Newton's laws of motion in nLab

Contents

Contents

Idea

Newton (1687) formulated three law of mechanics (today called “classical mechanics”) whose statement constitutes the beginning of physics as a science rooted in mathematics, in contrast to the more naively descriptive “natural philosophy”.

Statements

In modern formulation, Newton’s three laws express the following:

Newton’s first and second law

The first and second law together says that equations of motion in physics are given by differential equations of second order.

This implies that initial value data for equations of motion is given by 0th and by 1st derivatives. This data is in physics traditionally is referred to as canonical coordinates and canonical momenta, respectively. In modern mathematical physics this is axiomatized by the notion of phase space being a symplectic manifold.

Newton’s third law

The third Law expresses an invariance under the equations of motion under the Galilei group.

With the advent of the “theory of special relativity” this law was later refined to postulate invariance under the Poincaré group. With the advent of the theory of general relativity it was further refined to postulate only local symmetry under the Poincaré group, which leads to the modern formulation of Einstein gravity.

• Newtonian mechanics

• equations of motion

  • Hamilton's equations, de Donder-Weyl-Hamilton equation

  • Einstein's equations

References

• Isaac Newton, Philosophiæ naturalis principia mathematica (1687)

• Wikipedia, Newton’s laws of motion