Newton’s Laws and Euler’s laws

[member 529879]

I was looking at the Wikipedia page for Newton’s laws and read that they applied only to point particles. If we want to use F = ma on a larger object, if I read correctly, we must instead use Euler’s laws.

I have a few questions about this.

If this is correct why are we taught to use Newton’s laws to find the motion of rigid bodies? Wouldn’t we actually be using Euler’s laws?

How exactly did Newton come to the conclusion that point masses were subject to his laws? It’s not like he could just test them with point particles. I would have thought that that Euler’s f = ma(cm) would have been the more useful and simpler to verify law.

 

[BvU]

Hi,

the lemma is clear enough:

This can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star.

 

[member 529879]

Hi,

the lemma is clear enough:

I don’t quite understand why something like a planet can be idealized as a point particle without Euler’s laws. The motion of a planet can be described by f = ma because eulers law says that all bodies can be described by f = ma(cm), but how do we know that a planet can be idealized as a point particle without reference to Euler?

 

[BvU]

how do we know that a planet can be idealized as a point particle

As long as we accept the position of the center of mass of the planet (usually the center of the 'sphere') as a good enough description of the position of the planet as a whole

And that goes a long way.

 

[jbriggs444]

I don’t quite understand why something like a planet can be idealized as a point particle without Euler’s laws. The motion of a planet can be described by f = ma because eulers law says that all bodies can be described by f = ma(cm), but how do we know that a planet can be idealized as a point particle without reference to Euler?

If you were to calculate the angular momentum of a planet rotating about its polar axis with that of the planet's center of mass revolving around a star you should quickly notice that the quantities differ by several orders of magnitude.

Edit: In addition, they couple poorly -- there is not much torque causing the planet to dump its angular momentum of rotation into angular momentum of revolution.

 

[DrStupid]

I was looking at the Wikipedia page for Newton’s laws and read that they applied only to point particles.

That's the case for the 1st law but not the 2nd and 3rd law which even apply to open systems.

If we want to use F = ma on a larger object, if I read correctly, we must instead use Euler’s laws.

That's correct, because acceleration is not well defined for a larger object. Different parts of the same rotating object have different accelerations. But the first law actually is F=dp/dt and that can be used for larger objects as well as for the derivation of Euler's laws.

How exactly did Newton come to the conclusion that point masses were subject to his laws?

Before we discuss how he came to this conclusion we should check if he came to this conclusion.

 

[member 529879]

That's the case for the 1st law but not the 2nd and 3rd law which even apply to open systems.
That's correct, because acceleration is not well defined for a larger object. Different parts of the same rotating object have different accelerations. But the first law actually is F=dp/dt and that can be used for larger objects as well as for the derivation of Euler's laws.
Before we discuss how he came to this conclusion we should check if he came to this conclusion.

Was that not the conclusion he came to? Everywhere I look, it says Newton’s laws describe point particles, and eulers laws extends Newton’s laws to extended objects.

 

[DrStupid]

Was that not the conclusion he came to?

Do you have a corresponding references? In Lex 1 & 3 he is talking about bodies and in Lex 2 he dosn't make any restrictions at all. Of course "body" includes a point mass as a special case, but I'm not aware of any sources where Newton mentioned it.

Everywhere I look, it says Newton’s laws describe point particles, and eulers laws extends Newton’s laws to extended objects.

Newton's laws apply to point masses but at least Lex 2 & 3 are not limited to them. I don't see in which way the are extended by Euler's laws. I would reather say that Euler's laws are a special case for closed systems.

 

[member 529879]

Do you have a corresponding references? In Lex 1 & 3 he is talking about bodies and in Lex 2 he dosn't make any restrictions at all. Of course "body" includes a point mass as a special case, but I'm not aware of any sources where Newton mentioned it.
“Newton's laws apply to point masses but at least Lex 2 & 3 are not limited to them. I don't see in which way the are extended by Euler's laws. I would reather say that Euler's laws are a special case for closed systems.

From Wikipedia: “In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion”

I’m seeing stuff like this everywhere. Not directly from Newton. Is this information incorrect, or am I just misinterpreting it?

Also, if Newton’s laws already apply to all bodies, what exactly did Eulers law accomplish?

 

[A.T.]

...what exactly did Eulers law accomplish?

Confuse you?

 

[Drakkith]

If this is correct why are we taught to use Newton’s laws to find the motion of rigid bodies? Wouldn’t we actually be using Euler’s laws?

From what I've read, you're using both. Euler took Newton's laws and put them in the mathematical form that we use today. But since this was just a reformulation of Newton's existing laws, Euler's equations are usually called Newton's laws. Give this article a look: http://www.icmp.lviv.ua/journal/zbirnyk.41/001/art01.pdf

 

[DrStupid]

Is this information incorrect, or am I just misinterpreting it?

