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Symmetry and conservation laws of differential equations

Department of Physics, University of Notre Dame - Notre Dame, IN 46556, USA

(ricevuto il 22 Ottobre 1996; approvato il 3 Dicembre 1996)

Summary. — A method is presented for associating a constant of the motion with the

(pseudo)symmetries of certain differential equations. The method requires the existence of an invariant differential form but does not require, as does Noether’s theorem, that the differential equations follow from a variational principle. The method does not require any integrations. Applications to divergence-free equations and to Hamiltonian systems are discussed.

PACS 02.30.Hg – Ordinary differential equations. PACS 46.90 – Other topics in classical mechanics.

1. – Introduction

The use of groups of symmetries in the solution or reduction of sets of differential equations dates from the work of Lie and has been extensively studied [1, 2]. For special types of ordinary differential equations and symmetries this reduction assumes a particularly simple form by associating with the symmetry a conservation law, that is, a globally defined function invariant under the flow generated by the differential equations. This is done through Noether’s theorem for differential equations that can be derived from a variational principle [1]. A very well-known example of this is in the Hamiltonian formulation of mechanics where the generators of the canonical symmetries of the Hamiltonian are invariant functions under the motion of the system [3], that is, they are constants of the motion. A perhaps less well-known example [4] is from magnetostatics where a symmetry of the vector potential leads to a function that is invariant under motion along the field lines of the corresponding magnetic field.

For more general sets of ordinary differential equations there is no general relation between global symmetries and global constants of the motion. There are a number of more specialized results available [4-6]. In this paper I describe a method which allows, under some circumstances, the direct construction of global invariant functions (constants of the motion or COMs) from symmetries of the differential equations without requiring any integration. In these considerations it is important to keep in mind the distinction between local and global results. Global invariant functions are defined over the entire state space (the space of dependent variables) and therefore

constrain the solutions of the differential equations for all values of the independent variable, that is, they carry information about the asymptotic behavior of the solutions. Local invariant functions, defined on open subsets of the state space, carry information about the solutions only over a limited range of the independent variable. It is easy to find trivial relations between local symmetries and local invariant functions. Furthermore, it is known [7] that any system of differential equations can locally be put in Hamiltonian form so that the usual Hamiltonian relation between symmetries and COMs applies locally. Our interest therefore is to find relations between globally defined symmetries and global COMs. Hereafter, all symmetries and COMs will be understood to be global ones.

In sect. 2 the appropriate notion of symmetry is discussed and the method for relating symmetries to COMs is given. The method requires, in addition to symmetries, the existence of a differential form invariant under the differential equations of interest. The discussion can easily be given in the coordinate-free notation of differential geometry but I use instead the older, but perhaps more familiar, tensor notation.

There are two natural areas of application. Differential equations defined by divergence-free vector fields, such as in magnetostatics in three-space, have an invariant volume form. These are discussed in sect. 3. In Hamiltonian mechanics there is an invariant two-form. In sect. 4 implications for Hamiltonian systems are discussed. Section 5 reports comments and discussion.

2. – Symmetries and invariant functions

Any set of ordinary differential equations can be put in the form of a set of autonomous first-order equations,

dxi

dt 4 V

i_(x1_{, R , x}n_{) ,}

i 41, R, n (1)

in some appropriate set of variables x which coordinatize a state space which is a differential manifold. Under general smooth changes of variables the V transforms as a contravariant vector field [8]. V is assumed to be smooth (normally C infinity). The solutions of (1) (the integral curves of V) are non-intersecting parameterized curves x(t). The set of all x corresponding to a particular integral curve is an integral manifold of V (just the integral curve forgetting the parameterization). Two vector fields differing by a multiplicative non-zero function

(

)

Under smooth coordinate transformations y 4f(x) the components of V transform by Wi (y) 4 ¯y i ¯xjV j_{(x) .} (2)

The coordinate transformation is a symmetry of V if the Wi _{and the V}i _{are the same}

functions, Wi_{(x) 4V}i(x) for all x and i.

