Newtonian Mechanics ○𓆪 Entries

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Newtonian Mechanics ○𓆪｜Definition｜1st｜20251119205401-00-⌔

Classical mechanics - Wikipedia#Newtonian_mechanics

Newtonian mechanics

A force in physics is any action that causes an object’s velocity to change; that is, to accelerate. A force originates from within a field, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton’s second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature.¹ Either interpretation has the same mathematical consequences, historically known as “Newton’s second law”:

• ✤ $\mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=\frac{\mathrm{d}(m\mathbf{v})}{\mathrm{d}t}.$

The quantity m v is called the (canonical) momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is a = d v/d t, the second law can be written in the simplified and more familiar form:

• ✤ $\mathbf{F}=m\mathbf{a}.$

So long as the force acting on a particle is known, Newton’s second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton’s second law to obtain an ordinary differential equation, which is called the equation of motion.

As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example:

• ✤ ${\mathbf{F}}_{\mathrm{R}}=-\lambda \mathbf{v},$

where λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is

• ✤ $-\lambda \mathbf{v}=m\mathbf{a}=m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}.$

This can be integrated to obtain

• ✤ $\mathbf{v}={\mathbf{v}}_0{e}^{-\lambda t/m}$

where v${}_0$ is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the conservation of energy), and the particle is slowing down. This expression can be further integrated to obtain the position r of the particle as a function of time.

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Newton’s third law can be used to deduce the forces acting on a particle when in a closed system. If it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, −F, on A. For conservative forces, this means that the line integral around a closed loop is zero. The strong form of Newton’s third law requires that F and −F act along the line connecting A and B, and these forces are defined as central forces. However, central forces are an approximation since objects that are at rest are only at rest with respect to one another.² This limitation to Newton’s third law can be shown using the Coulomb force, where charges must remain stationary with respect to a nonaccelerating frame of reference.³ When dealing with non-central forces like the Lorentz force, the weak form of Newton’s third law is used by identifying conservation of momentum. Illustrations of the weak form of Newton’s third law can be found for magnetic forces like the Lorentz force while discussing the curl or cross product of vectors. Thus, the forces acting on objects cannot be identified without accounting for relative acceleration and direction by utilizing reference frames.⁴

Printed 2026-06-28.

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Footnotes

1. Thornton, Stephen T.; Marion, Jerry B. (2004). Classical dynamics of particles and systems (5. ed.). Belmont, CA: Brooks/Cole. pp. 50. ISBN 978-0-534-40896-1. ↩

2. Rynasiewicz, Robert; Zalta, Edward N. (2022). Newton’s Views on Space, Time, and Motion (Spring 2022 ed.). Metaphysics Research Lab, Stanford University: The Stanford Encyclopedia of Philosophy. ↩

3. Taylor, John (2005). Classical Mechanics. University Science Books. pp. 133–138. ISBN 1-891389-22-X. ↩

4. Griffiths, David (2023). Introduction to Electrodynamics (4th ed.). Cambridge: Cambridge University Press. pp. 316–318. ↩

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