"락스 쌍 (Lax pair)"의 두 판 사이의 차이

2012년 6월 19일 (화) 12:00 판

많은 적분가능
a pair of \(N\times N\) matrices L(x,p) and M(x,p) such that the Lax equation \(\dot{L}=\{L,M\}\) is equivalent to the Hamiltonian equations of the mechanical system
\(\dot{q}_i=\{q_i,H\}=\partial H/\partial p_i\)
\(\dot{p}_i=\{q_i,H\}-\partial H/\partial q_i\)
integrals of motion can be derived from the trace of powers of L
\(\frac{d}{dt}\operatorname{tr}(L^p)=\operatorname{tr}(p [L,M]L^{p-1})=p\operatorname{tr}(LML^{p-1}-ML^{p})=0\)
따라서 \(\operatorname{tr}(L^p)\) 는 보존량이 된다

spectral curve
\(\{(k,z)\in\mathbb{C}\times\mathbb{C}:\det(kI-L(z))=0\}\)
integrals of motion
\(\operatorname{tr} L(z)=\sum_{n}L_{n}z^{n} \)
for examples, look at Introduction to classical integrable systems, chapter 3 http://goo.gl/LaawC

L is an isospectral deformation of L(0) if L(t) has the same eigenvalues for all t
\(L(t)v(t)=\lambda v(t)\)
Record their derivative by a matrix
v'(t)=B(t)v(t)
Differentiate \(L(t)v(t)=\lambda v(t)\)
L'(t)v(t)+L(t)v'(t)=\lambda v'(t)
L'(t)v'(t)=[B(t),L(t)]v(t)
L'(t)=[B[t],L(t)
So B(t) and L(t) are a Lax pair

@@ 12번째 줄: / 12번째 줄: @@
 * spectral parameter
-* <math>H(q,p)</math><br>
+<h5>기호</h5>
+*  해밀토니안 <math>H(q,p)</math><br>
 **  the coordinates <math>q=(q_1,\cdots,q_N)</math><br>
 **  the momenta <math>p=(p_1,\cdots,p_N)</math><br>
 ** <math>\{q_i,p_i\}=\delta_{ij}</math><br>
 **  the equation of motion<br><math>\dot{q}_i=\{q_i,H\}=\partial H/\partial p_i</math><br><math>\dot{p}_i=\{q_i,H\}-\partial H/\partial q_i</math><br>
-* For an integrable system, sometimes there exists a Lax pair
+<h5>락스 쌍</h5>
+* 많은 적분가능
 *  a pair of <math>N\times N</math> matrices L(x,p) and M(x,p) such that the Lax equation <math>\dot{L}=\{L,M\}</math> is equivalent to the Hamiltonian equations of the mechanical system<br><math>\dot{q}_i=\{q_i,H\}=\partial H/\partial p_i</math><br><math>\dot{p}_i=\{q_i,H\}-\partial H/\partial q_i</math><br>
-*  integrals of motion can be derived from the trace of powers of L<br>
+*  integrals of motion can be derived from the trace of powers of L<br><math>\frac{d}{dt}\operatorname{tr}(L^p)=\operatorname{tr}(p [L,M]L^{p-1})=p\operatorname{tr}(LML^{p-1}-ML^{p})=0</math><br> 따라서 <math>\operatorname{tr}(L^p)</math> 는 보존량이 된다<br>
@@ 37번째 줄: / 50번째 줄: @@
+<h5>isospectral deformation</h5>
-<h5 style="line-height: 2em; margin: 0px;">isospectral deformation</h5>
 *  L is an isospectral deformation of L(0) if  L(t) has the same eigenvalues for all t<br>