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Given a smooth manifold $M$ and another $S$, consider a smooth function $\psi: S \times M \rightarrow \mathbb{R}$, and use this to define $\psi_s:M\rightarrow \mathbb{R}$ by $\phi_s(p):= \psi(s,p)$. …

Given arbitrary $X,Y \in \mathfrak{su}(4)$, I want to maximize either of the following functions: $\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$ an …

What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. minimize over $V_1, V_2$: $\min_{V_1, V_2} | V_1 \otimes V_2 - U|$ in …

Given a smooth manifold $M$ and a smooth Lagrangian $\mathcal{L}(x,\dot{x})$ on $M$, the curves which make stationary the corresponding action are those which solve the Euler-Lagrange equations. If a …

I have a Lagrangian on $SU(n)$, which is not invariant. Given the Lagrangian $\mathcal{L}[U_t, \dot{U}_t] = \langle \dot{U}_t, \nabla J \big|_{U_t} \rangle$ I need to find the curves of stationary a …

Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ fo …

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form: $\frac{d U_t}{dt} = (A + w(t)B)U_t$ where $A,B \in \mathfrak{su}(n)$ generate the alge …

Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group? Specifically the case of a right invariant affine distribution: $D_{U} = \ …

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the …

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ …

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not i …

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth rea …

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a …

I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re …

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest …