Lacking intuition with partial derivatives

[Chenkel]

TL;DR Summary

I am having trouble understanding partial derivatives in the context of Euler-Lagrange equations and Lagrangian mechanics. My usual intuition for derivatives, based on changes over time, is not working in this case and I am struggling to understand changes in relation to position and velocity. I am reaching out for help to understand the correct physical interpretation of partial derivatives.

Hello everyone,

I seem to be majorly lacking in regards to intuition with partial derivatives. I was studying the Euler-Lagrange equations and realized that my normal intuition with derivatives seems to lead me to contradictory or non sensical interpretations when reading partial derivatives.

Let me give an example to clarify with a really simple illustration of my misunderstanding.

I'm studying this article regarding Lagrangian mechanics, and understood most things regarding stationary action principle, and I also realize that the stationary action principle is satisfied if the Euler-Lagrangian equation is satisfied.

However, when I try to intuitively understand what the Euler-Lagrange equations mean in terms of rates of change, I start thinking "how does potential energy change wrt position" and "how does kinetic energy change wrt velocity" but I'm used to only thinking how things change wrt time, and the idea of something changing with wrt position or velocity doesn't make a huge amount of sense to me.

I'm having a little difficulty posting latex in the initial post on my phone, so I will post a follow up post that will give a practical illustration of part of my misunderstanding.

If anyone can help me with the correct physical interpretation of partial derivatives I will be incredibly grateful.

Looking forward to your feedback, thank you!

 

[Dale]

the idea of something changing with wrt position … doesn't make a huge amount of sense to me.

Have you never been in a room and found that it was cold in one part of the room and warm in another?

 

[A.T.]

... the idea of something changing with wrt position or velocity doesn't make a huge amount of sense to me.

Take a bike ride through the hills. The elevation changes with position and the wind in your face changes with velocity.

 

[Chenkel]

I think I may have had a misunderstanding based on some rusty interpretations of the math, and perhaps my intuition regarding partial derivatives isn't as deeply flawed as I first thought, but I think I could still use some clarification possibly if anyone would like to share their insights.

Let me explain my logic of how I understand the Euler-Lagrange equation and see where my interpretation can be improved.

In the article I see the following equation:$${\frac {d} {dt}}{\frac {\partial T} {\partial \dot x}}=-{\frac {\partial V}{\partial x}}$$The way I interpret this is described as follows: I view V as a function of position, and I interpret T as a function of velocity.

When I see ${\partial \dot x}$ I interpret it as a change in velocity, so according to my logic ${\partial \dot x}$ has units of meters per second and is the change that occurs to velocity over dt seconds.

When you multiply dt by ${\partial \dot x}$ in the denominator on the left side for the Euler-Lagrange equations you get meters in the denominator, so the left side is in kinetic energy per meter, and the right side is in potential energy per meter.

Perhaps I'm question my logic in treating partials as differentials, according to my interpretation of infinitesimals, it doesn't seem necessarily like flawed logic to treat ${\partial \dot x}$ as a distinct infinitesimal quantity at time t when considering an infinitesimal time dt.

In summary I feel like I have a general feel for the equation, but I am questioning my logic.

Please feel free to respond if you can shed some light on the matter, thank you!

 

[Chenkel]

Have you never been in a room and found that it was cold in one part of the room and warm in another?

That does make some sense to me, I'm guessing that would be the partial derivative of temperature wrt to position?

But doesn't it all come back to time? Because (according to my logic) position can only change if time changes.

 

[Chenkel]

I accidentally posted a link to a different site instead of the article on Lagrangian mechanics, but I just corrected that, my apologies for the inconvenience.

 

[Chenkel]

When you multiply dt by ∂x˙ in the denominator on the left side for the Euler-Lagrange equations you get meters in the denominator, so the left side is in kinetic energy per meter, and the right side is in potential energy per meter.

Perhaps I didn't state that as well as I could, based on my interpretation, both sides are in energy per meter, but the equation only holds true (I believe) if the negative change in potential energy is equal to the change in kinetic energy for any infinitesimal amount of time dt, and I believe that is what the equation shows, so more or less I see it as a statement of conservation of energy.

 

[Mark44]

In the article I see the following equation: $${\frac {d} {dt}}{\frac {\partial T} {\partial \dot x}}=-{\frac {\partial V}{\partial x}}$$The way I interpret this is described as follows: I view V as a function of position, and I interpret T as a function of velocity.

That's only part of the story. Since the equation involves partial derivatives, it's implied that both T and V are functions of more than one variable. So both T and V are functions of both position and velocity.

I start thinking "how does potential energy change wrt position" and "how does kinetic energy change wrt velocity" but I'm used to only thinking how things change wrt time, and the idea of something changing with wrt position or velocity doesn't make a huge amount of sense to me.

