Fine distinctions: F vs ma, E vs H, etc

  • Thread starter pellman
  • Start date
In summary, the conversation discusses the distinction between the equations F = ma and H = E, and how they represent fundamental physical laws rather than tautologies. The speaker also emphasizes the importance of recognizing the distinction between force and acceleration, and the Hamiltonian and energy, in accurately describing and predicting the behavior of physical systems.
  • #1
pellman
684
5
I would maintain the following and I am curious if others disagree.

When we say "F = ma" this is not to say that the force F is identified with ma. If we define the force function (and how we define what we mean by "force" may be a point of contention) as

[tex]F=-\nabla V[/tex]

then F is some function of the position and velocity and, maybe, the positions and velocities of other particles. The acceleration, on the other hand, is the second time derivative of the position function. These two quantities are not the same thing. When we say "F = ma" what we mean is that of all possible functions x(t), the physically realizable ones are only those that satisfy

[tex]\frac{d^2 x}{dt^2}=\frac{F}{m}=-\frac{1}{m}\nabla V[/tex]


Similarly I would maintain that the Hamiltonian is not the same thing as the energy of a system. Rather the statement H = E is a constraint on the allowed states of the system.

This distinction is more clearly seen in quantum theory. When we write

[tex]H|\psi(t)\rangle = i d/dt |\psi(t)\rangle[/tex]

this is not to say that H = i d/dt. H and i d/dt are two distinct operators. Rather, we are saying that only those time-dependent states [tex]|\psi(t)\rangle[/tex] which satisfy the above condition describe our system. Any other function mapping the real numbers (time) to the Hilbert space of states is not physically allowed.

That is, F = ma, H = E, H = i d/dt are meaningful equalities, not tautologies.

I think this is actually an important distinction and failure to see it sometimes results in confusion. But does anyone disagree?
 
Physics news on Phys.org
  • #2


I would agree with your statement that F = ma and H = E are not tautologies, but rather meaningful equalities. This is because they represent fundamental physical laws and principles that have been derived through careful observation and experimentation.

In the case of F = ma, this equation represents Newton's second law of motion, which states that the force acting on an object is equal to its mass times its acceleration. This is a fundamental principle in classical mechanics and has been confirmed through countless experiments.

Similarly, H = E represents the principle of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. This is a fundamental concept in physics and has been supported by numerous experiments and observations.

I also agree with your point about the distinction between the force and acceleration, and the Hamiltonian and energy. While they are related, they are not the same thing and have different physical interpretations.

Overall, I believe that understanding and recognizing these distinctions is crucial in accurately describing and predicting the behavior of physical systems. So, I do not disagree with your statements.
 

What is the difference between F and ma?

Force (F) and mass times acceleration (ma) are both related to the concept of force, but in slightly different ways. F represents the overall force acting on an object, while ma represents the force required to accelerate that object. In other words, F is a measurement of the total force, while ma is a measurement of the force required to produce a specific acceleration.

What is the difference between E and H?

E and H are both related to the concept of electromagnetic fields. E represents the electric field, which is the force that a charged particle experiences in an electric field. H represents the magnetic field, which is the force that a moving charged particle experiences in a magnetic field. In other words, E is related to stationary charges, while H is related to moving charges.

How are F, ma, E, and H related?

All four of these concepts are related to the concept of force, but in different ways. F and ma are both measurements of force, but F is the overall force acting on an object, while ma is the force required to produce a specific acceleration. E and H are both related to electromagnetic fields, with E representing the electric field and H representing the magnetic field. In the context of electromagnetism, E and H can be thought of as components of the overall force acting on a charged particle.

Why are fine distinctions between F and ma, E and H important?

Fine distinctions between these concepts are important because they help us better understand the fundamental forces and principles that govern the universe. By understanding the differences between F and ma, E and H, we can more accurately describe and predict the behavior of objects and particles in different situations. Additionally, these fine distinctions can also lead to new discoveries and advancements in various fields of science and technology.

How can we apply our understanding of F, ma, E, and H in real-world situations?

Our understanding of F, ma, E, and H can be applied in a variety of real-world situations, such as designing and building structures, creating new technologies, and studying natural phenomena. For example, engineers may use their understanding of these concepts to design buildings that can withstand strong winds or earthquakes, while scientists may use them to study the behavior of particles in particle accelerators. Additionally, our understanding of these concepts can also help us develop new technologies, such as electric motors and generators, which rely on the principles of electromagnetism.

Similar threads

Replies
2
Views
4K
Replies
7
Views
653
  • Quantum Physics
Replies
13
Views
695
  • Advanced Physics Homework Help
Replies
0
Views
617
Replies
2
Views
540
  • Advanced Physics Homework Help
Replies
10
Views
460
  • Calculus and Beyond Homework Help
Replies
3
Views
515
  • Advanced Physics Homework Help
Replies
1
Views
206
Replies
2
Views
1K
Replies
19
Views
1K
Back
Top