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how to get canonical equations from Hamiltonian Principle
Ok here's some fun stuff...
I start off with the first order term of the taylor series representation for the function where S[q,p] has a local minimum. Thus, we need delta S to be zero (all higher orders need to be neglegible).
This is folled by a total differentiation in epsilon and rearranging which includes one partial integration assuming that the variations vanish at the boundaries t1 and t2.
Ops, forgot to mention: replacing the Lagrange function by the Hamiltonian and pq* expression and executing the differential with new variables is called "Legendre transformation" and works as follows:
df = u dx + v dy
replace: g = f - ux
find: dg = v dy + x du
any questions? ;) Feel free to ask
Length: 176
Rating: 0.00 (0 ratings)
Tags: hamiltonian canonical equations from variation principle Legendre Transformation
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Euler-Lagrange equation from Hamiltonian Principle
yes... it's that easy :)
But I forgot to mention that [dL/dq* dq](t1..t2) = 0
It's the same trick as with the canonical equations in my other video, don't miss that one :)
Question for you guys: why is the hamiltonian principle applicable?
Length: 158
Rating: 0.00 (0 ratings)
Tags: euler lagrange equation derived from hamiltonian principle
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Hamiltonian live at Junkyard (1)
Hamiltonian live at Junkyard (1)
Length: 177
Rating: 1.00 (1 ratings)
Tags: Hamiltonian Junkyard
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Hamiltonian mean-field (HMF) simulation #1
Simulation, displaying formation of a 'macro-particle', nonzero M=mean(exp i q), and jittery center-of-mass motion.
Length: 13
Rating: 0.00 (0 ratings)
Tags: Simulation Chaos Hamiltonian
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Intestinal Fortitude - From Flying Spaghetti to Donut
This video illustrates the Shrink-On-No-Overlaps algorithm (SONO) applied to a
maximally-compact hamiltonian walk on a 14x14x14 cubic lattice.
SONO was conceived by Piotr Pieranski. The shrinking-knot animation was furnished by Sylwester Przybyl and Piotr Pieranski. See http://fizyka.phys.put.poznan.pl/~pieransk/Hamiltonian%20knots.html
The program that created randomly-generated (up to a certain degree) compact hamiltonian walks on a lattice was written by Alexander Borovinskiy. The knot analysis (e.g. using the mathematics of knot invariants) was performed by Rhonald Lua. This work was done under the direction of Alexander Grosberg at the University of Minnesota.
For related work, check out the following publications online:
http://www.pnas.org/cgi/content/full/101/37/13431
http://compbiol.plosjournals.org/perlserv/?request=get-document&doi=10.1371%2Fjournal.pcbi.0020045
Length: 82
Rating: 5.00 (2 ratings)
Tags: knots topology lattice cubic hamiltonian polymers morphing transformer
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Hamiltonian live at Junkyard (2)
Hamiltonian live at Junkyard (2)
Length: 136
Rating: 0.00 (0 ratings)
Tags: Hamiltonian Junkyard
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A solution to the Lane-Emden System on bounded domain
The Lane-Emden system is an elliptic system of Hamiltonian type (so-called Hamiltonian PDEs). Its associated functional is strongly indefinite, which means that each of its critical points has infinite Morse index and hence is a saddle point & unstable.
Length: 26
Rating: 0.00 (0 ratings)
Tags: Lane-Emden system; Hamiltonian PDEs
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