Editing Newton's laws of motion (section)

===Lagrangian===
[[Lagrangian mechanics]] differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant.<ref name=":2">{{Cite book|last1=José|first1=Jorge V.|url=https://www.worldcat.org/oclc/857769535|title=Classical dynamics: A Contemporary Approach|last2=Saletan|first2=Eugene J.|date=1998|publisher=Cambridge University Press|isbn=978-1-139-64890-5|location=Cambridge [England]|oclc=857769535|author-link=Jorge V. José}}</ref>{{Rp|page=109}} It is traditional in Lagrangian mechanics to denote position with <math>q</math> and velocity with <math>\dot{q}</math>. The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies:
<math display="block">L(q,\dot{q}) = T - V,</math>
where the kinetic energy is
<math display="block">T = \frac{1}{2}m\dot{q}^2</math>
and the potential energy is some function of the position, <math>V(q)</math>. The physical path that the particle will take between an initial point <math>q_i</math> and a final point <math>q_f</math> is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. [[Calculus of variations]] provides the mathematical tools for finding this path.<ref name="Boas" />{{Rp|page=485}} Applying the calculus of variations to the task of finding the path yields the [[Euler–Lagrange equation]] for the particle,
<math display="block">\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\right) = \frac{\partial L}{\partial q}.</math>
Evaluating the [[partial derivative]]s of the Lagrangian gives
<math display="block">\frac{d}{dt} (m \dot{q}) = -\frac{dV}{dq},</math>
which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy.<ref name=":1">{{Cite book|last=Gbur|first=Greg|url=https://www.worldcat.org/oclc/704518582|title=Mathematical Methods for Optical Physics and Engineering|date=2011|publisher=Cambridge University Press|isbn=978-0-511-91510-9|location=Cambridge, U.K.|oclc=704518582|author-link=Greg Gbur}}</ref>{{Rp|page=737}}

[[Course of Theoretical Physics|Landau and Lifshitz]] argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws.<ref name="Landau">{{cite book
 |author-first1=Lev D. |author-last1=Landau |author-link1=Lev Landau
 |author-first2=Evgeny M. |author-last2=Lifshitz |author-link2=Evgeny Lifshitz
 |translator-first1=J. B. |translator-last1=Sykes
 |translator-first2=J. S. |translator-last2=Bell |translator-link2=John Stewart Bell
 |series=[[Course of Theoretical Physics]]
 |date=1969
 |title=Mechanics
 |edition=2nd |volume=1 |page=vii
 |publisher=[[Pergamon Press]]
 |isbn=978-0-080-06466-6
 |oclc=898931862
 |quote=Only with this approach, indeed, can the exposition form a logical whole and avoid tautological definitions of the fundamental mechanical quantities. It is, moreover, essentially simpler, and leads to the most complete and direct means of solving problems in mechanics.
}}</ref> Lagrangian mechanics provides a convenient framework in which to prove [[Noether's theorem]], which relates symmetries and conservation laws.<ref>{{cite book |last=Byers |first=Nina |author-link=Nina Byers |chapter=Emmy Noether |title=Out of the Shadows: Contributions of 20th Century Women to Physics |year=2006 |editor1-first=Nina |editor1-last=Byers |editor2-first=Gary |editor2-last=Williams |place=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-82197-1 |url-access=registration |url=https://archive.org/details/outofshadowscont0000unse |oclc=1150964892 |pages=83–96}}</ref> The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption.<ref name=":2" />{{Rp|page=124}}