# What the equal sign means in physics

• B

For example F=ma means that the definition of force is m*a or the quantity of left side equals to the quantity of right side or both ? or kinetic energy..we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it ?

For example F=ma means that the definition of force...
No, but you can use it as definition (because of the equality) for the force needed to accelerate a mass, or to stop one. However, this might not help you to understand force, as it only applies to this one case. What is the force needed to stretch a spring? Or the force between two magnets? But according to Newton's laws of motion, by which force is defined by a change of the velocity of a mass, this definition will do in may cases.
... is m*a or the quantity of left side equals to the quantity of right side or both ? ...
Both, since equality is symmetric.
... or kinetic energy..we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it ?
It is a dependence between kinetic energy and velocity found in experiments (or by theoretical deduction). It is one way to calculate kinetic energy in the special case of a moved mass. It's the same as with the force: just a certain case.

That's kind of a philosophical question so it's tricky. I'm actually reading the famous Feynman's Lecture on Physics and he too writes about this. I'll try to explain what I understood (hoping it will be correct).

F = ma is not a definition, it is a law. The difference is that a law allows us to make predictions about the results of experiments using mathematics; definitions can't do that. A definition of a force can be something like this: "A force is something you have to apply in order to keep a spring (or a wire) strecthed". An other definition may be: "A force is something you have to apply in a certain manner to keep an object moving of circular motion."

(These are some poor definition of a force so I apologize to all the physicist here! :-D)

You see, that definition are vague things in physics, because if you try to get too rigorous you will be stuck with philosophy. In physics definitions assure only that we are talking roughly about the same thing so that we can understand each other.

We need not to dive into philosophy here.

F=ma says that the thing on the left is the same as the thing on the right. If the force (F) is 12 Newtons at 30 degrees above the horizontal then the equation states that ma will also evaluate to 12 Newtons at 30 degrees above the horizontal.

This equation applies in a certain circumstance. The circumstance is that F denotes the total of all forces on an object, m denotes the mass of that object and a denotes the acceleration of that object.

• Dale
what about kinetic energy and the formula 1/2mu^2 ? we define it as this product ? i am getting a little bit frustrated because the equal sign..from a math point of view every equality should be a definition right ?

what about kinetic energy and the formula 1/2mu^2 ? we define it as this product ? i am getting a little bit frustrated because the equal sign..from a math point of view every equality should be a definition right ?
No. ##(a^2-b^2)=(a+b)(a-b)## is not a definition.

thx for the answer because in one book it said definition of kinetic energy and then K=1/2mu^2

Wikipedia says:
"Energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object."
In this sense, kinetic energy is the energy, which applies work to a body to make it move. Thus an acceleration is involved and we can try to deduce the formula from what we know about acceleration and work. Work will lead us to force, and so we can find the formula for kinetic energy. Thus all these quantities depend on each other in a way. One could ask for the most elementary qualities and end up, e.g. with the MKSA system: meter, kilogram, second, Ampère. Another way is to end up with all fundamental constants in our universe. But whatever we do, we start with a description of a physical quality which we then try to define in a way, which makes it possible to calculate also quantities. Those definition can vary from case to case. E.g. it makes sense to define energy by ##\frac{1}{2}mv^2## if we deal with cars in motion, but not much sense to express it by ##\frac{1}{2}CU^2## in this case.

For example F=ma means that the definition of force is m*a or the quantity of left side equals to the quantity of right side or both ?

Equations, by themselves, don't have particular meanings.

There is a subtle distinction between "force" and "units of force". Statements about forces or units of force are made using words. Equations may be used to abbreviate the words, but written by themselves, equations are ambiguous.

Some equations are appear so often in one particular context, that its common to see the equation presented without the words that explain the context. It is hoped that experts will know the appropriate context just by seeing the equation. For example, @jbriggs444 gave the words describing the usual context for F = MA.

The equation F = MA is also handy as an aide to remembering the definition of a Newton as a unit of force, but you have to remember the context that M is a kilogram mass and A is 1 meter per second per second.

