajectories is

f ;



;



þ f

oPoP



þ f k



¼ 0;

3.7 Examples from Physics 93

where in the second term a second rank tensor o

H=ðoPoPÞ in T(M) is projected

on the 1-forms on both sides. This expression can uniquely be resolved for the k,if

oPoP





6¼ 0; ð3:89Þ

which condition indeed follows from (3.85) and from the linear independence of

the f

(k)

. Hence, the constraint forces are correctly determined.

If the distribution D deﬁned by (3.86) is involutive, that is, if df

ðkÞ

ð1Þ

^

ðlÞ

¼ 0; k ¼ 1; ...; l, then the constraints are called holonomic. In this case the

constraints are completely integrable and the distribution of velocities D deﬁnes a

submanifold N of M as its integral manifold to which the whole motion of the

system is conﬁned.

If D is not involutive, the constraints are called nonholonomic. The classical

example is a rolling disc without slipping on an inclined plane (Chaplygin).

Though nonholonomic motion is rather the standard case in everydays life (see the

just mentioned example) it comprises a huge in large parts unexplored ﬁeld, and

even the motion of simple (but tricky) toys like the ‘rattleback’ is poorly under-

stood. There is in general a ‘bracket formulation’ of the equations of motion [12],

but the brackets do not obey property 3 of Poisson brackets, the Jacobi identity

(3.42).

3.7.5 Thermodynamics

The thermodynamic equilibrium state of a gas of N particles is described by its

volume v and temperature t. The thermodynamic phase space is M ¼ R

in this

case. The amount of heat put into an ideal gas is

dQ ¼ cðtÞdt þ

dv; ð3:90Þ

where c(t) is the heat capacity at constant volume which is a function of tem-

perature only and R is the gas constant. A change of the thermodynamic state is

called adiabatic, if no heat is exchanged, that is, if dQ = 0.

The question is, whether dQ = 0 deﬁnes uniquely paths through the phase

space. Since (3.90) is a 1-form in R

, dQ = 0 is a Pfafﬁan equation, for which

ddQ ¼ðoðRt=vÞ= o tÞdt ^ dv 6¼ 0; but since ddQ ^ dQ = 0, the equation is com-

pletely integrable (as any 1-form in two dimensions, since dx ^ x is a 3-form for

every 1-form x). Hence, the answer to the question is positive, and there is an

adiabatic ﬂow /

(t, v)(s 2 R is some curve parameter) through the phase space:

through every point (t, v) there is exactly one adiabatic trajectory. Consequently,

there are functions, constant on trajectories and hence invariant under the adiabatic

ﬂow.

94 3 Manifolds

Nature or microscopic reasoning in Statistical Physics tells us, that as part of

Second Law of thermodynamics it always holds that entropy s, given by ds ¼

dQ=t is such an adiabatic invariant, that is, ds is always a total differential, and

s /

¼ s:

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order constraints. I and II. Phys. Rev. D 56, 1135–1155 (1997)

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3.7 Examples from Physics 95