On this website, we make a clear distinction between classical and Newtonian mechanics .

In the definition of his Second Law of Motion, or “Lex Secunda”, Newton stated that the time derivative of the momentum of a body is equal to the acting force.

In his famous work “Philosophiae Naturalis Principia Mathematica” Isaac Newton writes:

Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

Translated: “The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

The “Lex Secunda” also allows a variable mass

If the momentum is differentiated, two terms result:

The first term describes the change in speed. The second term provides a possible variation in mass.

\[\vec{F}=\frac{d(m\vec{v})}{dt}\hspace{20pt} \Rightarrow \hspace{20pt}\vec{F}=m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}\hspace{20pt}(1)\]

If the mass is regarded as a constant (as is the case at low speeds), then the second term of relation (1) is zero and the “Lex Secunda” is reduced to: 

\[\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}\]

The acceleration is then directly proportional to the force. In this condensed form, the “Lex Secunda” describes classical mechanics.

The “Lex Secunda” is compatible with relativity

If, however, the mass is regarded as a variable (as is the case at high speeds), then the law must be applied with both terms of relationship (1):

\[\vec{F}=m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}\hspace{20pt}(1)\]

In this generalized form the “Lex Secunda” is compatible with relativistic mechanics.

On this website,

Classical mechanics is called the part of mechanics which assumes that the mass is constant. Therefore it makes use only of the first term of relation (1) of the Second Law of Motion.

Newtonian mechanics” refers to the part of mechanics that also involves a variation in mass. Consequently it uses both terms of relation (1) of the Second Law of Motion. We will see that, starting from this, the formulas of the Special Theory of Relativity can be derived alternatively (see “Sequence of Relativistic Proofs“).

(See e.g. the relativistic derivations of mass, energy and acceleration).

Therefore, relation (1) represents the link between Newtonian and relativistic mechanics.

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