Are Newton's three laws of motion essentially correct

[Antonio Lao]

InvariantBrian,

No. But let me be more specific about it. In analytical mechanics, there are many ways that force is defined.

One definition is that force is the negative gradient of a scalar potential function.

$$F = -\vec{\nabla} V$$

This force depends only on the coordinates of the object to which the force is applied. It does not depends on the velocity of the object. And generalized forces can be defined as the derivative of the Lagrangian with respect to the generalized coordinates.

$$F_i = \frac{\partial L}{\partial q_i}$$

while the generalized momenta are given by

$$p_i = \frac{\partial L}{\partial \.{q}_i}$$

where $\.{q}_i$ are the generalized velocities.

The first leads to Newton's 2nd law of motion.

 

[InvariantBrian]

I have no problem with what your saying. My only point is that Newton's third law is a genenal expression of the conservation of momentum if one remembers that force is defined as the change in momentum.

Newton's third law is still correct today in that general sense. Applying to in a modern way may call for some modification of this very general but important idea..agreed?

 

[InvariantBrian]

I'm mean...i guess your saying that force is defined many ways...fine...
but force as defined in Newton's laws of motion logically entails that the meaning of the third law..is a general expression of the conservation of momentum.

if force is defined other ways fine.

my point is that the third law is correct as it was defined by Newton...
that's all.

I...appreciate the discussion...

 

[Antonio Lao]

InvariantBrian,

I agree with what you said. But in almost all experiments of modern quantum physics, it is the change in energy that can be detected not so much as the force. While in classical mechanics, forces are disguised in all forms of vibrations and oscillations (damped or undamped). Although, We can still find the remnant of a direct application of the 3rd law in the science of rocket propulsions. But these necessarily need another law of conservation, the law of conservation of angular momentum besides the law of energy conservation.

There are really three major important laws of conservation.

1. Energy
2. linear momentum ( this is the one you are mostly concerned with, i guess).
3. angular momentum

Note: As a physical model, the 3rd law is very useful for linear systems. But, unfortunately, reality is mostly nonlinear. I could even say that it is really chaotic. But, the good thing is that the powerful energy principle is ideally applicable to all systems (linear or nonlinear or chaotic).

Energy (E) is just the "product" of force (F) and distance (d).

$$E = Fd$$

Vector analysis will help you understand the various ways of making a "product" of two vector quantities such as force and distance and velocity and linear momentum and angular momentum, etc.

 

[InvariantBrian]

My point is that Newton's third law is neither linear nor agular. It is general.
It states that generally momentum is conserved. And generally speaking, that is correct.

The third law is very simply stated:

action=reaction

and if one bears in mind that force is defined is at the change in momentum.

it follows that the third law genenally expresses the conservation of momentum.

It is a principle that can be used to derive the more specific laws of linear and angular momentum.

As a general statement, it is correct...in so far as it is generally true that momentum is conserved.

Does that makes sense?

Newton's laws were general principles he used to derive equations. The three laws themselves are not equations. They are used to derived equations.

 

[Antonio Lao]

InvariantBrian,

Once you are faced with an engineering problem, then you will start to realize that Newton's 3rd law is not enough to model physical reality. Specially in situation where static forces are in equilibrium, there is no obvious momentum because all velocity components are zero and hence all linear momenta are zero. In this particular case,
one cannot apply Newton's 3rd law directly. But forces still exist and they depend only on the coordinates instead of velocities.

 

[InvariantBrian]

InvariantBrian,

No. But let me be more specific about it. In analytical mechanics, there are many ways that force is defined.

One definition is that force is the negative gradient of a scalar potential function.

$$F = -\vec{\nabla} V$$

This force depends only on the coordinates of the object to which the force is applied. It does not depends on the velocity of the object. And generalized forces can be defined as the derivative of the Lagrangian with respect to the generalized coordinates.

$$F_i = \frac{\partial L}{\partial q_i}$$

while the generalized momenta are given by

$$p_i = \frac{\partial L}{\partial \.{q}_i}$$

where $\.{q}_i$ are the generalized velocities.

The first leads to Newton's 2nd law of motion.

but that doesn't make Newton's second law wrong, force is
still always the change of momentum with respect to time in a system.
If that wasn't true once then that is the exception to Newton's second
law. In other words if we have moon orbiting a planet, then the gradient of
the potential is equal to the change of momentum with respect to time on
the moon.

 

[InvariantBrian]

InvariantBrian,

Once you are faced with an engineering problem, then you will start to realize that Newton's 3rd law is not enough to model physical reality. Specially in situation where static forces are in equilibrium, there is no obvious momentum because all velocity components are zero and hence all linear momenta are zero. In this particular case, one cannot apply Newton's 3rd law directly. But forces still exist and they depend only on the coordinates instead of velocities.

I agree. But Newton third law is a general principle that serves as a guide in solving particular engineering problems. As a guiding intuition regarding the general equilibrium of momentum, it is correct.

In fact it may be formulated this way...

The Third law in effect states the net force acting on a closed sysem = 0

 

[Antonio Lao]

but that doesn't make Newton's second law wrong, force is
still always the change of momentum with respect to time in a system.