A contravariant vector field S generates a one-parameter family of transformations x(x0, s) as solutions of the differential equations dx/ds 4S(x) and S is said to be a

generated by S are symmetries of V. In this case the transformations generated by S transform solutions of equations (1) into solutions of (1), that is, they map integral curves of V onto integral curves of V. By considering the infintesimal transformations generated by S, one shows that S is a symmetry of V if and only if the Lie derivative of V along S vanishes, LSV 40 where LSV is defined by

(LSV)i4 Sj ¯Vi ¯xj 2 V j ¯S i ¯xj . (3)

A weaker notion of symmetry is when, under the transformation (2), Wi

(x) 4 b(x) Vi(x) for some scalar function b and all x and i. The transformation will then be called, following [5], a pseudosymmetry of V. Such transformations map the integral manifolds of V onto integral manifolds of V, but they do not preserve the parameterization of the integral curves. The transformations generated by a vector field S are pseudosymmetries of V if and only if LSV(x) 4l(x) V(x), for some scalar

function l(x).

Given a function f(x), its rate of change induced by eqs. (1) is given by

df dt 4 V i ¯f ¯xi fLVf , (4)

where the last equality defines the Lie derivative of the function f in the direction of the vector field V. f is a constant of the motion (COM) of V if df/dt 40, that is if LVf 40.

The notion of Lie differentiation can be extended to any tensor field of arbitrary covariant and contravariant rank. If T is any tensor field and S a contravariant vector field, then the Lie derivative of T along S is a tensor of the same type as T, and is defined [8] by (5) (LST) i1, R , ip j1, R , jq4 S k ¯T i1, R , ip j1, R , jq ¯xk 1 Tk , j2, R , jq i1, R , ip ¯S k ¯xj1 1 R 1 Tj1, R , jq 21, k i1, R ip ¯S k ¯xjq 2 2Tj1, R , jq k , i2, R , ip ¯S i1 ¯xk 2 R 2 Tj1, R , jq i1, R ip 21, k ¯S ip ¯xk .

This clearly reduces to (3) and (4) when T is, respectively, a contravariant vector field and a scalar. As before, T is invariant under the transformations generated by S, if LST 40. Three properties of the Lie derivative, which follow from (5), will be useful.

The first is that Lie differentiation commutes with contraction of indices. That is, if one contracts a covariant and a contravariant index of a tensor and takes the Lie derivative of the resulting tensor, then that is the same as Lie-differentiating the original tensor and then contracting the indices. The second useful property is that Lie differentiation is a derivation with respect to direct product of tensors, that is, LS(R 7 T) 4

(LSR) 7 T 1R7 (LST). The third is that if T is antisymmetric (symmetric) under the

interchange of a pair of contravariant or a pair of covariant indices, then the Lie derivative of T has the same property. A rank-q covariant tensor field which is antisymmetric in all indices will be called (from differential geometry) a q-form. The Lie derivative of a q-form is a q-form.

The association of a COM with pseudosymmetries is through the following:

Let V be a vector field and v be a p form invariant under V. If S1, R , Sp 21 are

p 21 pseudosymmetries of V, then f4S1i1S2i2RSp 21 ip 21_Vip_v

i1, i2, R , ip is an invariant

function (COM) of V.

The proof is straightforward. We have, by assumption, that LVv 40, and LSiV 4 liV , for i 41, R, p. We want to show that LVf 40. From the definition of f and

because the Lie derivative is a derivation that commutes with contraction of indices we have that LVf 4 (LVS1)i1S2i2RVipvi1, R , ip1 1S1i1(LVS2) i2_RVip_v i1, R , ip1 R 1 S1 i1_S 2 i2_R(L VV) ip_v i1, R ip1 S1 i1_S 2 i2_RVip_(L Vv)i1, R , ip.

