 #1
Apashanka
 429
 15
 Homework Statement:

Given initial displacement ##X_0## and displacement at any time ##t## as ##x##.
Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time.
For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11....##and so on.
From this, is there any method to find ##x(t)## as an explicit function of time??(##X_0=##constt,initial fixed coord.)
Another part from above we find ##dx=(\partial_{X_0}f_t(X_0))dX_0## and hence in 3D ##d^3x=\Pi_i (\partial_{X_i} f_t(X_i)d^3X##
Therefore at any time the volume will be ##\Pi_i f_t(X_i)## isn't it??
 Relevant Equations:

Given initial displacement ##X_0## and displacement at any time ##t## as ##x##.
Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time.
For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11....##and so on.
From this, is there any method to find ##x(t)## as an explicit function of time??(##X_0=##constt,initial fixed coord.)
Another part from above we find ##dx=(\partial_{X_0}f_t(X_0))dX_0## and hence in 3D ##d^3x=\Pi_i (\partial_{X_i} f_t(X_i)d^3X##
Therefore at any time the volume will be ##\Pi_i f_t(X_i)## isn't it??
Given initial displacement ##X_0## and displacement at any time ##t## as ##x##.
Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time.
For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11...##and so on.
From this, is there any method to find ##x(t)## as an explicit function of time??(##X_0=##constt,initial fixed coord.)
Another part from above we find ##dx=(\partial_{X_0}f_t(X_0))dX_0## and hence in 3D ##d^3x=\Pi_i (\partial_{X_i} f_t(X_i)d^3X##
Therefore at any time the volume will be ##\Pi_i f_t(X_i)## isn't it??
Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time.
For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11...##and so on.
From this, is there any method to find ##x(t)## as an explicit function of time??(##X_0=##constt,initial fixed coord.)
Another part from above we find ##dx=(\partial_{X_0}f_t(X_0))dX_0## and hence in 3D ##d^3x=\Pi_i (\partial_{X_i} f_t(X_i)d^3X##
Therefore at any time the volume will be ##\Pi_i f_t(X_i)## isn't it??