In my opinion it is incorrect. I do not see any reason why Newton's laws of motion in their original form should be not adequate to characterise the motion of rigid bodies and deformable bodies. There is a problem with the first law which needs to be limited to non-rotating rigid bodies in order to get a well defined state of rest or uniform motion. However, this law is not required to characterise the motion of bodies. The second and third law are sufficient and they are perfectly valid for any kind of system in classical mechanics, including rigid and deformable bodies.

Also, if Newton’s laws already applied to all bodies, what exactly did Eulers law accomplish?

They are easier to use. Newton's laws of motion in their original form describe the momentum of bodies and how it changes. You need an additional step to turn it into velocity and acceleration. This step is already included in Euler's law.

 

[member 529879]

In my opinion it is incorrect. I do not see any reason why Newton's laws of motion in their original form should be not adequate to characterise the motion of rigid bodies and deformable bodies. There is a problem with the first law which needs to be limited to non-rotating rigid bodies in order to get a well defined state of rest or uniform motion. However, this law is not required to characterise the motion of bodies. The second and third law are sufficient and they are perfectly valid for any kind of system in classical mechanics, including rigid and deformable bodies.
They are easier to use. Newton's laws of motion in their original form describe the momentum of bodies and how it changes. You need an additional step to turn it into velocity and acceleration. This step is already included in Euler's law.

How does Newton’s second law apply to deformable bodies. After reading through some of the Principia, I see that it does not specify point particles, but it also isn’t clear what the velocity should be in a case where the velocity of all parts of an object are not the same.

 

[DrStupid]

How does Newton’s second law apply to deformable bodies. After reading through some of the Principia, I see that it does not specify point particles, but it also isn’t clear what the velocity should be in a case where the velocity of all parts of an object are not the same.

There is no velocity in the second law.

 

[member 529879]

There is no velocity in the second law.

But there is momentum which is equal to mv. For the second law to be useful for non rigid bodies do you not need a well defined change in velocity over time or acceleration? If different parts of the object have different accelerations then how do you go from the momentum of the object to its acceleration?

 

[DrStupid]

But there is momentum which is equal to mv.

According to the explanation following this definition the momentum of a body is equal to the sum of the momentums of its parts:

p = \sum\limits_i {m_i v_i }

That's all you need to handle larger objects with Newton's laws of motion.

For the second law to be useful for non rigid bodies do you not need a well defined change in velocity over time or acceleration?

No, you don’t. The second law also applies if the parts have different accelerations:

F = \dot p = \sum\limits_i {m_i \cdot a_i }

If different parts of the object have different accelerations then how do you go from the momentum of the object to its acceleration?

Just use Newton’s definition of momentum to define a total velocity for the object

v = \frac{p}{m} = \frac{1}{m}\sum\limits_i {m_i v_i }

This is the velocity of the center of mass and the second law gives you the corresponding acceleration.

 

[member 529879]

v = \frac{p}{m} = \frac{1}{m}\sum\limits_i {m_i v_i }

This is the velocity of the center of mass and the second law gives you the corresponding acceleration.

I’m still a bit confused about this part. I don’t see where it comes directly from Newton. I couldn’t find anything from the principa about the center of mass in this context. And if v in the equation represent the velocity of the center of mass, it seems like you’re basically using euler’s first law, p = mv(cm).

 

[DrStupid]

I don’t see where it comes directly from Newton.

Here you go: http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/26

Def. II:

“Quantitas motus est mensura ejusdem orta ex Velocitate et quantitate Materiæ conjunctim.”

= "The quantity of motion [that’s Newton’s term for momentum] results from measuring velocity and quantity of matter [that’s Newton’s term for mass] together."

That means in modern notation

p: = m \cdot v

“Motus totius est summa motuum in partibus singulis, […]”

= "The total motion is the sum of the motions of the individual parts,"

That means

p = \sum\limits_i {p_i } = \sum\limits_i {m_i v_i }

Combining both equations and solving for v results in

v = \frac{1}{m}\sum\limits_i {m_i v_i }

I couldn’t find anything from the principa about the center of mass in this context.

The center of mass is given by

r = \frac{1}{m}\sum\limits_i {m_i r_i }

Differentiation with respect to time results in the velocity of the center of mass

v = \frac{1}{m}\sum\limits_i {m_i v_i }

This is the velocity as used in Newton’s definition of momentum (see above).

And if v in the equation represent the velocity of the center of mass, it seems like you’re basically using euler’s first law, p = mv(cm).

It seems so because Euler’s first law is identical with Newton’s definition II.