In this equation each of the first p 21 terms has a factor of the form LVSj4 2 LSjV 4

2 ljV so that v is contracted with two factors of V and each term vanishes by the

antisymmetry of v. The p-th term vanishes since LVV 40, and the last term because LVv 40. Hence LVf 40. Note that if the pseudosymmetries are linearly dependent

then f 40 by the antisymmetry of v. 3. – Divergence-zero vector fields

For the above result to apply the differential equations of interest must have an invariant form of some rank associated with them. In this section we consider differential equations defined by divergence-zero vector fields in three dimensions. These have an invariant three-form. Such equations arise naturally in a number of settings. For example, in the steady flow of an incompressible fluid the velocity field has zero divergence and eqs. (1) give the motion (in time) of small volume elements of the fluid. A COM has level surfaces across which there is no fluid flow. Another example concerns finding the field lines of a magnetic field. The magnetic field has zero divergence and the solutions of eqs. (1) are then the magnetic field lines in terms of an arbitrary parameter t. A COM of these equations has level surfaces which are called, in the fusion plasma literature [4], good flux surfaces. Since charged particles tend to spiral around lines of magnetic field, charge transport across good flux surfaces tends to be very slow compared to transport along field lines. This is important in plasma confinement problems.

Let eijkbe the completely antisymmetric tensor whose elements are all 61 in some

Euclidean coordinates (x , y , z). A straightforward application of eq. (5) shows that LVe 4 ( Div V) e, where Div V4

¯Vx ¯x 1

¯Vy ¯y 1

¯Vz

¯z . If Div V 40 then e is an invariant three-form. Let R and T generate pseudosymmetries of V. By the above theorem f 4 RiTjVkeijk is a COM of V. In Euclidean three-vector notation f 4 (R3T)QV. That

two vector fields are needed to generate a COM seems rather restrictive, but because R and T need generate only pseudosymmetries, rather than symmetries, of V there is more latitude.

As an example consider the case where rotations about the z-axis and translations along the z-axis are pseudosymmetries of V. In Euclidean coordinates the generators are R 4 (2y, x, 0) and T4 (0, 0, 1) and the COM is f4xVx

1 yVy. In the cylindrical coordinates (r , f , z) which are natural to this case f 4rVr_.

For this it is best to use cylindrical coordinates. So R 4 (0, 1, 0) and T4 (0, 0, 1). The conditions that R be a pseudosymmetry are just that ¯V

a

¯f 4 l1(r , f , z) V

a_{(r , f , z),}

for each cylindrical component of V. This requires that Va

(r , f , z) 4 m1(r , f , z) na(r , z), where m14 exp

k



l

have the form Va

(r , f , z) 4m2(r , f , z) na(r , f). These two forms are compatible if,

and only if, there is a function m such that

Va_{(r , f , z) 4m(r, f, z) n}a(r) , _{a 4r, f, z .} (6)

Now m must be chosen so that V is divergenceless, Div V 41 r ¯rVr ¯r 1 ¯Vf ¯f 1 ¯Vz ¯z 4 0 , where the divergence has been expressed in terms of the contravariant (not the, so-called, physical) components of V. This, with eq. (6) requires that m satisfy

(7) m(r , f , z) r ¯rnr_(r) ¯r 1 n r_(r) ¯m(r , f , z) ¯r 1 n f_(r) ¯m(r , f , z) ¯f 1 1nz(r) ¯m(r , f , z) ¯z 4 0 . The solutions of (7) are of the form

m(r , f , z) 4 m0 rnr(r)K

g





h

where K(f , z) must be periodic in f . It is easy to choose n and K so that m is C infinity and bounded at infinity. For example, the choices nr

4 r2, nf

4 r2, nz

4 1 , K 4exp [ cos f2z2] lead to m(r , f , z) 4 m0

r3exp [ cos (f 2r)2 (z11Or) 2

] and to a vector field V which vanishes at infinity and is C infinity everywhere.

