User hoj201 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T21:27:26Z http://mathoverflow.net/feeds/user/16852 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125618/what-is-the-name-of-this-product-of-lie-groups What is the name of this product of Lie groups hoj201 2013-03-26T12:16:15Z 2013-03-26T12:33:59Z <p>Let $G$ be a Lie group and $V$ be a vector space. Let $\rho_{l} : G \times V \to V$ be a left representation and $\rho_r:V \times G \to V$ be a right representation which <em>commutes</em> with $\rho_l$ in the sense that $\rho_l(g_1 , \rho_r(v , g_2) ) = \rho_r( \rho_l(g_1, v) , g_2)$. Then the group multiplication $(g_1,v_1) \cdot (g_2,v_2) = (g_1 \cdot g_2 \quad,\quad \rho_r(v_1,g_2) + \rho_l(g_1,v_2))$ makes the set $G \times V$ into a Lie group. The identity is $(e_G, e_H)$ and the inverse of $(g,h)$ is $(g^{-1}, - \rho_l[ g^{-1} , \rho_r(h,g^{-1} ) ] )$ and associativity can be verified by hand. This Lie group appears to be some sort of sum of a left semi-direct product with a right semi-direct product. Does it have a name?</p> http://mathoverflow.net/questions/78698/dissipative-hamiltonian-system-with-a-periodic-force Dissipative Hamiltonian System with a Periodic Force hoj201 2011-10-20T19:51:33Z 2011-10-20T19:51:33Z <p>Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for $Y = \omega^{\sharp}(F)$ we have that $Y[H] &lt; 0$ everywhere outside a point $x_0 \in P$. This makes $x_0$ a stable point of the dissipative Hamiltonian system $(P,\omega,H,F)$. Now let $f: \mathbb{R} \times P \to T^*P$ be a time-periodic force. My question: Does there exists a periodic orbit (near $x_0$) for the periodically forced system $(P,\omega,H, F + \epsilon f)$ for sufficiently small $\epsilon$ ? I'm sure the answer is yes, but how big can $\epsilon$ be?</p> http://mathoverflow.net/questions/71836/is-the-set-of-average-position-preserving-transformations-a-lie-group Is the set of average-position preserving transformations a Lie group hoj201 2011-08-01T21:53:37Z 2011-08-01T21:53:37Z <p>Let $M$ be a compact subset in $\mathbb{R}^n$ and $\mu$ a volume form on $M$. Let $x_i$ denote the function corresponding to the $i$-coordinate. Does the set of diffeomorphisms satisfying $$\int_M{x_i \mu } = \int_M{ \varphi_*(x_i) \mu }, \quad i = 1,\dots,n$$ form a Lie group? In other words, is the set of transformations that preserve the $\mu$-average position a Lie group? Certainly the isotropy group of the $\mu$-average position is a Lie-Group, however this set seems to incude a little more than the isotropy group.</p> http://mathoverflow.net/questions/78698/dissipative-hamiltonian-system-with-a-periodic-force Comment by hoj201 hoj201 2011-12-08T22:37:43Z 2011-12-08T22:37:43Z Thankyou Jaap. I will take a look. Your interpretation of the notation is correct. Secondly, I guess you are right. There is no reason to discriminate the unstable case. For dissipative systems I'd expect to find a stable periodic orbit from a stable point. Under time-reversal a stable equilibria would become an unstable one, and the corresponding stable limit cycle would become unstable as well. http://mathoverflow.net/questions/71836/is-the-set-of-average-position-preserving-transformations-a-lie-group Comment by hoj201 hoj201 2011-08-02T02:12:51Z 2011-08-02T02:12:51Z Thanks for the quick response guys. I'm not too concerned with the dimensionality concerns, so my question is regarding psuedo-groups. Does this set fail the closure axiom? Is my intuition my enemy? If not, which axiom has failed?