Fundamental Equations of Classical Physics

[Xyius]

So I want to get all the Equations that describe classical Physics together because I think it would make a kick *** poster. Tell me if I am missing anything or if you don't think one should be on here.
[PLAIN]http://img694.imageshack.us/img694/9646/classicalequations.gif

 

[AuraCrystal]

Newton's law of gravity, wave equation and basic wave relations, and perhaps some stuff from fluid mechanics and optics

 

[Xyius]

Ah good point! I will have to look through my textbooks later to get all the stuff. Law of gravity is definitely going on there as well as the law of reflection and refraction.

 

[Philip Wood]

I think you should replace your inequality by an equation

dU = T dS - PdV

U, T, S, P, V are all functions of state.

Although establishing the equation usually proceeds via the notion of reversible cycles, we finish with an equation which tells us about the system itself, and not about any particular process, be it reversible or irreversible.

 

[Born2bwire]

For classical EM you should also include the Lorentz force in my opinion. The laws of refraction and reflection can be derived from Maxwell's equations. For Newton's laws, the force is equal to the time derivative of the momentum, not the product of mass and the derivative of velocity. The latter is not valid for situations where the velocity and mass of an object change with time (for example, a rocket that is gradually expelling its fuel).

 

[Xyius]

Excellent points! See, this is why I wanted to post it here first!

 

[cjl]

I would replace your F = ma with F = dp/dt. I think it makes a little more sense as the fundamental relation, since m is not always constant.

 

[Mordred]

http://api.viglink.com/api/click?fo...hysics/laws/&jsonp=vglnk_jsonp_13152008512151

this site will help

 

[cmos]

1) I would remove the Combined Law of Thermodynamics since it follows from the First Two.

2) Perhaps it's more a personal preference, but I would use the differential form of the Maxwell Equations. Of course, use the "microscopic equations," in which only fields to appear are the fundamental E and B fields.

3) Do add Newtonian gravitation as said above.

4) Do not add optical refraction/reflection or the wave equation, as these follow from the Maxwell equations. I would even argue against the Lorentz force for the same reason.

5) To me, it would seem more aesthetic if you reversed the overall order; i.e. Mechanics, Electromagnetism, then Thermodynamics.

 

[Philip Wood]

4) Do not add optical refraction/reflection or the wave equation, as these follow from the Maxwell equations. I would even argue against the Lorentz force for the same reason.

Maxwell's equations don't explicitly involve force, so I don't see how we can deduce the Lorentz force equation from them. Unless, that is, we assume a specific connection between E and B and force. In that case we are tacitly assuming the Lorentz force equation. I'm happy to be put right if I've missed something.

My own take is that I like to consider the Lorentz force equation as the defining equation for E and B. That raises the issue of whether or not defining equations ought to be on the list. What about dW = F.ds, for example?

 

[chrisbaird]

For classical electromagnetics, Maxwell's equations, the Lorentz force law, and Newton's laws are the complete set of fundamental equations, all others can be derived from these. The Lorentz force law is definitely not contained in Maxwell's equations because they say nothing about forces. I would argue that the Lorentz force law is more than a definition (even though historically it was viewed on as such). The electromagnetic fields exist on their own without forces on objects being present (as in free traveling em waves), and the electromagnetic forces can be calculated in certain situations without ever conjuring up fields. They are therefore physical entities independent of each other, not just handy definitions, and therefore the Lorentz force law is fundamental.

 

[Xyius]

I chose the integral form of Maxwells equations PURELY for aesthetic reasons. ;]
I don't think defining equations should go on the list as their meanings are derived from the fundamental equations. F.ds is just a term that appears when messing around with Newtons second law and it is just by convenience we define a name for it. So in essence, those defined equations are derived from the fundamental equations.

So this is what I have so far.
1. Maxwells Equations
2. Lorentz Force
3. Newtons Laws
4. First, Second and third law of thermodynamics, Omitting the combined law as that can be derived from these laws as cmos pointed out.
5. Newtons law of gravity.

That's a grand total of 11 Equations. Elegant indeed!

