conservation laws: systematic construction, noether's...

Conservation Laws: Systematic Construction,Noether’s Theorem, Applications,

and Symbolic Computations.

Alexey Shevyakov

Department of Mathematics and Statistics,University of Saskatchewan, Saskatoon, Canada

Physics seminarNovember 10, 2011

A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 1 / 33

Outline

1 Conservation LawsDefinition and ExamplesApplications

2 Variational Principles

3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem

4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example

5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations

6 Conclusions


Outline






6 Conclusions


Conservation Laws: Some Definitions

Notation:

The total derivative by x :

The partial derivative by x assuming all chain rules carried out as needed.

Dependent variables should be taken care of.

Example: u = u(x , y), g = g(u), then

Dx [xg ] ≡ ∂

∂x[xg(u(x , y))]

=∂

∂x[xg ] +

∂

∂g[xg ]

∂

∂u[g(u)]

∂

∂xu(x , y).



Conservation law:

Given: some model.Independent variables: x = (t, x , y , ...); dependent variables: u = (u, v , ...).

A conservation law: a divergence expression equal to zero

DtΘ(x,u, ...) + DxΨ1(x,u, ...) + Dy Ψ2(x,u, ...) + · · · = 0.



Conservation law:

Given: some model.Independent variables: x = (t, x , y , ...); dependent variables: u = (u, v , ...).

A conservation law: a divergence expression equal to zero

DtΘ(x,u, ...) + DxΨ1(x,u, ...) + Dy Ψ2(x,u, ...) + · · · = 0.

Time-independent:

Di Ψi ≡ divx,y,...Ψ = 0.

Time-dependent:

DtΘ + divx,y,...Ψ = 0.

A conserved quantity:

Dt

∫V

Θ dV = 0.


ODE Models: Conserved Quantities

An ODE:

Dependent variable: u = u(t);A conservation law

DtF (t, u, u′, ...) =d

dtDtF (t, u, u′, ...) = 0.

yields a conserved quantity (constant of motion):

F (t, u, u′, ...) = C = const.



An ODE:

Dependent variable: u = u(t);A conservation law

DtF (t, u, u′, ...) =d

dtDtF (t, u, u′, ...) = 0.

yields a conserved quantity (constant of motion):

F (t, u, u′, ...) = C = const.

Example: Harmonic oscillator, spring-mass system

Independent variable: t, dependent: x(t).

ODE: x(t) + ω2x(t) = 0; ω2 = k/m = const.

Conservation law:d

dt

(mx2(t)

2+

kx2(t)

2

)= 0.

Conserved quantity: energy.



Example: ODE integration

An ODE:

K ′′′(x) =−2 (K ′′(x))

2K(x)− (K ′(x))

2K ′′(x)

K(x)K ′(x).

Three independent conserved quantities:

KK ′′

(K ′)2= C1,

KK ′′ ln K

(K ′)2− ln K ′ = C2,

xKK ′′ + KK ′

(K ′)2− x = C3.

yield complete ODE integration.


PDE Models

Example 1: small string oscillations, 1D wave equation

Independent variables: x , t; dependent: u(x , t).

utt = c2uxx , c2 = T/ρ.

Conservation laws:

Momentum: Dt(ρut)−Dx(Tux) = 0;Conserved quantity: total momentum M =

∫ρutdx = const.

Energy: Dt

(ρu2

t

2+

Tu2x

2

)−Dx(Tutux) = 0;

Conserved quantity: total energy E =∫ (ρu2

t

2+

Tu2x

2

)dx = const.


PDE Models

Example 2: Adiabatic motion of an ideal gas in 3D

Independent variables: t; x = (x1, x2, x3) ∈ D ⊂ R3.

Dependent: ρ(x , t), v 1(x , t), v 2(x , t), v 3(x , t), p(x , t).

Equations:Dtρ+ Dj(ρv j) = 0,

ρ(Dt + v jDj)v i + Di p = 0, i = 1, 2, 3,

ρ(Dt + v jDj)p + γρpDjvj = 0.

Conservation laws:

Mass: Dtρ+ Dj(ρv j) = 0,

Momentum: Dt(ρv i ) + Dj(ρv i v j + pδij) = 0, i = 1, 2, 3,

Energy: Dt(E) + Dj

(v j(E + p)

)= 0, E = 1

2ρ|v |2 +

p

γ − 1.

Angular momentum, etc.


Applications of Conservation Laws

ODEs

Constants of motion.

Integration.


Applications of Conservation Laws

ODEs


Integration.

PDEs

Direct physical meaning.


Differential constraints (div B = 0, etc.).