It is worth noting that if R and T were required to be symmetries rather than pseudosymmetries, then, in contrast to (8), we would have m 4m0, a constant. The

divergence condition (7) would then require that nr

4 aOr which is singular unless a 40. The most general form for V would then be Vr4 0 , Vf4 nf(r), Vz_{4 n}z(r), so considerable latitude has been gained by needing only pseudosymmetries.

4. – Hamiltonian systems

Here we consider only systems with time-independent Hamiltonians. For Hamiltonian systems there is a natural invariant two-form v. In any canonical set of coordinates, x 4 (q1, p1, R , qn, pn), it is represented by an n-block diagonal matrix

with 2 32 blocks of the form v4

u

21

0

v

coordinates). It is a standard result in Hamiltonian mechanics that both v and g are invariant under canonical transformations.

The equations of motion are of the form (1) where Vi

4 gij ¯H

¯xj . Another standard

result from mechanics is that the solutions of these equations are one-parameter (time) families of canonical transformations. Hence the two-form v is invariant under V and the method should apply. So, if S is a pseudosymmetry of V, then

f 4SiVjvij4 Sigjk ¯H ¯xkvij4 S i ¯H ¯xi (9)

is a COM of the Hamilton equations of motion. Note that if S is a symmetry of H (LSH 40) then from (9) f40 and the method is ineffectual. This is in contrast to the

standard result in mechanics that a generator of a canonical symmetry of H is a constant of the motion. To get a non-zero COM we must find a pseudosymmetry of the equations of motion which is not a symmetry of H. A simple example of this is the quartic oscillator H 4p21 q4whose equations of motion (dq/dt 42p, dp/dt44q3) have the scaling transformations generated by S 4 (q, 2p) as a pseudosymmetry. In this case, from (9), we see that f 44H. The fact that H is a constant of the motion usually follows from invariance under time translations in Hamiltonian mechanics but here it follows from a pseudosymmetry of the equations of motion.

The use of pseudosymmetries to generate constants of the motion may be limited in Hamiltonian systems because, once we have any such COM, it can be used to generate a canonical symmetry of H. That is, even though pseudosymmetries of the equations of motion may not be (and in fact include) canonical symmetries of the Hamiltonian, any COM resulting from a pseudosymmetry also results from some canonical symmetry of H.

5. – Discussion

The method presented here requires that the set of differential equations of interest have an associated invariant p-form. That may seem like a very strong requirement but something like it is almost inevitable. This is because the set of differential equations and its pseudosymmetries are described by contravariant vector fields while the COM to be found is a scalar. The only general way of constructing a scalar from contravariant vectors is to contract them with some covariant tensor—in this case an invariant p-form. Given this, the method is interesting only if there are naturally occurring classes of equations with invariant forms of some order and two examples of this have been presented.

The case of divergence-free vector fields in three dimensions requires two pseudosymmetries. I have pointed out that this is considerably less restrictive than requiring two symmetries. Furthermore, there are no requirements on the pseudosymmetries other than that they be independent. In the particular example given in sect. 3, of translational and rotational pseudosymmetries, the two commute (the Lie derivative of one with respect to the other vanishes) but this is not necessary for the theorem. It does allow coordinates to be chosen in which the generators of the two pseudosymmetries are both unit vector fields and this considerably simplifies the investigation of what class of equations has these pseudosymmetries.

In Hamiltonian systems there is a natural invariant two-form, thus the theorem associates with each pseudosymmetry of the Hamilton equations of motion a COM, but this may be of limited use since that same COM could also be associated with a

canonical symmetry of the Hamiltonian in the conventional way. Any system of equations, defined on a star-shaped region of Rn_{, with a closed, non-degenerate,}

invariant two-form must be Hamiltonian in character (by the Poincaré Lemma). So, a system of equations could have an invariant two-form, and still not be Hamiltonian, if the two-form were to be either non-closed or degenerate. I know of no natural class of such equations.

R E F E R E N C E S

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