Editing Newton's laws of motion (section)

==Relation to other formulations of classical physics==
Classical mechanics can be mathematically formulated in multiple different ways, other than the "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, [[Lagrangian mechanics]] helps make apparent the connection between symmetries and conservation laws, and it is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere.<ref name=":2"/>{{Rp|48}} [[Hamiltonian mechanics]] is convenient for [[statistical physics]],<ref>{{Cite book|last1=Ehrenfest|first1=Paul|url=https://www.worldcat.org/oclc/20934820|title=The Conceptual Foundations of the Statistical Approach in Mechanics|last2=Ehrenfest|first2=Tatiana|date=1990|publisher=Dover Publications|isbn=0-486-66250-0|location=New York|pages=18|oclc=20934820|author-link=Paul Ehrenfest|author-link2=Tatiana Ehrenfest-Afanaseva|orig-date=1959}}</ref><ref name=":5">{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091}}</ref>{{Rp|57}} leads to further insight about symmetry,<ref name=":2"/>{{Rp|251}} and can be developed into sophisticated techniques for [[perturbation theory]].<ref name=":2"/>{{Rp|284}} Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion.

===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}}

===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>

===Hamilton–Jacobi===
The [[Hamilton–Jacobi equation]] provides yet another formulation of classical mechanics, one which makes it mathematically analogous to [[wave optics]].<ref name=":2" />{{Rp|page=284}}<ref>{{Cite journal|last=Houchmandzadeh|first=Bahram|date=May 2020|title=The Hamilton–Jacobi equation: An alternative approach|url=http://aapt.scitation.org/doi/10.1119/10.0000781|journal=[[American Journal of Physics]]|language=en|volume=88|issue=5|pages=353–359|doi=10.1119/10.0000781|arxiv=1910.09414 |bibcode=2020AmJPh..88..353H |s2cid=204800598 |issn=0002-9505}}</ref> This formulation also uses Hamiltonian functions, but in a different way than the formulation described above. The paths taken by bodies or collections of bodies are deduced from a function <math>S(\mathbf{q}_1,\mathbf{q}_2,\ldots,t)</math> of positions <math>\mathbf{q}_i</math> and time <math>t</math>. The Hamiltonian is incorporated into the Hamilton–Jacobi equation, a [[differential equation]] for <math>S</math>. Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant <math>S</math>, analogously to how a light ray propagates in the direction perpendicular to its wavefront. This is simplest to express for the case of a single point mass, in which <math>S</math> is a function <math>S(\mathbf{q},t)</math>, and the point mass moves in the direction along which <math>S</math> changes most steeply. In other words, the momentum of the point mass is the [[gradient]] of <math>S</math>:
<math display="block"> \mathbf{v} = \frac{1}{m} \mathbf{\nabla} S.</math>
The Hamilton–Jacobi equation for a point mass is
<math display="block"> - \frac{\partial S}{\partial t} = H\left(\mathbf{q}, \mathbf{\nabla} S, t \right).</math>
The relation to Newton's laws can be seen by considering a point mass moving in a time-independent potential <math>V(\mathbf{q})</math>, in which case the Hamilton–Jacobi equation becomes
<math display="block">-\frac{\partial S}{\partial t} = \frac{1}{2m} \left(\mathbf{\nabla} S\right)^2 + V(\mathbf{q}).</math>
Taking the gradient of both sides, this becomes
<math display="block">-\mathbf{\nabla}\frac{\partial S}{\partial t} = \frac{1}{2m} \mathbf{\nabla} \left(\mathbf{\nabla} S\right)^2 + \mathbf{\nabla} V.</math>
Interchanging the order of the partial derivatives on the left-hand side, and using the [[power rule|power]] and [[chain rule]]s on the first term on the right-hand side,
<math display="block">-\frac{\partial}{\partial t}\mathbf{\nabla} S = \frac{1}{m} \left(\mathbf{\nabla} S \cdot \mathbf{\nabla}\right) \mathbf{\nabla} S + \mathbf{\nabla} V.</math>
Gathering together the terms that depend upon the gradient of <math>S</math>,
<math display="block">\left[\frac{\partial}{\partial t} + \frac{1}{m} \left(\mathbf{\nabla} S \cdot \mathbf{\nabla}\right)\right] \mathbf{\nabla} S = -\mathbf{\nabla} V.</math>
This is another re-expression of Newton's second law.<ref>{{Cite journal|last=Rosen|first=Nathan|author-link=Nathan Rosen|date=February 1965|title=Mixed States in Classical Mechanics|url=http://aapt.scitation.org/doi/10.1119/1.1971282|journal=[[American Journal of Physics]]|language=en|volume=33|issue=2|pages=146–150|doi=10.1119/1.1971282|bibcode=1965AmJPh..33..146R |issn=0002-9505}}</ref> The expression in brackets is a [[Material derivative|''total'' or ''material'' derivative]] as mentioned above,<ref>{{Cite journal |last=Weiner |first=J. H. |date=November 1974 |title=Hydrodynamic Analogy to the Hamilton–Jacobi Equation |url=http://aapt.scitation.org/doi/10.1119/1.1987920 |journal=[[American Journal of Physics]] |language=en |volume=42 |issue=11 |pages=1026–1028 |doi=10.1119/1.1987920 |bibcode=1974AmJPh..42.1026W |issn=0002-9505}}</ref> in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place:
<math display="block">\left[\frac{\partial}{\partial t} + \frac{1}{m} \left(\mathbf{\nabla} S \cdot \mathbf{\nabla}\right)\right] = \left[\frac{\partial}{\partial t} + \mathbf{v}\cdot\mathbf{\nabla}\right] = \frac{d}{dt}.</math>