Lie derivative

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Problem 3.100 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Let $V, W$ be smooth vector fields defined in $\mathbb {R}^{n}$ . Let $f,g : \mathbb {R}^{n} \to \mathbb {R}$ be smooth functions. Consider now the pairs $(V,f), (W,g)$ . One defines a commutator of such pairs as

[(V,f),(W,g)] := ([V,W], L_{V}g - L_{W}f)

where $L_{V}(\cdot )$ denotes the Lie derivative. Now $[V,W]$ is a vector field again and $L_{V}g - L_{V}f$ is a smooth function again. The Lie derivative is linear and obeys the product rule. Consider $n=2$ . Let

V = u_{2}\frac{\partial }{\partial u_{1}} - u_{1}\frac{\partial }{\partial u_{2}}, \qquad W = u_{1}\frac{\partial }{\partial u_{1}} + u_{2}\frac{\partial }{\partial u_{2}}

and $f(u_{1},u_{2}) = g(u_{1},u_{2}) = u_{1}^{2}+u_{2}^{2}$ . Calculate the commutator.

Problem 4.59 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 smooth vector fields in $\mathbb {R}^{2}$

V = V_1(\overrightarrow {x})\frac{\partial }{\partial x_1} + V_2(\overrightarrow {x})\frac{\partial }{\partial x_2}, \quad W = W_1(\overrightarrow {x})\frac{\partial }{\partial x_1} + W_2(\overrightarrow {x})\frac{\partial }{\partial x_2} .

Assume that $[V,W] = 0$ . Calculate the Lie derivatives

L_V(V_1W_2-V_2W_1), \quad L_W(V_1W_2-V_2W_1) .

Discuss.

Problem 4.67 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 Lie derivative of a differential form $\omega$ with respect to a vector field $W$ can be expressed as

L_W \omega \equiv W \rfloor d\omega + d(W \rfloor \omega )

where $d$ is the exterior derivative and $\rfloor$ denotes the contraction. Let $\alpha$ be a smooth differential one-form and $V$ be smooth vector field. Assume that

L_V \alpha = f\alpha

where $f$ is a smooth function and $L_V(.)$ denotes the Lie derivative. Define the function $F$ as

F := V \rfloor \alpha

Show that

dF = f\alpha - V \rfloor d\alpha .

Problem 4.118 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 4.2 "Supplementary Problems"

(i)(ii)(iii)(iv)

Consider the two-dimensional sphere

S_1^2+S_2^2+S_3^2 = S^2

where $S > 0$ is the radius of the sphere. Consider the symplectic structure on this sphere with the symplectic differential two form

\omega := -\frac{1}{2S^2} \sum _{j,k,\ell =1}^{3} \epsilon _{jk\ell }S_j dS_k \wedge dS_\ell

( $\epsilon_{123} = 1$ ) and the Hamilton vector fields

V_{S_j} := \sum _{k,\ell =1}^{3} \epsilon _{jk\ell }S_k\frac{\partial }{\partial S_\ell } .

The Poisson bracket is defined by

[S_j,S_k]_{PB} := -V_{S_j}S_k .

(i)

Calculate $[S_j,S_k]_{PB}$ .

(ii)

Calculate $V_{S_j}\rfloor \omega$ .

(iii)

Calculate $V_{S_j}\rfloor V_{S_k}\rfloor \omega$ .

(iv)

Calculate the Lie derivative $L_{V_{S_j}}\omega$ .

References

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