- Hamiltonian mechanics
**Hamiltonian mechanics**is a re-formulation ofclassical mechanics that was introduced in 1833 by Irish mathematicianWilliam Rowan Hamilton . It arose fromLagrangian mechanics , a previous re-formulation ofclassical mechanics introduced byJoseph Louis Lagrange in 1788, but can be formulated "without" recourse to Lagrangian mechanics using "symplectic spaces" (See Mathematical formalism, below). The Hamiltonian method differs from the Lagrangian method in that instead of expressing second-order differential constraints on an "n"-dimensionalcoordinate space , it expresses first-order constraints on a 2"n"-dimensionalphase space . [*Citation | last1=LaValle | first1=Steven M. | chapter=§13.4.4 Hamiltonian mechanics | chapter-url=http://planning.cs.uiuc.edu/node707.html | title=Planning Algorithms | publisher=*]Cambridge University Press | isbn=978-0-521-86205-9 | year=2006.As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at classical mechanics. Generally, these equations do not provide a more convenient way of solving a particular problem. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to

quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.**implified overview of uses**For a closed system the sum of the kinetic and

potential energy in the system is represented by a set ofdifferential equation s known as the "Hamilton equations" for that system. Hamiltonians can be used to describe such simple systems as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time. Hamiltonians can also be employed to model the energy of other more complex dynamic systems such as planetary orbits and in quantum mechanics. [*[*]*http://ocw.mit.edu/ans7870/18/18.013a/textbook/chapter16/section03.html The Hamiltonian*] MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007The Hamilton equations are generally written as follows:

:$dot\; p\; =\; -frac\{partial\; mathcal\{H\{partial\; q\}$:$dot\; q\; =~~frac\{partial\; mathcal\{H\{partial\; p\}$

In the above equations, the dot denotes the ordinary derivative with respect to time of the functions"p = p(t)" (called generalized momenta) and "q = q(t)" (called

generalized coordinates ), taking values in some vector space,and "$mathcal\{H\}$ = $mathcal\{H\}(p,q,t)$" is the so-called Hamiltonian, or (scalar valued) Hamiltonian function. Thus, a little more explicitly, one can equivalently write:$frac\{mathrm\; d\}\{mathrm\; dt\}p(t)\; =\; -frac\{partial\}\{partial\; q\}mathcal\{H\}(p(t),q(t),t)$:$frac\{mathrm\; d\}\{mathrm\; dt\}q(t)\; =~~frac\{partial\}\{partial\; p\}mathcal\{H\}(p(t),q(t),t)$

and specify the domain of values in which the parameter "t" ("time") varies.

For a detailed derivation of these equations from

Lagrangian mechanics , see below.**Basic physical interpretation**The simplest interpretation of the Hamilton Equations is as follows, applying them to a one-dimensional system consisting of one particle of mass "m" and exhibiting

conservation of energy :The Hamiltonian "$mathcal\{H\}$" represents theenergy of the system,which is the sum ofkinetic andpotential energy , traditionally denoted "T" & "V", respectively. Here "q" is the "x"-coordinate and "p" is the momentum, "mv." Then: $mathcal\{H\}\; =\; T\; +\; V\; ,\; quad\; T\; =\; frac\{p^2\}\{2m\}\; ,\; quad\; V\; =\; V(q)\; =\; V(x).$Note that "T" is a function of "p" alone, while "V" is a function of "x" (or "q") alone.

Now the time-derivative of the momentum "p" equals the "Newtonian force", and so here the first Hamilton Equation means that the force on the particle equals the rate at which it loses potential energy with respect to changes in "x," its location. (Force equals the negative

gradient of potential energy.)The time-derivative of "q" here means the velocity: the second Hamilton Equation here means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum. (For the derivative with respect to "p" of "p

^{2}/2m" equals "p/m = mv/m = v.")**Using Hamilton's equations**#First write out the

Lagrangian "L" = "T" – "V". Express "T" and "V" as though you were going to use Lagrange's equation.

#Calculate the momenta by differentiating the Lagrangian with respect to velocity.

#Express the velocities in terms of the momenta by inverting the expressions in step (2).

#Calculate the Hamiltonian using the usual definition, $mathcal\{H\}\; =\; sum\_i\; p\_i\; \{dot\; q\_i\}\; -\; mathcal\{L\}$. Substitute for the velocities using the results in step (3).

#Apply Hamilton's equations.**Notes**Hamilton's equations are appealing in view of their beautiful simplicity and (slightly "broken")

symmetry . They have been analyzed under almost every imaginable angle of view, from basic physics up tosymplectic geometry . A lot is known about solutions of these equations, yet the exact general case solution of theequations of motion cannot be given explicitly for a system of more than two massive point particles. The finding of conserved quantities plays an important role in the search for solutions or information about their nature. In models with an infinite number of degrees of freedom, this is of course even more complicated.An interesting and promising area of research is the study ofintegrable system s, where an infinite number of independent conserved quantities can be constructed.**Deriving Hamilton's equations**We can derive Hamilton's equations by looking at how the Lagrangian changes as you change the time and the positions and velocities of particles.

