# Why no higher derivative in physics?

• I
• Ksitov
In summary: Higher derivatives are needed to describe the behaviour of a function relatively far away from the current point, as the interaction between nearby points is weaker.In summary, Higher derivatives are needed to describe the behaviour of a function relatively far away from the current point.
Ksitov
Dear all,

I'm asking why there is no higher derivative than two in physics ? I never encountered a third (time or space) derivative in physics.

Thank you.

Regards.

Ksitov said:
I'm asking why there is no higher derivative than two in physics ?
Google for jerk, jounce, crackle and pop.

Why EDP are always in term of second derivatives ? (diffusive - convective - wave equations)

Even in General Relativity, Riemann curvative tensor is "the derivative" of the Christoffel symbols which is "the derivative" of the metric tensor => order 2 in term of spatial derivative.

Ksitov said:
I knew these quantities but I'm speaking about equations.
You can use these quantities in equations too.

Ksitov said:
I'm asking why there is no higher derivative than two in physics ?
You are asking the forbidden "why" question, for which there is never a proper answer. All we can say is that 'we find that' the vast majority of phenomena can be characterised in equations requiring just a second derivative. Variables such as Jerk etc. are part of Science but not needed for most purposes so you don't come across them too often.

Stephen Tashi, symbolipoint and cnh1995
The fun part is, that either the first and second derivative is considered (sometimes third, as in curve sketching) or it is required to have infinitely often differentiable functions. So implicitly the other derivatives are often used, as smooth functions play a big role in physics.

cnh1995
Third and higher derivatives of position are used in things like camshafts. In order for the valve to have the maximum area under the valve changes in acceleration have to be carefully managed. As an example when the cam goes past the maximum valve opening the acceleration changes sign.

Cheers

symbolipoint
Look for "wave equation in stiff string," "Korteweg-de Vries (KdV) equation," and similar.

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There are equations in the theory of elastic bending of plates where there's a double Laplacian operator ##(\nabla^2 )^2## acting on an unknown function - a fourth order PDE. Another example is the equation of motion of thin liquid films for low-Reynolds number flow.

The question is nevertheless reasonable. Derivatives beyond 2 are really rare in physics. I think ultimately the reason for this is Locality. The physics at (x,t) is determined mostly by events at (x+dx, t-dt). The further away stuff is, the weaker the interaction. But higher derivatives ultimately describe the behaviour of a function relatively far away from the current point.

Gigaz said:
But higher derivatives ultimately describe the behaviour of a function relatively far away from the current point.
This is not true. Differentiability is always a local property, no matter how often it is possible. "Relatively far away" is definitely a misinformation.

sophiecentaur said:
You are asking the forbidden "why" question, for which there is never a proper answer. All we can say is that 'we find that' the vast majority of phenomena can be characterised in equations requiring just a second derivative. Variables such as Jerk etc. are part of Science but not needed for most purposes so you don't come across them too often.
I don't recall seeing snap (jounce), crackle, and pop in servo drive manuals and parameter lists, but jerk is available on all modern servos, and (although often called "s-curve") in many induction motor VFDs, too. Jerk increases overall motion profile move time, but is worth the trade-off in applications where high positioning accuracy is required, and also helps prevent machinery from tearing itself apart where high inertia loads are involved.

There are plenty of situations, where higher derivatives appear. But maybe the scope of the question can be reduced to why there are usually no higher derivatives than the first in the Lagrangian? This will turn up quite a lot of results on Google.

Every car journey exhibits a third order time derivative as your accelerator pedal takes time from one setting to another.

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Telemachus
fresh_42 said:
This is not true. Differentiability is always a local property, no matter how often it is possible. "Relatively far away" is definitely a misinformation.

Yes, in math. But have you done numerical differention, perhaps even for solving partial differential equations? For higher derivatives at x0, you need to evaluate the function at points further away from x0. And ultimately, the infinitesimally small doesn't really exist in physics.