This is far from a dumb question. The equal sign can be quite misleading.

For example, consider Maxwell's famous equation ∇ x E = -∂B/∂t. What does it say?
It says that if I have -∂B/∂t I must have ∇ x E = -∂B/∂t.
It does NOT say that if I have ∇ x E I must have -∂B/∂t.
There are many ways to get ∇ x E. Another way is a battery of emf ##\mathcal E## connected to a resistor R forming an enclosed plane. The circulation of E (due entirely to the emf produced by the battery) = ##\mathcal E## = iR with i the current. So by Stokes' theorem, which is entirely a mathematical relation, ∇ x E integrated over the plane = iR also, and so ∇ x E must be non-zero on average over the plane.

In general, any source of emf will result in a finite ∇ x E, of which electromagnetic induction is just one example. Maxwell wrote his equations as the basis of his hypothesis of electromagnetic waves. He assumed no additional sources of emf.

Analogy: apple = fruit always holds. Does that mean fruit = apple always holds?

Comments welcome! I'm aware there is some controversy here. I can learn just like everybody else, and I have - plenty of times in this forum.

This is far from a dumb question. The equal sign can be quite misleading.

For example, consider Maxwell's famous equation ∇ x E = -∂B/∂t. What does it say?
It says that if I have -∂B/∂t I must have ∇ x E = -∂B/∂t.
It does NOT say that if I have ∇ x E I must have -∂B/∂t.
There are many ways to get ∇ x E. Another way is a battery of emf ##\mathcal E## connected to a resistor R forming an enclosed plane. The circulation of E (due entirely to the emf produced by the battery) = ##\mathcal E## = iR with i the current. So by Stokes' theorem, which is entirely a mathematical relation, ∇ x E integrated over the plane = iR also, and so ∇ x E must be non-zero on average over the plane.

In general, any source of emf will result in a finite ∇ x E, of which electromagnetic induction is just one example. Maxwell wrote his equations as the basis of his hypothesis of electromagnetic waves. He assumed no additional sources of emf.

Analogy: apple = fruit always holds. Does that mean fruit = apple always holds?

Comments welcome! I'm aware there is some controversy here. I can learn just like everybody else, and I have - plenty of times in this forum.
Yes there is something controversial here. It is simply not true, that the equality sign has a direction as in an assignment like in programming code, which you seemed to interpret. This is wrong. An equality sign means something else as it does in codes.

The sign "=" means especially:
\begin{align}
\text{reflexivity: }&A = A \\
\text{symmetry: }&A=B \Longrightarrow B=A \\
\text{transitivity: }&A=B \wedge B=C \Longrightarrow A=C
\end{align}
and this holds in physics, too. If two physical quantities are equal, as in your examples, then this means: there is a setup in which those quantities can be considered equal; with all it's implications. It does not mean, the equality would make sense in all thinkable setups. The molecules in this room all are in motion and have mass. Thus they have a kinetic energy. But it doesn't make sense to compute this energy by ##\frac{1}{2}mv^2##. Nevertheless, it remains true for a single molecule at a certain time.

So the main difference between code, mathematics and physics are:
• code = stands for an assignment as ##k=k+1##
• math = stands for quantities which are considered equal as ##1=\frac{2}{2}##
• physics = stands for quantities which are considered equal in a certain setup; it simply doesn't apply in other setups.

• QuantumQuest and jbriggs444
If a battery is connected to a wire, and there is a current, the total circulation in the circuit IS zero. Note that the battery is pushing the current AGAINST thge electric field, inside the battery.

In fact, any electrostatic field (independent of time) is such that ∇ X E is zero. Maxwell equations don't just mean that they are true under some circumstances. If there are electric and magnetic fields around, then Maxwell equations are true, always.

