Editing Newton's laws of motion (section)

===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>