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.