Analysis: existence, uniqueness, stability.

Nonlocally related PDE systems, exact solutions. Potentials, stream functions, etc.:

V = (u, v), div V = ux + vy = 0,

u = Φy ,v = −Φx .

Modern numerical methods often require conserved forms.

An infinite number of conservation laws can indicate integrability / linearization.


Outline






6 Conclusions


Variational Principles

Action integral

J[U] =

∫Ω

L(x ,U, ∂U, . . . , ∂kU)dx .

Principle of extremal action

Variation of U: U(x)→ U(x) + εv(x).

Require: variation of action δJ ≡ J[U + εv ]− J[U] =∫

Ω(δL) dx = O(ε2).



Action integral

J[U] =

∫Ω

L(x ,U, ∂U, . . . , ∂kU)dx .





Variation of Lagrangian

δL = L(x ,U + εv , ∂U + ε∂v , . . . , ∂kU + ε∂kv)− L(x ,U, ∂U, . . . , ∂kU)

= ε

(∂L[U]

∂Uσvσ +

∂L[U]

∂Uσj

vσj + · · ·+ ∂L[U]

∂Uσj1···jk

vσj1···jk

)+ O(ε2)

= ε(vσEUσ (L[U]) + ...) + O(ε2),

where EUσ are the Euler operators.



Action integral

J[U] =

∫Ω

L(x ,U, ∂U, . . . , ∂kU)dx .





Euler-Lagrange equations:

Euσ (L[u]) =∂L[u]

∂uσ+ · · ·+ (−1)kDj1 · · ·Djk

∂L[u]

∂uσj1···jk= 0,

σ = 1, . . . ,m.



Example 1: Harmonic oscillator, x = x(t)

L =1

2mx2 − 1

2kx2, ExL = −m(x + ω2x) = 0.

Example 2: Wave equation for u(x , t)

L =1

2ρut

2 − 1

2T ux

2, EuL = −ρ(utt − c2uxx) = 0.

Example 3: Klein-Gordon nonlinear equations for u(x , t)

L = −1

2utux + h(u), EuL = utx + h′(u) = 0.

Many other non-dissipative systems have variational formulations.

A practical way to tell whether a DE system has a variational formulation: itslinearization must be self-adjoint (symmetric). Relatively few models are!


Outline






6 Conclusions


Symmetries of Differential Equations

Consider a general DE system

Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

with variables x = (x1, ..., xn), u = u(x) = (u1, ..., um).

Definition

A transformationx∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).

depending on a parameter a is a point symmetry of Rσ[u] if the equation is the same innew variables x∗, u∗.




Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N


Definition



Example 1: translations

The translationx∗ = x + C , t∗ = t, u∗ = u

leaves KdV invariant:

ut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .




Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N


Definition



Example 2: scaling

Same for the scaling:x∗ = αx , t∗ = α3t, u∗ = αu.

One hasut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .


Variational Symmetries


Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N

that follows from a variational principle with J[U] =∫

ΩL(x ,U, ∂U, . . . , ∂kU)dx .

Definition

A symmetry of Rσ[u] given by

x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).

is a variational symmetry of Rσ[u] if it preserves the action J[U].




Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N


ΩL(x ,U, ∂U, . . . , ∂kU)dx .

Definition




Example 1: translations for the wave equation

utt = (T/ρ)2uxx , L =1

2ρut

2 − 1

2T ux

2.

The translation x∗ = x + C , t∗ = t, u∗ = u is evidently a variational symmetry.




Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N


ΩL(x ,U, ∂U, . . . , ∂kU)dx .

Definition




Example 2: scaling for the wave equation

utt = (T/ρ)2uxx , L =1

2ρut

2 − 1

2T ux

2.

The scaling x∗ = x , t∗ = t, u∗ = u/α is not a variational symmetry: L∗ = α2L,J∗ = α2J.


Noether’s Theorem (restricted to point symmetries)

Theorem

Given:

a PDE system

Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N,

following from a variational principle;

a variational symmetry

(x i )∗ = f i (x , u; a) = x i + aξi (x , u) + O(a2),(uσ)∗ = gσ(x , u; a) = uσ + aησ(x , u) + O(a2).

Then the system Rσ[u] has a conservation law Di Φi [u] = 0.

In particular,Di Φ

i [u] ≡ Λσ[u]Rσ[u] = 0,

where the multipliers are given by

Λσ = ησ(x , u)− ∂uσ

∂xiξi (x , u).


Noether’s Theorem: Examples

Example: translation symmetry for the harmonic oscillator

Equation: x(t) + ω2x(t) = 0, ω2 = k/m.

