Multiple-scale analysis: Difference between revisions

Revision as of 01:41, 10 August 2009

Revision as of 02:20, 10 August 2009

Line 11:

:<math>\frac{dp}{dt}=-\frac{dH}{dq}, \qquad \frac{dq}{dt}=+\frac{dH}{dp},</math> {{pad|2em}} with {{pad|2em}} <math>H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \epsilon q^4,</math>

with ''q''&nbsp;=&nbsp;y(''t'') and ''p''&nbsp;=&nbsp;''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'',''q'') is a conserved quantity, a constant, equal to ''H''&nbsp;=&nbsp;½&nbsp;+&nbsp;¼&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded:

:<math>\left| y(t) \right| \le \sqrt{1 + \tfrac12 \epsilon}</math> {{pad|2em}} and {{pad|2em}} <math>\left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \epsilon}</math>{{pad|3em}} for all ''t''.

===Straightforward perturbation-series solution===