Gigaz said:
Yes, in math. But have you done numerical differention, perhaps even for solving partial differential equations? For higher derivatives at x0, you need to evaluate the function at points further away from x0. And ultimately, the infinitesimally small doesn't really exist in physics.
You are implicitly suggesting that there are two different concepts of differentiability in physics and in mathematics. This is also wrong! To cover one misinformation by the next one is nonsense.

You can have a perfect smooth function at a point which is completely wild and not even continuous "further away". To say you need points farther apart is a contradiction to the concept of a local phenomena and very misleading. E.g. in a often used Taylor expansion in physics, all derivatives are evaluated at the same point. It should not be told that "other points are needed", because it is simply wrong. And you do not need "infinitesimal small" or "dx". One only needs a small neighborhood of a point to have an approach, which normally does exist even in physics. This is already needed for the first derivative.

Telemachus
I have made simulations of the thin-film flow system that is described by a 4th order PDE, and the high order doesn't cause any significant trouble in the implicit finite differencing - the shape of the liquid surface approaches a paraboloid of revolution as it sets into static equilibrium (just as it physically should do to minimize surface energy).

Telemachus
fresh_42 said:
You are implicitly suggesting that there are two different concepts of differentiability in physics and in mathematics. This is also wrong! To cover one misinformation by the next one is nonsense.

You can have a perfect smooth function at a point which is completely wild and not even continuous "further away". To say you need points farther apart is a contradiction to the concept of a local phenomena and very misleading. E.g. in a often used Taylor expansion in physics, all derivatives are evaluated at the same point. It should not be told that "other points are needed", because it is simply wrong. And you do not need "infinitesimal small" or "dx". One only needs a small neighborhood of a point to have an approach, which normally does exist even in physics. This is already needed for the first derivative.

Yes, and if you want to have the second derivative, you'll have to use a larger neighborhood than what you took for the first derivative. Higher orders even more so. And as long as the taylor series converges, the derivatives will yield more information about the larger neighborhood of the point.

Let's say you're solving Navier-Stokes or pressure waves. Obviously, the derivatives are not "real derivatives" in a mathematical sense. Length scales below the size of an atom are unreasonable, then the equation doesn't apply anymore. The derivatives only occur in those equations because atoms are in fact pretty small.

First off, as previously mentioned, there are third and higher derivatives in physics. However, you are completely correct that they are rare-- jerk is occasionally used but beyond that I've never really seen a problem using snap (jounce), crackle or pop; perhaps their rare use is why they were named after a cereal slogan. I do not know exactly why they are generally trivial, but I do have a guess (of course, nobody can know exactly why something in physics exists).

I assume that the answer has something to do with energy. Energy underlies nevery concept in physics-- whether you are dealing with classical mechanics, quantum mechanics or relativity. Work (a concept extremely closely related to energy) is very easy to express in terms of force, a quantity associated with acceleration, the second derivative of position (W=F⋅d for a constant force or ∫F⋅dx for a varying force). Now, I could express this in terms of yank (Y, mass times jerk, the third derivative of position) as W=F0⋅d+Yt⋅d for a constant yank and force not varying wrt d, and in the general case W=∫∫Ydt⋅dx. I find the equation in terms of force much more appealing to use, don't you agree? In fact, you can also express kinematics for a constant jerk, but at that point I think I'd rather use calculus.

Gigaz said:
Yes, and if you want to have the second derivative, you'll have to use a larger neighborhood than what you took for the first derivative. Higher orders even more so. And as long as the taylor series converges, the derivatives will yield more information about the larger neighborhood of the point.
There is no minimum size for the required mathematical neighborhood. Any neighborhood of non-zero extent will do and can allow derivatives of all orders to be obtained (if said derivatives exist at all).

If you are interested in estimating a derivative using imperfectly accurate physical measurements then using a larger interval may be of some benefit. Is that the point that you are trying to make?

fresh_42 said:
This is also wrong! To cover one misinformation by the next one is nonsense.