 

[cmos]

Maxwell's equations don't explicitly involve force, so I don't see how we can deduce the Lorentz force equation from them. Unless, that is, we assume a specific connection between E and B and force. In that case we are tacitly assuming the Lorentz force equation. I'm happy to be put right if I've missed something.

My own take is that I like to consider the Lorentz force equation as the defining equation for E and B. That raises the issue of whether or not defining equations ought to be on the list. What about dW = F.ds, for example?

I'll first mention Poynting's Theorem just because people are usually more comfortable with the fact that the Theorem follows from the Maxwell Equations. Well, Poynting's Theorem is simply the electrodynamic statement of the conservation of energy. So if you buy the fact that this Theorem follows from the Maxwell Equations, then it is not surprising that one may also derive an electrodynamic statement of the conservation of momentum. Such a statement yields the Lorentz force density $\textbf{f}=\rho\textbf{E}+\textbf{j}\times\textbf{B}$. Integrating f over a delta-function (i.e. point) source yields the traditional Lorentz force.

This is done in must physics-oriented textbooks. For a quick reference, check out p. 59ff. of the Thidé's free textbook (http://www.plasma.uu.se/CED/Book).

Your last two sentences, is an interesting thought. My first thought is that such definitions follow from the analysis. Hmm...

 

[cmos]

I chose the integral form of Maxwells equations PURELY for aesthetic reasons. ;]

But consider this:
It may be more pleasing to replace the field fluxes with E and B explicitly. This way, you explicitly show the time-dependent coupling that occurs between E and B; which, IMO, is the beauty of Maxwell's theory. Of course, if you were to do this, then the differential form of the equations would be way less cluttered. :tongue2:

Also, I feel that the differential forms make more sense locally. The presence of sources cause the fields to change at every point in space; time variance of those sources and of the fields themselves cause the fields to change at every point in space and time. If I think about it in terms of integrals, I'd have to form Gaussian surfaces and closed loops at every unit of an arbitrarily discretized space.

Furthermore, the Helmholtz Theorem states that, given boundary conditions, a vector field is uniquely determined if its divergence and curl are known everywhere. Again, impetus for the differential forms.

But now I'm just ranting... :zzz:

 

[atyy]

I would replace your F = ma with F = dp/dt. I think it makes a little more sense as the fundamental relation, since m is not always constant.

The other reason might be that F=dp/dt is good in special relativity. OTOH, if one includes Newtonian gravity and Maxwell's equations, then the principle of relativity is violated anyway.

BTW, Feynman does his list of the complete equations of classical physics somewhere in his lectures. I'd imagine volume II, but am not sure.

 

[AuraCrystal]

Oh you might want to include Lagrange's equation:

$\frac{\partial \mathcal{L}}{\partial q_{i}} - \frac{d}{dt} \frac{\partial \mathcal{L}} {\partial \dot{q_{i}}}$

I know it's equivalent to $\mathbf{F}=m \mathbf{a}$ and the definitions of energies, but it's something you might want to consider. =]

 

[Andrew Mason]

I think you should replace your inequality by an equation

dU = T dS - PdV

U, T, S, P, V are all functions of state.

Although establishing the equation usually proceeds via the notion of reversible cycles, we finish with an equation which tells us about the system itself, and not about any particular process, be it reversible or irreversible.

But the problem is that this equality is only true in reversible processes, and since there are no perfectly reversible processes, it is never true. So I am not sure why you would include something that is never true as a fundamental equation of physics.

AM

 

[Philip Wood]

Au contraire, the equation is true for irreversible as well as reversible processes, because U, T, S, P, V are functions of state.

Pippard (Elements of Classical Thermodynamics) explains this point as follows: "...for any change in a fluid
dU = q +w
and dU = TdS - PdV.
It is only for a reversible change, however, that q = TdS and w = -PdV; neither of these equalities hold for an irreversible change – q$\neq$TdS and w$\neq$-PdV, but dU = TdS - PdV is still valid; if q = TdS -$\epsilon$ then w = -PdV + $\epsilon$."

 

[Andrew Mason]

Au contraire, the equation is true for irreversible as well as reversible processes, because U, T, S, P, V are functions of state.

