Limit of zeo mass for a dynamical system

[muzialis]

Hello All,

from considering the system of equations (differentiation is mean with respect to time for the function v = v(t) )



G' = a - v
v' = (1/m) (G - p (v))

Very briefly, they characterize the speed of a crack with an "effective mass" m under a generalized force G.

Dividing the top equation by the lower oneone can conclude that

dG / dV = m (a - v) / (G- p(v))

Considering the massless limit m-> 0 one then could obtain

dG(G- p(v)) = 0.



After a long preambe, my question : should one not be able to get to the massless limit right from the start by ignoring the term mv' (inertial term) from the start? If I try i do not recover the relationship dG(G- p(v)) = 0

Any help would really be the most appreciated

thanks

 

[gtoguy97]

in advance No, you cannot ignore the inertial term from the start. The equation for v' includes the inertial term (1/m), which affects the behavior of the system even when m is very small. Even if you start with the massless limit, the inertial term still needs to be taken into account. It's only when you take the limit as m-> 0 that you obtain the relationship dG(G- p(v)) = 0.