For the first, consider how a heavy weight's PE increases the higher it is raised, or how a spring's PE increases as one end is compressed toward the other.
For the second, consider how heating a quantity of some gas causes the gas molecules to move more rapidly, thereby increasing the KE of the gas. Neither example depends directly on time t.

Rather than thinking a derivative (or partial derivative) is a rate of change with respect to time, think about it as a rate of change of something with respect to some other attribute.

I'm not sure that thinking about things in terms of infinitesimals is all that helpful here if you're uncertain about what derivatives can represent.

 

[Chenkel]

That's only part of the story. Since the equation involves partial derivatives, it's implied that both T and V are functions of more than one variable. So both T and V are functions of both position and velocity.

Thanks again for all the info.

One thing I'm wondering is how T (which I believe is the kinetic energy) can be a function of not only velocity, but also position; and how V which is potential energy is not only a function of position but also a function of velocity.

That seems to contradict my interpretation of what I thought I learned, but perhaps I'm not looking at the problem using the proper lense.

 

[kuruman]

If you want a feeling of what $${\frac {d} {dt}}{\frac {\partial T} {\partial \dot x}}=-{\frac {\partial V}{\partial x}}$$ is an expression of, see if you can reduce what you see to familiar forms from introductory physics. The right-hand side is the x-component of a force derived from a potential $V$ which could be a function of many variables. The left hand side can also be transformed to something familiar for a particle that moves in three dimensions $${\frac {d} {dt}}\frac{\partial T} {\partial \dot x}={\frac {d} {dt}}\frac{\partial} {\partial \dot x}\left[\frac{1}{2}m(\dot x^2+\dot y^2+\dot z^2)\right]={\frac {d} {dt}}(m\dot x)=\frac {dp_x} {dt}.$$So the equation that you wrote down becomes $$\frac {dp_x} {dt}=F_x.$$This should look familiar and is clearly not a statement of energy conservation. What is it a statement of? I hope this helps your intuition a bit.

 

[Chenkel]

This should look familiar and is clearly not a statement of energy conservation. What is it a statement of? I hope this helps your intuition a bit.

I'm going to say that must be f = ma

So the Euler Lagrange equation must imply f = ma in terms of the Lagrangian (if I'm seeing things correctly.)

 

[Chenkel]

If you want a feeling of what $${\frac {d} {dt}}{\frac {\partial T} {\partial \dot x}}=-{\frac {\partial V}{\partial x}}$$ is an expression of, see if you can reduce what you see to familiar forms from introductory physics. The right-hand side is the x-component of a force derived from a potential $V$ which could be a function of many variables. The left hand side can also be transformed to something familiar for a particle that moves in three dimensions $${\frac {d} {dt}}\frac{\partial T} {\partial \dot x}={\frac {d} {dt}}\frac{\partial} {\partial \dot x}\left[\frac{1}{2}m(\dot x^2+\dot y^2+\dot z^2)\right]={\frac {d} {dt}}(m\dot x)=\frac {dp_x} {dt}.$$So the equation that you wrote down becomes $$\frac {dp_x} {dt}=F_x.$$This should look familiar and is clearly not a statement of energy conservation. What is it a statement of? I hope this helps your intuition a bit.

Also thank you for all the math equations, very helpful!

 

[A.T.]

But doesn't it all come back to time? Because (according to my logic) position can only change if time changes.

You can measure something at different positions at the same time.

 

[Chenkel]

You can measure something at different positions at the same time.

Would that require special relativity? I'm wondering how that would work.

 

[kuruman]

Surveyors do it all the time. It works because they move at non-relativistic speeds.

 

[Chenkel]

Surveyors do it all the time. It works because they move at non-relativistic speeds.

I don't understand how multiple distinct positions values of a singular "thing" can be measured at the same time in the same coordinate system.

I suppose one possible way would be to have the same "thing" but measure position in different coordinate systems. Is that what you are suggesting?

 

[Dale]

But doesn't it all come back to time? Because (according to my logic) position can only change if time changes.

Why does time need to change for it to be warm on one side of the room and cold on the other side of the room? Do you somehow think that physics only happens in one place at a time?

 

[Chenkel]

Why does time need to change for it to be warm on one side of the room and cold on the other side of the room? Do you somehow think that physics only happens in one place at a time?

I agree that it can definitely be a different temperature in one side of the room vs the other side.

I think what you are saying (if I'm understanding correctly) is that you can take a snapshot of the physical model at a particular time and sample the rate of change of temperature wrt to changes in position, that does make sense to me.

 

[Dale]

I think what you are saying (if I'm understanding correctly) is that you can take a snapshot of the physical model at a particular time and sample the rate of change of temperature wrt to changes in position, that does make sense to me.

Yes, that is what a partial derivative means. Hold everything else constant and just look at the change with respect to variations in that one parameter.