Are position, velocity, and acceleration total or partial derivatives, and when should we choose between them?

Question

I'm trying to understand the distinction between total and partial derivatives in the context of kinematics, and I feel confused about how to treat variables explicitly or implicitly as functions of each other.

Is velocity defined as the total or partial derivative of position with respect to time? In other words, is velocity always expressed as $$v = \frac{ds}{dt}$$ where $ s $ is the position as a function of time $t$, or are there cases where velocity involves partial derivatives?

Can velocity be expressed as a function of both time $ t $ and the distance $ s$ traveled along a curve, such that $$v = v(t, s)?$$ If so:

Is velocity in this case a partial derivative with respect to $ t $, or does it depend on the situation?

In general, when analyzing kinematics, how do we decide whether to treat velocity, acceleration, position, or time as dependent or independent variables? Is time always the independent variable, or can it sometimes be treated as dependent?

I'm struggling with the concepts of explicit and implicit dependence of variables. How do we decide when to use partial derivatives $\frac{\partial}{\partial t}$ versus total derivatives $\frac{d}{dt}$ in kinematics?

I’d appreciate it if someone could help clarify these ideas with examples or an intuitive explanation. I feel like understanding this distinction would help me better grasp when to apply the appropriate type of derivative in physics problems.

Answer 1

Is velocity defined as the total or partial derivative of position with respect to time?

Total derivatives occur when you, explicitly or implicitly, have a function of "really" only one variable.

In basic kinematics (as opposed to, say, continuum mechanics and fluid dynamics), we always follow individual particles. For such a single particle, it can be labelled with a position $r$ at a time $t$; that is, its position can be considered a function of time: $r(t)$.

The velocity of the particle is defined as the (vectorial) rate of change of its position, that is, $v(t) \equiv \frac{d r(t)}{dt} (t)$. Since position $r$ is function of only one variable, it is a total derivative.

How do we decide when to use partial derivatives versus total derivatives in kinematics?

To reiterate, total derivatives occur when you have a function, explicitly or implicitly, of one variable. For example, if you want to know the rate of change of $f(w, u(w), v(w))$ with respect to $w$, then this is implicitly a function which depends "only" on $w$. You want (using the chain rule) $\frac{df}{dw} = \frac{\partial f}{\partial w} + \frac{\partial f}{\partial u} \frac{du}{dw} + \frac{\partial f}{\partial v} \frac{dv}{dw}$. If this is a function which is instead $f(w, u, v)$, then if we want the rate of change with respect to $w$, this is a partial derivative.

In physics, you'll likely see partial derivatives come up when you're writing a position in coordinates, e.g. Cartesian $x,y,z$ or spherical $r, \phi, \theta$. You take partial derivatives when you need to calculate the rate of change in that particular coordinate direction, such as when calculating a gradient.

Can velocity be expressed as a function of both time $t$ and the distance $s$ traveled along a curve, such that $v=v(t,s)$? Is velocity in this case a partial derivative with respect to $t$, or does it depend on the situation?

You may indeed see such an expression where there is a velocity as a function of both time and position (not "distance along a curve"!). This occurs in continuum or fluid dynamics, and does not occur in basic kinematics.

The Lagrangian and Eulerian treatments of fluid dynamics treat the fluid flow differently. In the Eulerian specification, the measuring device and the frame of reference are both fixed. We view all the properties of the fluid as fields. That means we describe all the fluid properties as functions of position $r$ and time $t$: e.g. velocity, pressure, density are represented as functions $\mathbf{u}(\mathbf{r},t)$, $\mathbf{p}(\mathbf{r},t)$, $\mathbf{\rho}(\mathbf{r},t)$.

Note this description of velocity is sensible for fluid dynamics and not basic particle mechanics since we can define for a fluid a velocity at every point in space (that the fluid occupies), at a given instant in time. (And the velocity is also given at fixed points in space as time varies.)

On the other hand, in the Lagrangian specification, quantities are given for a particular moving particle at varying times (similar to basic kinematics): we follow individual fluid parcels through time. So in the Lagrangian description, the fluid parcels are labelled e.g. by $n$, and the properties such as velocity, pressure and density are written corresponding to each parcel: $\mathbf{u}_n (t)$, $\mathbf{p}_n(t)$, $\mathbf{\rho}_n(t)$.

To relate the Lagrangian to the Eulerian specification, for the velocity field, instead of labelling by $n$ we could label each fluid parcel by some time-independent vector field $\mathbf{x}_0$. (For example, $\mathbf{x}_0$ could be the position of the centre of mass of the parcels at some initial time $t_0$.) Then, some function $\mathbf{X}(\mathbf{x}_0, t)$ gives the position of the particle $\mathbf{x}_0$ at time $t$, and the Eulerian and Lagrangian specifications are related by $\mathbf{u}( \mathbf{X}(\mathbf{x}_0, t), t) = \frac{\partial \mathbf{X}}{\partial t} (\mathbf{x}_0, t) = \frac{d \mathbf{X}_n}{dt} (t)$.

Answer 2

This is a very broad question and it's hard to figure out an answer covering all possible situations. However, the concept of implicit and explicit variables are usually related the equations satisfied by the variables.

For example, in Newtonian mechanics, we have the second law, that could be written as

$$ {d p\over dt} = F $$

If we are thinking about a point particle, there is no reason to think that $p$ is a variable of position. Of course it changes depending where it was measured in the particle trajectory, but to write $p=p(x,y,z,t)$ would be equivalent to say it is a property of space (like a momentum field), and that is not the case for point particles.

In fluid dynamics, you'll find the material derivative, where you could find a term $\mathbf u(x,y,z,t)$ known as the flow velocity. This, in turn, is a field, interpreted as a property of the space (or the fluid).

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If you want to get more intuition about total and partial derivatives, I would recommend you to read more about these operations in mathematics, because they have precise geometrical meanings. Usually we operate with partial derivatives, because it generalize the most common properties of derivatives (like approximating finite differences, Newtonian coefficient, etc) of one variable functions. Total derivatives plays a role when someone needs to take into account the full variation of a function (like $df$) with respect with some variable.