Symmetry:t∗ = t + a, ξ = 1;x∗ = x , η = 0,

Multiplier (integrating factor): Λ = η − x(t)ξ = −x ;

Conservation law:

ΛR = −x(x(t) + ω2x(t)) = − 1

m

d

dt

(mx2(t)

2+

kx2(t)

2

)= 0.


Limitations of Noether’s Theorem

The given DE system, as written, must be variational.

Numbers of PDEs and dependent variables must coincide.

Dissipative systems are not variational.

If single PDE, must be of even order.

If a PDE system is not variational, artifices sometimes can make it variational!








Example 1: The use of multipliers.

The PDEutt + H ′(ux)uxx + H(ux) = 0,

as written, does not admit a variational principle. However, the equivalent PDE

ex [utt + H ′(ux)uxx + H(ux)] = 0,

does follow from a variational principle!








Example 2: The use of a transformation.

The PDEexutt − e3x(u + ux)2(u + 2ux + uxx) = 0,

as written, does not admit a variational principle. But the point transformation x∗ = x ,t∗ = t, u∗(x∗, t∗) = y(x , t) = exu(x , t), maps it into the self-adjoint PDE

ytt − (yx)2yxx = 0,

which is the Euler–Lagrange equation for the Lagrangian L[Y ] = y 2t /2− y 4

x /12.








Example 3: The use of a differential substitution.

The KdV equationut + uux + uxxx = 0,

as written, does not admit a variational principle. However, a substitution u = vx yields avariational PDE

vxt + vxvxx + vxxxx = 0.


Limitations of Noether’s Theorem (ctd.)

. . .

The use of an artificial additional equation. Any PDE system can be madevariational, by appending an adjoint equation!

Example:

The diffusion equation ut − uxx = 0 is dissipative, hence not self-adjoint.

Its adjoint equation: wt + wxx = 0.

But the PDE systemut − uxx = 0,

ut + uxx = 0

is self-adjoint!

Lagrangian:L = utw − wtu + 2uxwx .


Outline






6 Conclusions


The Idea of the Direct Construction Method

Definition

The Euler operator with respect to U j :

EU j =∂

∂U j−Di

∂

∂U ji

+ · · ·+ (−1)sDi1 . . .Dis

∂

∂U ji1...is

+ · · · , j = 1, . . . ,m.

Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).

Theorem

The equationsEU j F (x ,U, ∂U, . . . , ∂sU) ≡ 0, j = 1, . . . ,m

hold for arbitrary U(x) if and only if

F (x ,U, ∂U, . . . , ∂sU) ≡ Di Ψi

for some functions Ψi (x ,U, ...).


The Idea of the Direct Construction Method

Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).

Direct Construction Method

Specify dependence of multipliers: Λσ = Λσ(x ,U, ...), σ = 1, ...,N.

Solve the set of determining equations

EU j (ΛσRσ) ≡ 0, j = 1, . . . ,m,

for arbitrary U(x) (off of solution set!) to find all such sets of multipliers.

Find the corresponding fluxes Φi (x ,U, ...) satisfying the identity

ΛσRσ ≡ Di Φi .

Each set of fluxes and multipliers yields a local conservation law

Di Φi (x , u, ...) = 0,

holding for all solutions u(x) of the given PDE system.


Completeness

Completeness of the Direct Construction Method

For the majority of physical DE systems (in particular, all systems in solved form),all conservation laws follow from linear combinations of equations!

ΛσRσ ≡ Di Φi .


A Detailed Example

Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):

R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.

Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).


A Detailed Example




Determining equations:

EU

[ξ(x , t,U,V )(Vt − (U2 + 1)Ux − U) + φ(x , t,U,V )(Ut − Vx)

]≡ 0,

EV


]≡ 0.

Euler operators:

EU =∂

∂U−Dx

∂

∂Ux−Dt

∂

∂Ut,

EV =∂

∂V−Dx

∂

∂Vx−Dt

∂

∂Vt.


A Detailed Example




Determining equations:

EU


]≡ 0,

EV


]≡ 0.

Split determining equations:

φV − ξU = 0, φU − (U2 + 1)ξV = 0,

φx − ξt − UξV = 0, (U2 + 1)ξx − φt − UξU − ξ = 0.


