# What is force? The basic intuition

Conventional definition of force is :

An agency (push or pull) that tends to change state of rest to motion or vice versa.

I don't find this definition satisfactory.

Second definition comes from newton's second law :

$$F = ma$$

But as feynman has said that this law relates two fundamental virtues and is used to define one in terms of other, which is not idea of physics ( I am not quoting the exact wording)

Then what actually force is? I want to know basic intuition idea behind it (even in modern physics)

• Forces are better described with the vector equation $F=\dot p$, which is how Newton expressed force. It's a more faithful definition, and is more stands the test of time better when considering relativistic effects. If you don't see the insight immediately through $F= \dot p$, it is more plainly the rate of change of momentum - an interaction with an object's inertia, if you will. – sangstar Dec 1 '17 at 2:11
• @sangstar You probably should move this comment to an answer. – safesphere Dec 1 '17 at 4:35
• Your second definition is missing the sum symbol: $$\sum F=ma$$ This is a quite crucial detail. Now both of the definitions you mention are equivalent: A force tries to cause acceleration (change of motion). – Steeven Dec 1 '17 at 5:56
• It relates $dp/dt$ (movement and inertia/mass/resistance to movement) to action over distance effects that depend on geometry and properties of the matter (charge, mass, ...) like the Lorentz force, gravitational force,... Not sure how the multiphysics setting look, since the charge-movement combination is described by another system of equations. – Emil Dec 1 '17 at 7:17

Force is defined as instantaneous rate of change of momentum, but it is not merely a product of mass and acceleration as said by Feynman also in his lecture :

The real content of Newton’s laws is this: that the force is supposed to have some independent properties, in addition to the law $F=m\mathbf a$ but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law$F=m\mathbf a$ is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.

It says that forces have material origin and not a vague concept,as it is seen evidently, like we often say that force is a pull or push, which implies that we have a physical notion attached to the concept and it greatly simplifies our quest to understand and predict nature.

One way to understand force is to say that the net force is defined as the rate of change - or time derivative - of momentum. That is exactly what a force is in all physics because that is what it is defined to be. Then Newton's second law is a definition. ($\bf{F} = m\bf{a}$ because the time derivative of momentum ($\dot{\bf{p}}$) is mathematically the same as $m\bf{a}$ for a constant mass.)
Now your question becomes "What is momentum?" Well, that changes a little in some fields of physics, but the typical definition is $\bf{p} = m \bf{\dot{x}}$ where $\bf{x}$ is position. That is, momentum is the mass times the time derivative of position.