All Questions
18 questions
3
votes
2
answers
347
views
modeling mechanical systems with Lie algebras
I'm not a mathematician, so please forgive my obvious naivety.
I'm interested in generating a representation in vector space of a two dimensional armature, or set of rigid linked elements, with each ...
- 33
3
votes
0
answers
51
views
Maximizing a function on $SU(4)$ similar to Von Neumann Trace Inequality
Given arbitrary $X,Y \in \mathfrak{su}(4)$, I want to maximize either of the following functions:
$\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$
...
- 2,099
7
votes
1
answer
271
views
Closest point in $SU(n) \otimes SU(n)$ to $SU(n^2)$
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 ...
- 2,099
2
votes
0
answers
94
views
Maximize a tricky function on $SU(n)$
Given non-zero $\xi \in \mathfrak{su}(n)$ and $G \in SU(n)$, consider the function:
$Q(U) = Tr(G^{\dagger}U)GU^{\dagger} - Tr(U^{\dagger}G) UG^{\dagger}$
(which just happens to be the gradient of $|...
3
votes
0
answers
189
views
Non-invariant Lagrangian on SU(n)
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 ...
- 2,099
0
votes
0
answers
71
views
Curves in $\mathfrak{su}(n)$ with specific property
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 $s \in[0,T]$ ...
- 2,099
2
votes
0
answers
115
views
Are singular critical points isolated for control systems on compact semisimple Lie groups
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 ...
- 2,099
3
votes
1
answer
134
views
Singular curves of affine distributions on a Lie group
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} = \{...
- 2,099
3
votes
0
answers
70
views
Attainability of Global Optima In Optimal Control
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 ...
- 2,099
2
votes
0
answers
90
views
Singularities of the Quantum propagator (baby version)
Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...
- 2,099
3
votes
2
answers
136
views
Level sets on $SU(4)$
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 ...
- 2,099
2
votes
1
answer
236
views
Simultaneous integral equation on $SU(n)$
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 ...
- 2,099
3
votes
1
answer
115
views
Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?
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 ...
- 2,099
0
votes
1
answer
138
views
Gradient on $SU(n)$
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 ...
- 2,099
2
votes
1
answer
143
views
Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$
Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...
- 2,099
1
vote
1
answer
172
views
Existence of a Lie algebra element orthogonal to the adjoint orbit of another element
Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...
- 2,099
2
votes
0
answers
423
views
First Variation of Dyson Series/Magnus Expansion
Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
- 2,099
0
votes
1
answer
130
views
equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
- 2,099