A Detailed Example




Solution: five sets of multipliers (ξ, φ) =

0 1

t x − 12t2

1 −t

ex+ 12U2+V Uex+ 1

2U2+V

ex+ 12U2−V −Uex+ 1

2U2−V


A Detailed Example




Resulting five conservation laws:

Dtu −Dxv = 0,

Dt [(x − 12t2)u + tv ] + Dx [( 1

2t2 − x)v − t( 1

3u3 + u)] = 0,

Dt [v − tu] + Dx [tv − ( 13u3 + u)] = 0,

Dt [ex+ 1

2u2+v ] + Dx [−uex+ 1

2u2+v ] = 0,

Dt [ex+ 1

2u2−v ] + Dx [uex+ 1

2u2−v ] = 0.

To obtain further conservation laws, extend the multiplier ansatz...


Outline






6 Conclusions


Symbolic Software for Computation of Conservation Laws

Example of use of the GeM package for Maple for the KdV.

Use the module: with(GeM):

Declare variables: gem_decl_vars(indeps=[x,t], deps=[U(x,t)]);

Declare the equation:

gem_decl_eqs([diff(U(x,t),t)=U(x,t)*diff(U(x,t),x)

+diff(U(x,t),x,x,x)],

solve_for=[diff(U(x,t),t)]);

Generate determining equations:

det_eqs:=gem_conslaw_det_eqs([x,t, U(x,t), diff(U(x,t),x),

diff(U(x,t),x,x), diff(U(x,t),x,x,x)]):

Reduce the overdetermined system:

CL_multipliers:=gem_conslaw_multipliers();

simplified_eqs:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1);


Symbolic Software for Computation of Conservation Laws

Example of use of the GeM package for Maple for the KdV.

Solve determining equations:

multipliers_sol:=pdsolve(simplified_eqs[Solved]);

Obtain corresponding conservation law fluxes/densities:

gem_get_CL_fluxes(multipliers_sol, method=*****);


Examples of Symbolic Computation of Conservation Laws

Example 1

KdV equation:ut + uux + uxxx = 0.


Example 2: Conserved Form of Surfactant Dynamics Equations

Ref.: C. Kallendorf, A.S., M. Oberlack, Y.Wang, 2011

Surfactants = surface active agents.

Two-phase interface described by a level set function Φ; S =

x ∈ R3 : Φ(x) = 0

.

Unit normal: n = − ∇Φ

|∇Φ| . Concentration: c.



Ref.: C. Kallendorf, A.S., M. Oberlack, Y.Wang, 2011

Surfactants = surface active agents.

Two-phase interface described by a level set function Φ; S =

x ∈ R3 : Φ(x) = 0

.

Unit normal: n = − ∇Φ

|∇Φ| . Concentration: c.

Surfactant Dynamics Equations:

Incompressibility of the flow:∇ · u = 0.

Interface transport by the flow:

Φt + u · ∇ (Φ) = 0.

Surfactant transport:

ct + ui ∂c

∂x i− cni nj

∂ui

∂x j− α(δij − ni nj)

∂

∂x j

((δik − ni nk)

∂c

∂xk

)= 0.



Result

The surfactant dynamics system can be written in a fully conserved form:

∇ · u = 0.

Φt + u · ∇ (Φ) = 0.

∂

∂t(c F(Φ) |∇Φ|) +

∂

∂x i

(Ai F(Φ) |∇Φ|

)= 0,

where

Ai = cui − α(

(δik − ni nk)∂c

∂xk

), i = 1, 2, 3,

and F is an arbitrary sufficiently smooth function.


Conclusions and Open Problems

Conclusions

Divergence-type conservation laws are useful in analysis and numerics.

Conservation laws can be obtained systematically through the Direct ConstructionMethod, which employs multipliers and Euler operators.

The method is implemented in a symbolic package GeM for Maple.

For variational DE systems, conservation laws correspond to variational symmetries.

Noether’s theorem is usually not a preferred way to derive unknown conservationlaws.

Open problems

Computations become hard for conservation laws of high orders.Can we tell anything in advance about highest order of the conservation law for agiven DE system?

For complicated DEs and multipliers, computation of fluxes/densities can becomechallenging.


Some references

Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2010).Applications of Symmetry Methods to Partial Differential Equations.Springer: Applied Mathematical Sciences, Vol. 168.

Anco, S.C. and Bluman, G.W. (1997).Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78,2869–2873.

Anco, S.C. and Bluman, G.W. (2002).Direct construction method for conservation laws of partial differential equations.Part I: Examples of conservation law classifications. Eur. J. Appl. Math. 13,545–566.

Cheviakov, A.F. (2007).GeM software package for computation of symmetries and conservation laws ofdifferential equations. Comput. Phys. Commun. 176, 48–61.

Kallendorf, C., Cheviakov, A.F., Oberlack, M., and Wang, Y.Conservation Laws of Surfactant Transport Equations. Submitted, 2011.


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