Newtons’s 2nd law

According to Newton s 2nd law of motion, the force, F, is expressed as a product of mass, m, and acceleration, a F = m a, having the dimension of [M LT 2]. According to Newton s law of gravitation, force is defined by F mi m2/r2, thus leading to completely another dimension [M2 L-2]. To remedy this, the gravitational constant G - a dimensional constant - had to be introduced to ensure the dimensional homogeneity of the latter equation ... 

Consider a star as a ball of gas which is held up by the pressure forces within it. How long would the star survive if the pressure forces were removed The answer is known as the free-fall or dynamical time (tg or tdyn)- A simple estimate is given by the time required for a body to fall through a distance of the order R (the stellar radius) under the influence of a (constant) gravitational acceleration equivalent to the surface gravity g = GM/R2 of a star of mass M. From Newton s 2nd law,... 

Alternative formulation to Newton s 2nd law was provided by Joseph-Louis Lagrange (1736-1813). He defined a Lagrangian L as... 

Inertial and gravitational mass In physical relations the base unit of mass occurs with two fundamentally different meanings. Mass can specify a quantity of a substance in this meaning, the quantity is, for example, included in the definition of density p with the dimension (L M). Mass can also be included as a proportionality factor between force and acceleration in Newton s 2nd law, and in this context, inertial mass is meant. 

Scheck, F. 1994. Mechanics from Newton s laws to deterministic chaos, ed. 2nd corr. and enl.. Springer Verlag. 

Newton used this equation as the basis of mechanics, hence it is sometimes called Newton s equation [4]. Note that Newton s 1. and 3. laws merely emerge as particular simplifications of the 2nd law and are thus not explicitly stated. 

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