Pippard (Elements of Classical Thermodynamics) explains this point as follows: "...for any change in a fluid
dU = q +w
and dU = TdS - PdV.
It is only for a reversible change, however, that q = TdS and w = -PdV; neither of these equalities hold for an irreversible change – q$\neq$TdS and w$\neq$-PdV, but dU = TdS - PdV is still valid; if q = TdS -$\epsilon$ then w = -PdV + $\epsilon$."

Ok. So long as PdV is not the work done by the system, this relation holds. In other words, P is the internal pressure of the system and not (necessarily) the pressure that is producing the change in volume.

AM

 

[learypost]

I may be wrong, but if you assume all electric and magnetic fields are the results of charged particles and there motions, doesn't all of E&M really just boil down to Coulomb's Law? If you had an arbitrary arrangement of charged particles, I would think that Coulomb's Law and Newton's Laws would describe the entire history of the arrangement. I always thought the introduction of Maxwell's Equations was only necessary when considering continuous charge distributions (which don't actually exist).

Also, aren't the laws of Thermodynamics just a way of simplifying calculations of gases. I would imagine that Newton's Laws would really be enough to derive all of Thermodynamics (pressure, heat flow, etc. are all macroscopic properties which could in theory be derived from a complete analysis of every individual particle using Newton's Laws)

If the above is true, then all of Classical Mechanics comes down to Newton's Laws of Motion and Gravitation and Coulomb's Law.

 

[Andrew Mason]

I may be wrong, but if you assume all electric and magnetic fields are the results of charged particles and there motions, doesn't all of E&M really just boil down to Coulomb's Law? If you had an arbitrary arrangement of charged particles, I would think that Coulomb's Law and Newton's Laws would describe the entire history of the arrangement. I always thought the introduction of Maxwell's Equations was only necessary when considering continuous charge distributions (which don't actually exist).

I think you need Maxwell's equations. Otherwise, how do you handle electromagnetic radiation or magnetic fields?

Also, aren't the laws of Thermodynamics just a way of simplifying calculations of gases. I would imagine that Newton's Laws would really be enough to derive all of Thermodynamics (pressure, heat flow, etc. are all macroscopic properties which could in theory be derived from a complete analysis of every individual particle using Newton's Laws)

It was the failure of classical laws to explain thermodynamic phenomena that led to the development of quantum physics. So I don't think you could say that Newton's laws explain all of thermodynamics.

Newton's laws are sufficient to explain the kinetic theory of gases to a point. They do not explain heat capacities of diatomic and polyatomic gases, for example.

AM

 

[Dr_Pill]

Kinetic energy? mv²/2

Conservative Force as the negative gradient of potential energy?

F_x = - dU/dx

Hooke's Law for a spring :)


Perhaps some angular momentum stuff?

L = r X p

 

[jewbinson]

The wave equation? It seems pretty important for fluid dynamics and EM (for classical physics of course)

 

[cmos]

I may be wrong, but if you assume all electric and magnetic fields are the results of charged particles and there motions, doesn't all of E&M really just boil down to Coulomb's Law? If you had an arbitrary arrangement of charged particles, I would think that Coulomb's Law and Newton's Laws would describe the entire history of the arrangement. I always thought the introduction of Maxwell's Equations was only necessary when considering continuous charge distributions (which don't actually exist).

I think you need Maxwell's equations. Otherwise, how do you handle electromagnetic radiation or magnetic fields?
It was the failure of classical laws to explain thermodynamic phenomena that led to the development of quantum physics. So I don't think you could say that Newton's laws explain all of thermodynamics.

I agree with Andrew, but I just wanted to add that you can indeed formulate a theory of charged-particle dynamics as an action-at-a-distance theory by considering only the combination of Coulomb's Law, Biot & Savart's Law, and the Lorentz Force as the fundamental equations (cf. learypost only mentioning Coulomb's Law), in addition to Newton.

The problem, as Andrew brought up, is that this does not lead to the notion of EM radiation as a phenomenon to be separately studied in its own right. Indeed, it was Maxwell's serious belief of the EM field as a tangible physical entity that led to the unification of optics with electromagnetism.