As a general comment, I really really dislike these types of responses. They are the antithesis to learning, because they are beligerent in nature and don't even bother trying to understand the other side.

@Gigaz makes an interesting, and not entirely untruthful statement that in any approximation of a derivative, no matter how small your delta, the second derivative inevitably considers a neighborhood of 2*delta. You can use the limit of Delta going to zero, but that is evading the point, as the delta is never *actually* zero.

Isaac0427
rumborak said:
You can use the limit of Delta going to zero, but that is evading the point, as the delta is never *actually* zero.
If you are taking a limit, it is improper to talk about what the delta *actually* is. The delta is out-of-scope for any such comment since it is inside a quantifier.

It is not improper, since the delta is never zero. Infinitesimals are not zero, and that is the important point in the argument. @Gigaz has a perfectly valid point here.

rumborak said:
It is not improper, since the delta is never zero. Infinitesimals are not zero, and that is the important point in the argument. @Gigaz has a perfectly valid point here.
You are mistaken. In the traditional epsilon/delta definition of a a limit, the delta and the epsilon are inside quantifiers. They have no independent existence. It is a scope error to speak about their "actual" value.

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Stephen Tashi
rumborak said:
As a general comment, I really really dislike these types of responses. They are the antithesis to learning, because they are beligerent in nature and don't even bother trying to understand the other side.
Gigaz said:
But higher derivatives ultimately describe the behaviour of a function relatively far away from the current point.
fresh_42 said:
To say you need points farther apart is a contradiction to the concept of a local phenomena and very misleading.
To tell students, that the degree of differentiability takes you further away from the point in question, is a very bad way to look at it and ...
Gigaz said:
Yes, in math.
fresh_42 said:
... implicitly suggesting that there are two different concepts of differentiability in physics and in mathematics.
... makes it even worse. This is not what differentiability means.

Higher derivatives are not "rare" in physics in the sense that a whole bunch of physical phenomena that one encounters every day depend on behaviors described by higher derivatives. Some examples have already been given in the thread.

However, it is true that terms involving higher-order derivatives tend to get dropped whenever possible, and hence it's rarer to see them retained (or even acknowledged) in a problem. Hence, the real question is why can we often get away with dropping those terms? Here it may be valid to talk about ratios of terms, expressed in units of deltas, although it's probably getting into the weeds to argue about the effect of molecule sizes on higher-order derivatives in N-S...

hilbert2
olivermsun said:
Higher derivatives are not "rare" in physics in the sense that a whole bunch of physical phenomena that one encounters every day depend on behaviors described by higher derivatives. Some examples have already been given in the thread.

Yeah, for instance the streaks ("fingers") that rainwater forms when flowing down a window glass are described by the 4th order equation I mentioned earlier. Usually when there's this kind of higher order behavior somewhere, the differential equation is derived with an assumption that the solution function is a slowly varying function of position - therefore the function and its derivatives are not going to change a lot when moving a distance of 4 finite difference step lengths.

olivermsun
I think that the original question could be sharpened up a bit. It's not so much that higher derivatives are never used, but that the fundamental equations of physics don't involve them. Newton's equations of motion, Maxwell's equations, Einstein's field equations, Schrodinger's equation. In the case of Newtonian physics, this means that the state of a system, no matter how complicated, can be described by its trajectory in phase space, where phase space specifies the location and momentum of each particle. Since momentum involves velocity, the trajectory in phase space involves time derivatives of momentum, which means second derivatives of position. If there were higher-order derivatives involved, then you would need a different notion of phase space (where you would specify position, velocity and acceleration of each particle). Such a change would make a huge change to a lot of the formalism of physics (both classical and quantum). I don't know whether it's been studied much.