Newton's 2nd is correct for all inertial forces wherever they exist in a system. But there are also other forces: gravitational forces, electrical forces, magnetic forces, frictional forces, weak forces, strong forces, electromagnetic forces, elastic forces,
molecular forces, atomic forces, van der Waals forces, and many other which do not depend on linear momenta.

The Third law in effect states the net force acting on a closed sysem = 0

But the momenta are zero although the forces are not. These are the forces that depended on coordinates instead of velocities. The key point to bear in mind is that there are two basic energies in mechanics: the potential and kinetic. The kinetic is derived based on the concept of work. This is the one that uses a force that is the change in momentum with respect to time in order to derive work integral. But if the work integral is zero then the system is conservative. This means the 3rd law is true.
There are also nonconservative systems to think about. For example the electromagnetic force, which depends on the concept of charge instead of mass as in all of Newton's laws.

 

[Antonio Lao]

InvariantBrian,

Bottomline is that in any given system, there are many forces of interaction between components. Whichever is the force that dominate the system, its law can be shown correct by experiments.

For the solar system, Newton's mechanics is correct because in this system, both the inertial force and the gravity force dominate.

For the atomic system, it is dominated by the electromagnetic force. There, Newton's forces are not quite applicable. They can be practically negligible in experiments.

For thermodynamic system, it is the heat energy that dominate the system although there are masses, the inertial forces derived from these masses can be neglected in formulating the kinetic theory of heat.

 

[InvariantBrian]

If the force is concervative in nature then that means that the
only thing that matters in regards to the work is the where you started
with regards to the center and how far from the potential you are when you ended. If it isn't, then the work on the system (if there is any) will depend on some other quantity. However, in ALL CASES ENERGY OF THE SYSTEM IS CONCERVED! In the case of the magnetic force (the electric is a concervative force), then the net work of the stystem is 0. However, Newton's third law still applies because the
derivative of the momentum with respect to time is a direvative of a
vector quantity, not a scaler.


For atomic systems the electric force dominates (although there
are things dealing with quantum mechanics) and yes, we still apply
Newton's Laws without issue. It is mathmatically NO DIFFERENT THAN A
PLANET (ignoring quantum mechanical effects such as spin and energy
splitting due to that, but then we don't deal with Newton's Laws in the
same way)

For a thermodynaic system the idea of "heat" is really simply that, an
idea. When we apply the kenetic theory of gasses we are forced to use
Newton's laws (ignoring quantum effects and doing it classically, of
course)

 

[Antonio Lao]

InvariantBrian,

Please check out the virial theorem and let me know why you think the potential is only half of the kinetic by using Newton's 3rd law? Thanks.

 

[InvariantBrian]

The viral theorm works with the kenetic energy being double the
potential for 1/r potentials only. The general expression of it is KE = PE/(1-n) where n is the power of the r. In fact, with a central potential Newton's third laws works great (gravity?!).

 

[Antonio Lao]

InvariantBrian,

I would say half not double. The reason for this is that given $\alpha$ and $\beta$ for what values will the following be true?

$$\frac{\alpha - \beta}{\alpha \beta} = 1$$

This is where $\alpha = 1$ is the potential energy and $\beta = 1/2$ is the kinetic energy. For higher positive values for both, the following must be graphed in the 1st quadrant.

$$\alpha = \frac{\beta}{1 - \beta}$$

and

$$\beta = \frac{\alpha}{1 + \alpha}$$

these graphs show that if one variable increases from 0 to infinity, the other asymptotically approaches unity. And if they are equal then they are both identically zero.

 

[Antonio Lao]

This general covariance of alpha and beta might be capable of explaining the factor of 1/2 in kinetic energy for the total energy of an isolated system.

$$E = \frac{1}{2} mv^2 + V$$

where V is the energy potential. In relativity, as v -> c, the energy is $mc^2$. The factor of 1/2 becomes unity.

 

[Antonio Lao]

But for all relativistic mechanics, it is the square of energy that can describe the linear momentum of light

$$E^2 = c^2 \mathbf{p}^2 + m^2 c^4$$

when the mass is zero, such as that for a photon, its energy becomes purely kinetic, then its momentum is

$$p =\frac{E}{c}$$

this implies that when mass is zero, the momentum turns from a vector into a scalar at the maximum speed of c. In other words, the photon has none of that directional properties. Its is motionless. So applying Newton's 1st law, the photon is either at rest or moving at a constant speed of c in an inertial frame of reference.

 

[InvariantBrian]

The virial theorem says that <T> = <V>/2, so the kinetic energy is
double the potential for a 1/r potentialv Also,we know that Newton's Third law works for gravity, which is a 1/r potential.

 

[InvariantBrian]

In order to do this RIGHT you need to use 4 vector analysis which does
indeed have directions to it. The E = p*c is the magnitude of the
momentum. What is right is E = sqrt(p^2*c^2) where p^2 is really the
inner product of the regular momentum (as a vector).

 

[Antonio Lao]

Thanks for the math. Gonna work on it some more till I am eligible for a PhD.