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In Newton's Principa, $\sum \vec{F}_{ext} = \frac{d\vec{\rho}}{dt}$

If the momentum vector is in mutildimensions, wouldn't a more general equation be

$\sum \vec{F}_{ext} = \vec{\nabla}{\vec{p}}$

I realize we are only taking a single variable derivative with respect to t, but can I still do this?

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$d\mathbf{p}/dt$ doesn't depend on the dimensions of $\mathbf{p}$; each coordinate is differentiated separately to get a result vector with the same dimensions as $\mathbf{p}$. $\nabla\mathbf{p}$ also gives a vector result, but it means something completely different – in $\mathbb{R}^3$, $\langle\;d\mathbf{p}/dx,\;d\mathbf{p}/dy,\;d\mathbf{p}/dz\;\rangle$. –  rdhs Dec 29 '11 at 8:55

2 Answers 2

up vote 3 down vote accepted

Nope, the total momentum $\vec p$ (and similarly the force $\vec F$, which is its time derivative) is just one vector for the whole system (or whole space) so it makes no sense to differentiate it with respect to spatial coordinates, and $\nabla$ is a symbol for the differentiation with respect to $x,y,z$.

That doesn't mean that equations similar to those you are thinking about don't exist. In the mechanics of liquids, solids, and gases, one may talk about the "density" of forces and density of energy and momentum: we want to know not only the total momentum or force but also "where it is located". The density of momentum is combined with the flux of momentum which is expressed by the ($3\times 3$) stress tensor (which is generalized to a $4\times 4$ stress-energy tensor in relativity). There exist equations involving a gradient when one talks about the stress tensor although this page

http://en.wikipedia.org/wiki/Stress_(physics)

doesn't really offer too many of them... You may see the conservation law for the stress-energy tensor

http://en.wikipedia.org/wiki/Energy-momentum_density#Conservation_law

which is a generalization of Newton's second law along the lines you proposed. It has gradients but the right hand side is zero: that's because the force is immediately rewritten as the time derivative of the momentum carried by other parts of the physical system.

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OP starts from an equation

$$ F_i ~=~ \frac{d p_i}{dt^j}, \qquad (1) $$

which he acknowledges is flawed, because eq. (1) has an unmatched index $j$. I assume that OP's real question is, if people have thought of theories with multiple 'times' $t^j$, where the interpretation of $t^j$ is part of the question. Here are some comments that may help OP.

1) Some theories where $t^j$ are basically spatial directions are discussed in Luboš Motl's answer.

2) Several authors (in particular Itzhak Bars) have written papers about two-time-physics.

3) Infinitely many 'times' appear in integrable systems.

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I was thinking along similar lines, but theories with multiple time dimensions are pretty far out as these things go. I suspect jak had a simpler misconception in mind. (Still, interesting information) –  David Z Dec 29 '11 at 20:12
    
I was actually going to refer to the spatial derivatives being a function of time, so something like $x = x(t)$,$y = y(t)$ and $z = z(t)$ –  jip Aug 4 '12 at 17:42

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