Of course, saying "physics would be so much harder if the fundamental laws involved third derivatives" isn't really a satisfying reason why. It's not like God designed the universe so that beginner physicists wouldn't be overwhelmed.

stevendaryl said:
If there were higher-order derivatives involved, then you would need a different notion of phase space (where you would specify position, velocity and acceleration of each particle). Such a change would make a huge change to a lot of the formalism of physics (both classical and quantum). I don't know whether it's been studied much.

Actually, it occurs to me that you can formally treat higher-order derivatives as first-order derivatives involving more complex objects.

For example, suppose you have a third-order equation of motion:

$\frac{d^3 x}{dt^3} = F(x, \dot{x}, \ddot{x})$

Then you make that into a first-order equation for a vector-valued object:

$\frac{d \vec{x}}{dt} = \vec{Q}(\vec{x})$

where $\vec{x} = \langle x, \dot{x}, \ddot{x} \rangle$

and $\vec{Q}(\vec{x}) = \langle \dot{x}, \ddot{x}, F(x, \dot{x}, \ddot{x}) \rangle$

There's probably some sense in which that's cheating, but I'm not sure formally how you would define cheating.

hilbert2 said:
I have made simulations of the thin-film flow system that is described by a 4th order PDE, and the high order doesn't cause any significant trouble in the implicit finite differencing - the shape of the liquid surface approaches a paraboloid of revolution as it sets into static equilibrium (just as it physically should do to minimize surface energy).
Another example in fluid mechanics is the biharmonic equation for the stream function in creeping flow past a sphere.

Ksitov said:
Dear all,

I'm asking why there is no higher derivative than two in physics ? I never encountered a third (time or space) derivative in physics.

Thank you.

Regards.

I would say the short answer is that the all phenomena can be represented as a perturbation upon a differential equation having constant coefficients. The solutions of such a differential equation map out to functions that are the exponential functions (i.e., of the natural logarithm base, e) of the input variable scaled by the values that are the root of the polynomial equation that corresponds to the original differential equation such that the order of the differential is the power of the polynomial term (this is standard material covered by a course in differential equations). Now, for a polynomial with real coefficients, the roots must either be real values or pairs of complex conjugate values, and those pairs when multiplied together yield a quadratic factor; hence whatever is modeled mathematically can be resolved down to a function in which is no more than quadratic in nature, and hence, all that is needed to describe it is the use of at most the 2nd differential.

The Blasius equation relates to boundary layer flow on along a semi-infinite plate

2f''' + f''f = 0

## 1. Why are higher derivatives not included in physics equations?

Higher derivatives are not included in physics equations because they are not necessary to accurately describe the behavior of physical systems. The laws of physics are based on fundamental principles such as conservation of energy and momentum, and these principles can be described using first and second derivatives. Including higher derivatives would only complicate the equations without adding any significant insight or predictive power.

## 2. Can higher derivatives be used to improve the accuracy of physics equations?

No, including higher derivatives does not necessarily lead to more accurate equations. In fact, using higher derivatives can often introduce errors and inconsistencies into the equations. This is because higher derivatives are more sensitive to small changes in initial conditions, making it difficult to accurately predict the behavior of a system over time.

## 3. Are there any situations where higher derivatives are necessary in physics?

There are some rare cases where higher derivatives may be necessary to accurately describe a physical system. For example, in quantum mechanics, higher derivatives are used to describe the behavior of particles at the atomic and subatomic level. However, for macroscopic systems, higher derivatives are not needed and can be safely ignored.

## 4. Why do some mathematical models in physics include higher derivatives?

Some mathematical models in physics may include higher derivatives as a way to simplify the equations or make them easier to solve. However, these models are often just approximations and do not accurately reflect the true behavior of the physical system. In general, it is best to stick to the fundamental principles and use first and second derivatives in physics equations.

## 5. Are there any theories that allow for higher derivatives in physics?

There are some theoretical frameworks, such as string theory, that allow for the possibility of higher derivatives in physics equations. However, these theories are still highly debated and have not been confirmed by experimental evidence. In the current understanding of physics, higher derivatives are not necessary and are not included in most theories and equations.

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