How can engineers get away with splitting differentials in dynamics?

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In summary, This discusses how splitting the differential can sometimes work, but it can also be poor math. Engineers get away with it by practicing juggling.
  • #1
Trying2Learn
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TL;DR Summary
splitting the differential
In an introductory dynamics textbook, we often see this progression

v = ds/dt ---> dt = ds/v

a = dv/dt ---> dt = dv/a

Equating the dt, we get: vdv=ads

Now my question

On the one hand, this works for certain problems.
On the other hand, this is splitting the differential.

Could someone please explain

Why it works under certain conditions? How engineers get away with this?

If it is poor math to do this: why? Is it because one should never split the differential?

How can engineers get away with this?

I see that it does work, but only in ONE dimension.

This whole issue has always bothered me but I cannot state, with clarity, conviction, precision:
Why it is poor math to do this
Why we can get away with it.
 
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  • #2
Physicists are usually pretty casual in dealing with differentials :smile: .
In the 'progression' you quote, the expressions at the left are vector equations; the ones on the right are scalar expressions.
Beginning physicists should be careful not to accidentally hop back and forth, or they risk overlooking a Jacobian and other useful mathematical goodies.

##\ ##
 
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  • #3
Trying2Learn said:
TL;DR Summary: splitting the differential

In an introductory dynamics textbook, we often see this progression

v = ds/dt ---> dt = ds/v

a = dv/dt ---> dt = dv/a

Equating the dt, we get: vdv=ads

Now my question

On the one hand, this works for certain problems.
On the other hand, this is splitting the differential.

Could someone please explain

Why it works under certain conditions? How engineers get away with this?

If it is poor math to do this: why? Is it because one should never split the differential?

How can engineers get away with this?

I see that it does work, but only in ONE dimension.

This whole issue has always bothered me but I cannot state, with clarity, conviction, precision:
Why it is poor math to do this
Why we can get away with it.
A formal proof of why this jugglery with "d"'s actually works is a nightmare because they are not even defined as solids at this level.
$$
\dfrac{ds}{dt}=\lim_{h \to 0}\dfrac{s(t+h)-s(t)}{h}
$$
Now, how would you isolate ##dt## here? I like to avoid such steps by using Weierstraß's formula: $$ s(t+h)= s(t)+ s'(t) \cdot h + o(h)$$ with a remainder ##o(h)## that is quadratic in ##h## so it vanishes fast as ##h## goes to zero. With that formula, Weierstraß has out all the limit stuff in the ##o(h)## term and we can work with them as there was no limit stuff.

You should be careful with
BvU said:
Physicists are usually pretty casual in dealing with differentials :smile: .
because: they have practiced juggling! It can go wrong!
 
  • #4
fresh_42 said:
It can go wrong!
Tell me something I don't know :smile:

Been there, done that.

##\ ##
 
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Related to How can engineers get away with splitting differentials in dynamics?

What does it mean to split differentials in dynamics?

Splitting differentials in dynamics typically refers to the practice of separating variables and their differentials in differential equations to simplify the process of solving these equations. This can involve manipulating the equations to isolate differentials on one side, making it easier to integrate and find solutions.

Is splitting differentials mathematically rigorous?

Splitting differentials is not always mathematically rigorous in the strictest sense, especially if done informally. However, it can be justified under certain conditions and is often used as a heuristic or intuitive method in engineering to simplify complex problems. The key is to ensure that the manipulations are consistent with the rules of calculus and the physical context of the problem.

Why is splitting differentials useful in engineering dynamics?

Splitting differentials is useful in engineering dynamics because it simplifies the process of solving differential equations, which are prevalent in modeling dynamic systems. By separating variables and differentials, engineers can often reduce complex equations to more manageable forms, making it easier to find solutions or approximate behaviors of systems.

Are there any risks associated with splitting differentials in dynamics?

Yes, there are risks associated with splitting differentials, primarily related to the potential for introducing errors or making unjustified assumptions. If the process is not done carefully, it can lead to incorrect solutions or misinterpretations of the system's behavior. Engineers must ensure that their manipulations are valid and that they understand the underlying assumptions and limitations.

How can engineers ensure the validity of splitting differentials in their calculations?

Engineers can ensure the validity of splitting differentials by thoroughly understanding the mathematical principles behind their manipulations and by cross-checking their results with other methods or known solutions. They should also be aware of the physical context of the problem and ensure that their assumptions are reasonable. Consulting more rigorous mathematical methods or software tools for verification can also help ensure accuracy.

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