Displacement as a discrete function of time

[Apashanka]

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 3-D ##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 3-D ##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 3-D ##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??

 

[jbriggs444]

No.

 

[Apashanka]

No.

Then what will be will you please help??

 

[jbriggs444]

Then what will be will you please help??

As I understand it, you are interested in interpolation. There is no unique way of assigning a curve through a set of discrete points. Various approaches are possible such as linear interpolation, interpolating polynomials, cubic splines and least squares fit to a [insert function family here].

Typically, one wants to find a function family that is well motivated physically and then use a least squares approach to zero in on a suitable family member.

 

[Apashanka]

As I understand it, you are interested in interpolation. There is no unique way of assigning a curve through a set of discrete points. Various approaches are possible such as linear interpolation, interpolating polynomials, cubic splines and least squares fit to a [insert function family here].

Typically, one wants to find a function family that is well motivated physically and then use a least squares approach to zero in on a suitable family member.

Is the volume expression correct what I have posted above??

 

[jbriggs444]

No.

 

[Apashanka]

No.

Then what it will be ??

 

[jbriggs444]

See response #4 above.

 

[Apashanka]

See response #4 above.

I am asking for the volume expression e.g ##d^3x## in my first post

 

[jbriggs444]

I am asking for the volume expression e.g ##d^3x## in my first post

First you have to define x. As #4 points out, you cannot do that merely by stating a finite number of values. The idea that the third derivative of x (with respect to time?) is a "volume" is disconcerting as well.

 

[Apashanka]

First you have to define x. As #4 points out, you cannot do that merely by stating a finite number of values.

What I am asking is as ##x=f_t(X)## at any time ##t## then ##dx=\partial_X f_t(X) dX## and in 3-D it becomes ##d^3x=\Pi_i \partial_{X_i}f_t(X_i) d^3X## also sometimes called ##d^3x=Jd^3X##.
Integrating both sides can it be written ##v(t)=\Pi_i f_t(X_i)## for given ##Xi's##..??(##f_t(X)## known for any finite time)

 

[jbriggs444]

What I am asking is as ##x=f_t(X)## at any time ##t## then ##dx=\partial_X f_t(X) dX## and in 3-D it becomes ##d^3x=\Pi_i \partial_{X_i}f_t(X_i) d^3X## also sometimes called ##d^3x=Jd^3X##.
Integrating both sides can it be written ##v(t)=\Pi_i f_t(X_i)## for given ##Xi's##..??

I see a lot of notation with nary a definition in sight.

 

[haruspex]

What I am asking is as ##x=f_t(X)## at any time ##t## then ##dx=\partial_X f_t(X) dX## and in 3-D it becomes ##d^3x=\Pi_i \partial_{X_i}f_t(X_i) d^3X## also sometimes called ##d^3x=Jd^3X##.
Integrating both sides can it be written ##v(t)=\Pi_i f_t(X_i)## for given ##Xi's##..??(##f_t(X)## known for any finite time)

Presumably you mean ##x(X,t)=f_t(X)##, and since t is continuous that is effectively ##x(X,t)=f(t,X)##. So ##dx=\partial_X f(t,X) dX+\partial_t f(t,X) dt##.

But you seem not to be concerned with t as a variable, so let's make it simpler by getting rid of it: ##x(X)=f(X)##, ##dx=f'(X) dX##.

Your volume differential is then ##d^3x=\Pi_i f'(X_i) d^3X=Jd^3X##, but I'm not sure you can integrate that as ##v=\Pi_i f(X_i)##. Need to think about the bounds.

 

[Apashanka]

Presumably you mean ##x(X,t)=f_t(X)##, and since t is continuous that is effectively ##x(X,t)=f(t,X)##. So ##dx=\partial_X f(t,X) dX+\partial_t f(t,X) dt##.

But you seem not to be concerned with t as a variable, so let's make it simpler by getting rid of it: ##x(X)=f(X)##, ##dx=f'(X) dX##.

Your volume differential is then ##d^3x=\Pi_i f'(X_i) d^3X=Jd^3X##, but I'm not sure you can integrate that as ##v=\Pi_i f(X_i)##. Need to think about the bounds.

I am not taking ##x## as an explicit function of ##t## , although ##x## varies with ##t##...to take this into account ##x## at any ##t## is ##f_t(X)## where the functional dependence changes with time given ##X## fixed initial coordinate.
For exm for given ##X=2## say ##x## at ##t_1## varies as ##X^2##,at ##t_2## as ##X^2+2## ,at ##t_3## as ##X+15## and so on...

 

[haruspex]

I am not taking ##x## as an explicit function of ##t## , although ##x## varies with ##t##...to take this into account ##x## at any ##t## is ##f_t(X)## where the functional dependence changes with time given ##X## fixed initial coordinate.
For exm for given ##X=2## say ##x## at ##t_1## varies as ##X^2##,at ##t_2## as ##X^2+2## ,at ##t_3## as ##X+15## and so on...

So although t is time it is a discrete variable, and does not really enter into the matter as far as the thread is concerned.