Editing Newton's laws of motion (section)

===Hamiltonian===
[[File:Noether.jpg|thumb|upright|[[Emmy Noether]], whose 1915 proof of [[Noether's theorem|a celebrated theorem that relates symmetries and conservation laws]] was a key development in modern physics and can be conveniently stated in the language of Lagrangian or Hamiltonian mechanics]]
In [[Hamiltonian mechanics]], the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system.<ref name=":1" />{{Rp|page=742}} The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of the position and momentum variables are given by partial derivatives of the Hamiltonian, via [[Hamilton's equations]].<ref name=":2" />{{Rp|page=203}} The simplest example is a point mass <math>m</math> constrained to move in a straight line, under the effect of a potential. Writing <math>q</math> for the position coordinate and <math>p</math> for the body's momentum, the Hamiltonian is
<math display="block">\mathcal{H}(p,q) = \frac{p^2}{2m} + V(q).</math>
In this example, Hamilton's equations are
<math display="block">\frac{dq}{dt} = \frac{\partial\mathcal{H}}{\partial p}</math>
and
<math display="block">\frac{dp}{dt} = -\frac{\partial\mathcal{H}}{\partial q}.</math>
Evaluating these partial derivatives, the former equation becomes
<math display="block">\frac{dq}{dt} = \frac{p}{m},</math>
which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is
<math display="block">\frac{dp}{dt} = -\frac{dV}{dq},</math>
which, upon identifying the negative derivative of the potential with the force, is just Newton's second law once again.<ref name=":3" /><ref name=":1" />{{Rp|page=742}}

As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed.<ref name=":2" />{{Rp|page=251}}

Among the proposals to reform the standard introductory-physics curriculum is one that teaches the concept of energy before that of force, essentially "introductory Hamiltonian mechanics".<ref>{{Cite journal|last1=LeGresley|first1=Sarah E.|last2=Delgado|first2=Jennifer A.|last3=Bruner|first3=Christopher R.|last4=Murray|first4=Michael J.|last5=Fischer|first5=Christopher J.|date=2019-09-13|title=Calculus-enhanced energy-first curriculum for introductory physics improves student performance locally and in downstream courses|journal=[[Physical Review Physics Education Research]]|language=en|volume=15|issue=2|pages=020126|doi=10.1103/PhysRevPhysEducRes.15.020126|bibcode=2019PRPER..15b0126L |s2cid=203484310 |issn=2469-9896|doi-access=free|hdl=1808/29610|hdl-access=free}}</ref><ref>{{Cite journal|last=Ball|first=Philip|author-link=Philip Ball|date=2019-09-13|title=Teaching Energy Before Forces|url=https://physics.aps.org/articles/v12/100|journal=[[Physics (magazine)|Physics]]|language=en|volume=12|page=100 |doi=10.1103/Physics.12.100 |bibcode=2019PhyOJ..12..100B |s2cid=204188746 }}</ref>