Introduction to Symmetry Analysis

Brian Cantwell Department of Aeronautics and Astronautics

Stanford University

Chapter 8 - Ordinary Differential Equations

8.1 Extension of Lie Groups in the Plane

The Extended Transformation is a Group Two transformations of the extended group

Compose the two transformations

The last relation is rearranged to read

Differentiating F and G gives

Comparing the expressions in parentheses we have

The composed transformation is in exactly the same form as the original transformation!

Finite transformation of the second derivative

The twice extended finite transformation is

Finite transformation of higher derivatives

The p-th order extended group

where

Infinitesimal transformation of the first derivative

Recall the infinitesimal transformation of coordinates

where

Substitute

Expand and retain only the lowest order terms

Once-extended infinitesimal transformation in the plane

where the infinitesimal function fully written out is

where

Expand and retain only the lowest order terms

Infinitesimal transformation of the second derivative

The infinitesimal function transforming third derivatives

Expand and retain only the lowest order terms. The p times extended infinitesimal transformation is

The infinitesimal transformation of higher order derivatives

where

8.2 Expansion of an ODE in a Lie Series - the Invariance Condition for ODEs

The characteristic equations associated with extended groups are

Construction of the general first order ODE that admits a given group - the Ricatti Equation

Let the first integral of the group be In principle we can solve for either x or y. Assume we solve for y. The equation takes the following form.

where

The general solution of the Ricatti equation can always be determined if a particular solution of the equation can be found. A particular solution in this case is

To demonstrate take the differential of f.

Dividing by yields the Ricatti equation in terms of f. ξdx

Now let.

and work out the equation that governs h[x].

The general second-order ordinary differential equation

is invariant under the twice-extended group if and only if

Consider the simplest second-order ODE

The invariance condition is

Fully written out the invariance condition is

For invariance this equation must be satisfied subject to the condition that y is a solution of

The determining equations of the group are

These equations can be used to work out the unknown infinitesimals.

Assume that the infinitesimals can be written as a multivariate power series

Insert into the determining equations

The coefficients must satisfy the following algebraic system

Finally the infinitesimals are

The software package used on Yxx = 0

Example 8.2

Example 8.2 – Using the software

Solve yxx +1xyx − e

y = 0

Groups Xa = − x

2∂∂x

+ ∂∂y

Xb = x ln x[ ] ∂∂x

− 2 1+ ln x[ ]( ) ∂∂y

Lie algebra XiX j − X jXi =Xa Xb

Xa 0 −Xa

Xb Xa 0Xa is the ideal of the Lie algebra

Characteristic equations of Xa dxξ

= dyη

= dyxη 1{ }

⇒ dx−x / 2

= dy1

= dyxyx / 2

First two invariants dx−x / 2

= dy1

= dyxyx / 2

ln φ[ ] = y / 2 + ln x[ ]φ = xey/2

ln G[ ] = ln xyx[ ]G = xyx

Invariant groups are the same regardless of the sign

Example 8.2 - Solution

Solve yxx +1xyx − e

y = 0

Use the method of differential invariants

φ = xey/2

G = xyxFirst reduction

DG = ∂G∂x

dx + ∂G∂y

dy + ∂G∂yx

dyx

Dφ = ∂F∂x

dx + ∂F∂y

dy

DGDφ

=Gx +Gyyx +Gyx

yxxFx + Fyyx

= yx + xyxxey/2 + xyx

2ey/2

dGdφ

= yx + xyxxey/2 + xyx

2ey/2

=yx + x − 1

xyx + e

y⎛⎝⎜

⎞⎠⎟

ey/2 + xyx2ey/2

dGdφ

= xey/2

1+ xyx2

= φ

1+ G2

Second reduction integrate

1+ G2

⎛⎝⎜

⎞⎠⎟ dG = φdφ

G + G2

4− φ

2

2= C1 −1

G2 + 4G − 2φ 2 − 4 C1 −1( ) = 0

G φ[ ] = −2 ± 2 C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

Example 8.2 – Solution continued; reduce the order twice

So far we have G φ[ ] = −2 ± 2 C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

We still need to carry out one more integration.

φ = xey/2

G = xyx

dφ = ey/2dx + x2ey/2yxdx =

dxx

xey/2 + x2ey/2xyx

⎛⎝⎜

⎞⎠⎟ =

dxx

φ + φ2G⎛

⎝⎜⎞⎠⎟

dxx= dφ

φ 1+ G2

⎛⎝⎜

⎞⎠⎟

Nowxyx = G φ[ ]dy = G φ[ ]dx

x= G φ[ ] dφ

φ 1+ G φ[ ]2

⎛⎝⎜

⎞⎠⎟

Example 8.2 – Solution continued

G φ[ ] = −2 ± 2 C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

dy =−2 ± 2 C1 +

φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

1+−2 ± 2 C1 +

φ 2

2⎛⎝⎜

⎞⎠⎟

1/2⎛

⎝⎜

⎞

⎠⎟

2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

dφφ

dy =−2 ± 2 C1 +

φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

± C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

dφφ

= 2 ∓ 2

C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

dφφ

y = 2 ∓ 2

C1 +φ 2

2⎛⎝⎜

⎞⎠⎟

1/2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

dφφ

+C2φ=xey∫

C2 = y −2C1

C1 ∓1( )Ln φ[ ] ± Ln 2C1 + 2C1 2C1 +φ2( )1/2⎡

⎣⎤⎦( )

φ=xey/2

Example 8.2 – Solution continued

Problem 8.9

yxx −yxy2

+ 1xy

= 0

Groups Xa = x2 ∂∂x

+ xy ∂∂y

Lie algebra XiX j − X jXi =Xa Xb

Xa 0 −Xa

Xb Xa 0Xa is the ideal of the Lie algebra

Characteristic equations of Xa dxξ

= dyη

= dyxη 1{ }

⇒ dxx2 = dy

xy= dyxy − xyx

First two invariants dxx2

= dyxy

= dyxy − xyx

φ = y / xG = y − xyx

Xb = x ∂∂x

+ y2

∂∂y

Solve

φ = y / xG = y − xyxFirst reductionDG = −xyxdx + dy − xdyx

Dφ = − yx2 dx +

1xdy

DGDφ

=Gx +Gyyx +Gyx

yxxFx + Fyyx

= −yx + yx − xyxx− yx2 +

1xyx

= −xyxx− yx2 +

1xyx

dGdφ

= −x3yxx−y + xyx

=x3 yx

y2 −1xy

⎛⎝⎜

⎞⎠⎟

y − xyx=

x2

y2 xyx − y( )y − xyx

dGdφ

= − 1φ 2

Second reduction integrate

G = 1φ+C1

Problem 8.9 – reduce the order twice

φ = y / xG = y − xyx

G φ[ ] = 1φ+C1

y − xyx =xy+C1

Considerd x / y( )dx

= 1y− xyxy2

xyx = y − y2 d x / y( )

dx= y − x

y−C1

y2 d x / y( )dx

= xy+C1

Let f = x / yx2

f 2

d f( )dx

= f +C1

x2 d f( )dx

= f 3 +C1 f2

dff 3 +C1 f

2 = dxx2 = − 1

C1 f+ 1C1

2 LnC1 + ff

⎡⎣⎢

⎤⎦⎥= − 1

x+C2

C2 =1x+ 1

C12 Ln 1+C1φ[ ]− φ

C1

⎛⎝⎜

⎞⎠⎟φ=y/x

Problem 8.9 – Solution

Carry out one more integration