If a battery is connected to a wire, and there is current, the total circulation in the circuit IS zero.
No. There are two E fields in the battery, canceling each other: the electrostatic pointing from + to - and the non-conservative which points from - to +. The circulation of the static field is of course zero but that leaves the circulation of the non-conservative, and thus net, field = iR.
Note that the battery is pushing the current AGAINST the electric field, inside the battery.
Right. The non-conservative E field pushes against the static E field inside the battery, resulting in zero net E field inside the battery. You seem to think in terms of just one E field when in reality there are two.

Yes there is something controversial here. It is simply not true, that the equality sign has a direction as in an assignment like in programming code, which you seemed to interpret. This is wrong. An equality sign means something else as it does in codes.

The sign "=" means especially:
\begin{align}
\text{reflexivity: }&A = A \\
\text{symmetry: }&A=B \Longrightarrow B=A \\
\text{transitivity: }&A=B \wedge B=C \Longrightarrow A=C
\end{align}
and this holds in physics, too. If two physical quantities are equal, as in your examples, then this means: there is a setup in which those quantities can be considered equal; with all it's implications. It does not mean, the equality would make sense in all thinkable setups. The molecules in this room all are in motion and have mass. Thus they have a kinetic energy. But it doesn't make sense to compute this energy by ##\frac{1}{2}mv^2##. Nevertheless, it remains true for a single molecule at a certain time.

So the main difference between code, mathematics and physics are:
• code = stands for an assignment as ##k=k+1##
• math = stands for quantities which are considered equal as ##1=\frac{2}{2}##
• physics = stands for quantities which are considered equal in a certain setup; it simply doesn't apply in other setups.
Well, sir, I couldn't agree with you more.
The controversy I mentioned is strictly in the physics, not the logic. Believe me, the controversy exists, even among highly regarded physicists.

In fact, any electrostatic field (independent of time) is such that ∇ X E is zero. Maxwell equations don't just mean that they are true under some circumstances. If there are electric and magnetic fields around, then Maxwell equations are true, always.
That contradicts what I said? I agree with every word! Think about "electrostatic". Is this the only kind of field involved in the presence of ANY source of emf?

That contradicts what I said? I agree with every word! Think about "electrostatic". Is this the only kind of field involved in the presence of ANY source of emf?
If the total electric field inside the battery is zero, why is there a current in it? I am talking from
curent density J = conductivity (electric field)

It does NOT say that if I have ∇ x E I must have -∂B/∂t.
This is wrong. It most certainly does say exactly that. Indeed, that is the case.

• bhobba and Bystander
That contradicts what I said? I agree with every word! Think about "electrostatic". Is this the only kind of field involved in the presence of ANY source of emf?
"Electrostatic" just means that the electric field does not vary with time in whatever problem we're considering.

• bhobba
If the total electric field inside the battery is zero, why is there a current in it? I am talking from current density J = conductivity (electric field)
That relation does not apply to the static E field inside of the battery, or any other source of emf. Unlike in a resistor where the current does move in the direction of the static E field, and your relation applies.

"Electrostatic" just means that the electric field does not vary with time in whatever problem we're considering.
EDIT:
? In a ring of uniform resistance with a constant dB/dt field inside, both the the non-conservative and the static E fields would be constant over time.

P.S. the non-conservative firled is the one generated by emf. So perhaps a better name for it is the emf field.

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For example F=ma means that the definition of force is m*a or the quantity of left side equals to the quantity of right side or both ? or kinetic energy..we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it ?

Its an example of the modern general definition of energy which you probably haven't seen yet in your studies, but IMHO should be taught ASAP ie as soon as students have the mathematical background which is basic multi-variable calculus.

It is an example of Noethers Theorem: (the first link is about the theorem, the second is a bit about Noether herself, which as the title says is the most important mathematician you likely never have heard of):
http://www.mecheng.iisc.ernet.in/~suresh/me256/Lectures/Noether_theorem.pdf
https://www.nytimes.com/2012/03/27/...icant-mathematician-youve-never-heard-of.html

Specifically it says, regarding energy, if the system doesn't overall change with time (here it means the systems Lagrangian does not depend explicitly on time) then there exists a conserved quantity. The quantity for not being changed in time (the theorem says any symmetry leads to some conserved quantity and the resulting thing that is conserved are called different things depending on what symmetry is being considered ie what changes that leaves the Lagrangian unchanged) is called energy by definition. It is the modern definition of energy and by the theorem is of course conserved.

