Discrete symmetries of equations and dynamical systems

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Problem 4.2 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Consider the Hilbert space H=C4\mathcal{H} = \mathbb {C}^4 and the hermitian matrices

H=(2−10−1−12−100−12−1−10−12),A=(1000000000−100000),B=(000001000000000−1). H = \begin{pmatrix} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{pmatrix}, \quad A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} .

Using the Kronecker product we consider the 16×1616 \times 16 matrix

K=12(H⊗I4+I4⊗H)−λA⊗A−μB⊗B K = \frac{1}{2}(H \otimes I_4 + I_4 \otimes H) - \lambda A \otimes A - \mu B \otimes B

acting in the Hilbert space C16\mathbb {C}^{16}, where λ,μ≥0\lambda , \mu \geq 0. We consider KK as a Hamilton operator.

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(i)

Consider the permutation matrix

R=(0100001000011000). R = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix} .

Thus the inverse of RR exists. Calculate

(R⊗R)−1K(R⊗R) (R \otimes R)^{-1} K (R \otimes R)

and the commutator [R⊗R,K][R \otimes R, K]. Discuss the case λ=μ\lambda = \mu.

(ii)

Consider the permutation matrices

S=(0010010010000001),T=(1000000100100100). S = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} .

Find the commutators [S⊗S,K][S \otimes S, K] and [T⊗T,K][T \otimes T, K]. Discuss.

Problem 4.8 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Consider the system of linear equations Ax→=b→A\overrightarrow {x} = \overrightarrow {b}, where AA is an invertible 3×33 \times 3. Assume that the linear system is invariant under x1↔x2x_1 \leftrightarrow x_2, i.e.

Ax→=Ax→∣x1↔x2=b→. A\overrightarrow {x} = \left. A\overrightarrow {x} \right|_{x_1 \leftrightarrow x_2} = \overrightarrow {b} .

Can we conclude that x1=x2x_1 = x_2?

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Problem 4.16 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Consider the Hamilton function

H(p→,q→)=12(p12+p22)−(Aq12+Bq22+Cq12q2+Dq23) H(\overrightarrow {p}, \overrightarrow {q}) = \frac{1}{2}(p_1^2 + p_2^2) - (A q_1^2 + B q_2^2 + C q_1^2 q_2 + D q_2^3)

where q→=(q1,q2)∈R2\overrightarrow {q} = (q_1, q_2) \in \mathbb {R}^{2} and p→=(p1,p2)∈R2\overrightarrow {p} = (p_1, p_2) \in \mathbb {R}^{2} are the position vector and the momentum vector of the particle, respectively. The Hamilton function is a constant of motion, i.e. H(p→,q→)=EH(\overrightarrow {p}, \overrightarrow {q}) = E. Consider the real analytic diffeomorphism

r=∣q→∣θ=arctan⁡(q2/q1)u=r˙=(q1p1+q2p2)/∣q→∣v=rθ˙=(q1p2−q2p1)/∣q→∣ \begin{aligned} r & = \left|\overrightarrow {q}\right| \\ \theta & = \arctan (q_2/q_1) \\ u & = \dot{r} = (q_1 p_1 + q_2 p_2)/\left|\overrightarrow {q}\right| \\ v & = r\dot{\theta } = (q_1 p_2 - q_2 p_1)/\left|\overrightarrow {q}\right| \end{aligned}

which is a McGehee-type transformation of the second kind. The equations of motion take the form

r˙=uθ˙=v/ru˙=v2/r+2(Acos⁡2θ+Bsin⁡2θ)r+3(Ccos⁡2θ+Dsin⁡2θ)r2sin⁡θv˙=−uv/r+2(B−A)rsin⁡θcos⁡θ+(3(D−C)sin⁡2θ+C)r2cos⁡θ \begin{aligned} \dot{r} & = u \\ \dot{\theta } & = v/r \\ \dot{u} & = v^2/r + 2(A \cos ^2 \theta + B \sin ^2 \theta ) r + 3(C \cos ^2 \theta + D \sin ^2 \theta ) r^2 \sin \theta \\ \dot{v} & = -uv/r + 2(B-A) r \sin \theta \cos \theta + (3(D-C)\sin ^2\theta + C) r^2 \cos \theta \end{aligned}

and the energy takes the form

12(u2+v2)−(Acos⁡2θ+Bsin⁡2θ)r2−(Ccos⁡2θ+Dsin⁡2θ)r3sin⁡θ=E. \frac{1}{2}(u^2+v^2) - (A\cos ^2\theta + B\sin ^2\theta ) r^2 - (C\cos ^2\theta + D\sin ^2\theta ) r^3 \sin \theta = E .

Find the discrete symmetries of the vector field given by the autonomous system of differential equations.

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Problem 4.40 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Consider the Henon-Heiles Hamilton function

H(p1,p2,x1,x2)=12(p12+p22+x12+x22)+cx12x2−13x23 H(p_1,p_2,x_1,x_2) = \frac{1}{2}(p_1^2+p_2^2+x_1^2+x_2^2) + cx_1^2 x_2 - \frac{1}{3}x_2^3

with the equations of motion

d2x1dt2=−x1−2cx1x2,d2x2dt2=−x2−cx12+x22. \frac{d^2x_1}{dt^2} = -x_1 - 2cx_1x_2, \quad \frac{d^2x_2}{dt^2} = -x_2 - cx_1^2 + x_2^2 .

Show that the equations of motions are invariant for c=1c=1 under the 2π/32\pi /3 rotation

(x1′x2′)=(1/23/2−3/21/2)(x1x2). \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & 1/2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} .
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Problem 4.41 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Let x→,y→,z→\overrightarrow {x}, \overrightarrow {y}, \overrightarrow {z} be unit vectors in the Euclidean space R2\mathbb {R}^{2}. Find the maximum of the function

f(x→,y→,z→)=∥x→−y→∥2+∥y→−z→∥2+∥z→−x→∥2 f(\overrightarrow {x},\overrightarrow {y},\overrightarrow {z}) = \left\| \overrightarrow {x}-\overrightarrow {y}\right\| ^2 + \left\| \overrightarrow {y}-\overrightarrow {z}\right\| ^2 + \left\| \overrightarrow {z}-\overrightarrow {x}\right\| ^2

with respect to the constraints

∥x→∥=∥y→∥=∥z→∥=1 \left\| \overrightarrow {x}\right\| = \left\| \overrightarrow {y}\right\| = \left\| \overrightarrow {z}\right\| = 1

i.e., x→,y→,z→\overrightarrow {x}, \overrightarrow {y}, \overrightarrow {z} are unit vectors. Give an interpretation of the result. Solve the problem using the Lagrange multiplier method. Discuss the discrete symmetries of the problem.

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Problem 4.42 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

The D4D_4 symmetry consists of four rotations each of 90 degrees and four reflection mirrors with each angle between them being 45 degrees. Consider the system of second order ordinary differential equations

d2x1dt2=−x1+(μ−x12−x22)dx1dt \frac{d^2x_1}{dt^2} = -x_1 + (\mu - x_1^2 - x_2^2)\frac{dx_1}{dt} d2x2dt2=−x2+(μ−x22−x12)dx2dt \frac{d^2x_2}{dt^2} = -x_2 + (\mu - x_2^2 - x_1^2)\frac{dx_2}{dt}

where μ∈R\mu \in \mathbb {R} is a real bifurcation parameter. Show that as μ\mu varies through zero, there is a Hopf bifurcation with D4D_4 symmetry.

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References

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.