Acceleration as a function of time and position

[AliAhmed]

I'm quite curious as to whether there is an equation (whether it be in scalar, vector, or tensor form) that defines the acceleration of a particle as a function of the three space coordinates and time.

My curiosity arose when I was thinking of the one dimensional equation:
a = v*(dv/dx); where the acceleration and velocity are only defined in the x-direction.

I thought the generalization to three dimensions would be:
a = v_x*(dv_x/dx)i + v_y*(dv_y/dy)j + v_z*(dv_z/dz)k

To include the time variable I thought it might just be the material derivative of acceleration (like in fluid mechanics).

These assumptions are just based on my gut feeling (I have done no derivation whatsoever). Is there an actual equation?

 

[Vorde]

Well I don't know if there is a general equation (which there probably is), but in one dimension you are just looking for the second derivative of the equation of motion of the body.

In more than one dimension you can just split it up into an arbitrary amount of simultaneous parametric equations, and once again its the second derivative of each individual equation.

Once again there might be a general rule, but what I just said above will definitely work.

 

[elegysix]

I thought the generalization to three dimensions would be:
a = v_x*(dv_x/dx)i + v_y*(dv_y/dy)j + v_z*(dv_z/dz)k

In one dimension we can say dV/dt=(dx/dt)(dV/dx)=V_x(dV/dx) (but the sub x notation isn't needed because there is only one velocity, and its in x)

but for 3 dimensions, instead of x we use R

so dV/dt=(dR/dt)(dV/dR)=V*(dV/dR)

and

dV/dR = $( \partial V/ \partial x )i +( \partial V/ \partial y )j +( \partial V/ \partial z )k$

so

a = |V|*$\left[( \partial V/ \partial x )i +( \partial V/ \partial y )j +( \partial V/ \partial z )k \right]$

I believe you have to use the magnitude of V, rather than dotting V into dV/dR, for some reason. I can't explain/remember why, but I think this is how it is supposed to be. Could be wrong, idk. but that's my 2 cents, hope it helps

 

[AliAhmed]

a = |V|*$\left[( \partial V/ \partial x )i +( \partial V/ \partial y )j +( \partial V/ \partial z )k \right]$

- The equation seems right, but then again why wouldn't the material derivative of acceleration apply (as in fluid mechanics). Is there an assumption I am missing that doesn't apply to ordinary particles? The equation is:

$\vec{a}$ = $\frac{D\vec{v}}{Dt}$ = $\vec{v}$$\cdot$($\vec{\nabla}$$\vec{v}$) + $\frac{\partial \vec{v}}{\partial t}$

 

[PJC01]

I can confirm that there is indeed an equation that defines acceleration as a function of time and position. This equation is known as the acceleration vector function and is typically denoted as a(t) or a(x,y,z,t). It can be written in scalar, vector, or tensor form, depending on the specific situation and variables involved.

In general, the acceleration vector function takes into account the change in velocity over time and space, as well as the effects of external forces acting on the particle. It can be derived using principles of classical mechanics, such as Newton's laws of motion.

In the one-dimensional case you mentioned, the equation a = v*(dv/dx) is known as the acceleration equation of motion and is a special case of the more general acceleration vector function.

To include the time variable, the acceleration vector function would indeed involve the material derivative, which takes into account the change in acceleration over time as well.

I would suggest consulting a textbook on classical mechanics or speaking with a physics expert for a more detailed derivation and understanding of the acceleration vector function in three dimensions. It is a fundamental concept in physics and plays a crucial role in understanding the behavior of particles and objects in motion.