Now you asked 'we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it'. It is the definition of kinetic energy in classical mechanics. So it is tautologically true - definitions do not say anything. The equality is always true from the definition. But is it the energy of the system? The answer is sometimes yes - sometimes no. It all depends on what the conserved quantity in Noether's theorem is. In classical physics you can also have a potential energy term, and if that is not zero then the Kinetic Energy of the system is not its energy - its the sum of the kinetic and potential energy.

You mentioned F=MA. That too is a definition - hence tautologically true - its not saying anything really - or is it. I will let you think that one out yourself - do a post about it if you can't nut it out. BTW I couldn't - John Baez had to explain it to me - so it's not exactly trivial - but see how you go. The answer is also connected with Noethers Theorem and what is the most 'elegant', formulation of classical mechanics (it avoids the question I asked about what is F=MA and ties in nicely with Noether) - but I will leave that one with you should you desire to think about it.

Thanks
Bill

The thread is digressing from the original question - and, by doing so, illustrating that equations by themselves have no particular meaning. Certain equations become associated with particular contexts and, by tradition, are interpreted by the words used to describe those contexts ( e.g. Maxwell's equations, Newtons Laws of Motion etc.) There can be disagreements about the contexts. These disagreements are thrashed out in terms of the (verbal) content of the contexts. The disagreements can't be resolved just by looking at the symbols that represent the equation.

The original post asks if there is convention about the interpretation of equations in physics. Specifically, are they definitions or theorems? - or perhaps statements of empirical fact? The answer is that the mere symbolic statement of an equation (in particular the interpretation of "=") does not tell us whether the equation is a definition or theorem or empirical fact.

That's kind of a philosophical question so it's tricky. I'm actually reading the famous Feynman's Lecture on Physics and he too writes about this. I'll try to explain what I understood (hoping it will be correct). F = ma is not a definition, it is a law. The difference is that a law allows us to make predictions about the results of experiments using mathematics; definitions can't do that.

That is true, but while I do not like 'correcting' the great man, its not really the reason it's a law. Think a bit more about it and post if what's really happening doesn't become clear. The reason Feynman had not quite got it correct is how do you experimentally test it? Its quite subtle - but there is an answer - its not that it can be experimentally tested (it can BTW but exactly what is being tested is not that clear) - its - something else - it involves what we can consider laws. Think about it and if you do not see it post and I will explain it.

Thanks
Bill

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The thread is digressing from the original question - and, by doing so, illustrating that equations by themselves have no particular meaning.

That's not quite correct either. The meaning they have depends on context. For example given the free particle Lagrangian 1/2 mv^2 (why that is the free particle Lagrangian is elegantly explained in Landau - Mechanics - but the detail is not important here) its energy, by the definition of energy via Noether, is the kinetic energy. So we can say the kinetic energy is the energy of a free particle. That is a statement about nature. Can it be tested - again a subtle question. I think it can be - but ultimately what is being tested - I think its QM - but that is way off topic - it needs a new thread.

Thanks
Bill

That's not quite correct either. The meaning they have depends on context.
That's exactly my point. An equation by itself (e.g. y = 4z + 2w) conveys no information about whether it is a definition, a theorem, or an empirical fact. Only the tradition of associating certain ways of writing equations with particular contexts tells us how to interpret equations. There is no particular tradition associated with "N = YZ". There are traditions about interpreting "F = MA".

• russ_watters and bhobba
Indeed. Now we are starting to home in on what F=ma really says and why it is a law. It can be tested, but what is being tested is a bit subtle. If nobody posts what is going on I will a bit later - after people have thought about it a bit - have to head off to lunch now - will see how the thread is going when I get back.

Thanks
Bill

That's exactly my point. An equation by itself (e.g. y = 4z + 2w) conveys no information about whether it is a definition, a theorem, or an empirical fact.
An equation means that there is a relationship between the thing on the left and the thing on the right. Specifically, the relationship is that there is no difference between them.

The fact that an equation provides no information about whether the relationship is a definition or a theorem or an observed fact indicates that the distinction between those three is perhaps less important that you assume. We define things that coincide with our observed facts, and you can always reverse any proof so that the result is a premise and the premises are a result.

Anyway, the meaning of “=“ is clear. It can be used in a variety of contexts, but it means the same thing in all contexts.

• QuantumQuest, russ_watters, fresh_42 and 1 other person
Ok - people either are not interested in the meaning of F=MA as opposed to say Coulombs Law or can't see the solution - that's nothing to worry about - neither could I until John Baez explained it to me. Its really is off topic here so I will start a separate thread.

Thanks
Bill

In mathematics, the equal sign means that two items are the same in the algebra that has been defined, where many details may have been abstracted out. So two "equal" items may still be very different in many details.

In physics, the equal sign might mean that there is no way to physically distinguish between the two. In that case, they really should be considered identical in every way. They are the same thing. But there are also many occasions in physics where the equal sign is more like the mathematical abstract use, so one must be careful about interpreting it too strongly as the two items being identical in every way.

• Stephen Tashi, fresh_42 and bhobba
For example F=ma means that the definition of force is m*a or the quantity of left side equals to the quantity of right side or both ? or kinetic energy..we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it ?

The meaning of the equal sign in physics depends a lot on context, and the matter is not without its subtleties. For equations in physics to have physical meaning, you always need to be aware of the fine print which gives more explanation including:

- The definitions of the symbols in the equation
- The units of the physical quantities represented by the symbols
- Special conditions required for the equation to be exactly true
- Special conditions required for the equation to be true to an excellent approximation

It is rare that all the needed explanation is given along with an equation. But some of the explanations needed for F = ma are:

- F is the force in Newtons, m is the mass in kg, and a is the acceleration in m/s/s (or another consistent system of units
- The motion and forces are confined to one dimension (otherwise a vector equation is needed)
- Either the force is the net external force or only one external force is acting
- The mass of the object is not changing
- The acceleration refers to the center of mass of the object
- Quantum and relative effects may be neglected

No, but you can use it as definition (because of the equality) for the force needed to accelerate a mass, or to stop one. However, this might not help you to understand force, as it only applies to this one case. What is the force needed to stretch a spring? Or the force between two magnets? But according to Newton's laws of motion, by which force is defined by a change of the velocity of a mass, this definition will do in may cases.

Since the relevant F in F = ma is a net force, F = ma is only useful as a definition of force if we can be sure only one force is acting on the object. In practice, this may be a useful approximation, but without an outside understanding of what forces are, one quickly runs into a circular argument supporting the case that additional forces are not acting. In practice, F = ma is more useful as the definition of intertial mass. Inertial mass is the constant of proportionality between an applied net force and the resulting acceleration.

In mathematics, the equal sign means that two items are the same in the algebra that has been defined,
Yes. For example "=" for real numbers isn't the same relation as "=" for matrices.

Yes. For example "=" for real numbers isn't the same relation as "=" for matrices.
One can even discuss whether ##1=\frac{2}{2}## is a sign for "the same" or "the equivalent". It becomes more obvious if we have examples like ##= \in \mathbb{Z}/2\mathbb{Z}\,.##

That relation does not apply to the static E field inside of the battery, or any other source of emf. Unlike in a resistor where the current does move in the direction of the static E field, and your relation applies.
The specific relation between electric field and current should not matter in this case. If the net electric field inside the battery is zero, then there is no force on the charge carriers, and no current inside the battery.

The specific relation between electric field and current should not matter in this case. If the net electric field inside the battery is zero, then there is no force on the charge carriers, and no current inside the battery.
If the net electric field is zero there is no force on the charge carriers, no change in their average velocities and no change in the associated current.

The current through a circuit with zero electric field everywhere is indeterminate.

• QuantumQuest