$mathrm\{d\}\; mathcal\{L\}\; =\; sum\_i\; left\; (\; frac\{partial\; mathcal\{L\{partial\; q\_i\}\; mathrm\{d\}\; q\_i\; +\; frac\{partial\; mathcal\{L\{partial\; \{dot\; q\_i\; mathrm\{d\}\; \{dot\; q\_i\}\; ight\; )\; +\; frac\{partial\; mathcal\{L\{partial\; t\}\; mathrm\{d\}t$

Now the generalized momenta were defined as $p\_i\; =\; frac\{partial\; mathcal\{L\{partial\; \{dot\; q\_i$ and Lagrange's equations tell us that$frac\{mathrm\{d\{mathrm\{d\}t\}\; frac\{partial\; mathcal\{L\{partial\; \{dot\; q\_i\; -\; frac\{partial\; mathcal\{L\{partial\; q\_i\}\; =\; F\_i$where $F\_i$ is the generalized force. We can rearrange this to get$frac\{partial\; mathcal\{L\{partial\; q\_i\}\; =\; \{dot\; p\}\_i\; -\; F\_i$and substitute the result into the variation of the Lagrangian

$mathrm\{d\}mathcal\{L\}\; =\; sum\_i\; left\; [\; left(\; \{dot\; p\}\_i\; -\; F\_i\; ight)\; mathrm\{d\}\; q\_i\; +\; p\_i\; mathrm\{d\}\; \{dot\; q\_i\}\; ight]\; +\; frac\{partial\; mathcal\{L\{partial\; t\}mathrm\{d\}t$

We can rewrite this as

$mathrm\{d\}\; mathcal\{L\}\; =\; sum\_i\; left\; [\; left\; (\; \{dot\; p\}\_i\; -\; F\_i\; ight\; )\; mathrm\{d\}q\_i\; +\; mathrm\{d\}left\; (\; p\_i\; \{dot\; q\_i\}\; ight\; )\; -\; \{dot\; q\_i\}\; mathrm\{d\}\; p\_i\; ight\; ]\; +\; frac\{partial\; mathcal\{L\{partial\; t\}mathrm\{d\}t$

and rearrange again to get

$mathrm\{d\}\; left\; (\; sum\_i\; p\_i\; \{dot\; q\_i\}\; -\; mathcal\{L\}\; ight\; )\; =\; sum\_i\; left\; [\; left\; (\; F\_i-\{dot\; p\}\_i\; ight\; )\; mathrm\{d\}\; q\_i\; +\; \{dot\; q\_i\}\; mathrm\{d\}p\_i\; ight]\; -\; frac\{partial\; mathcal\{L\{partial\; t\}mathrm\{d\}t$

The term on the left-hand side is just the Hamiltonian that we have defined before, so we find that;

$mathrm\{d\}\; mathcal\{H\}\; =\; sum\_i\; left\; [\; left\; (\; F\_i-\{dot\; p\}\_i\; ight\; )\; mathrm\{d\}\; q\_i\; +\; \{dot\; q\_i\}\; mathrm\{d\}\; p\_i\; ight]\; -\; frac\{partial\; mathcal\{L\{partial\; t\}mathrm\{d\}t\; =\; sum\_i\; left\; [\; frac\{partial\; mathcal\{H\{partial\; q\_i\}\; mathrm\{d\}\; q\_i\; +\; frac\{partial\; mathcal\{H\{partial\; p\_i\}\; mathrm\{d\}\; p\_i\; ight\; ]\; +\; frac\{partial\; mathcal\{H\{partial\; t\}mathrm\{d\}t$

where the second equality holds because of the definition of the partial derivatives. Associating terms from both sides of the equation above yields Hamilton's equations:

:$frac\{partial\; mathcal\{H\{partial\; q\_j\}\; =\; F\_j\; -\; dot\{p\}\_j,\; qquadfrac\{partial\; mathcal\{H\{partial\; p\_j\}\; =\; dot\{q\}\_j,\; qquadfrac\{partial\; mathcal\{H\{partial\; t\; \}\; =\; -\; \{partial\; mathcal\{L\}\; over\; partial\; t\}.$

**As a reformulation of Lagrangian mechanics**Starting with

Lagrangian mechanics , the equations of motion are based ongeneralized coordinates :$left\{,\; q\_j\; |\; j=1,\; ldots,N\; ,\; ight\}$

and matching generalized velocities

:$left\{,\; dot\{q\}\_j\; |\; j=1,\; ldots\; ,N\; ,\; ight\}$

We write the

Lagrangian as:$mathcal\{L\}(q\_j,\; dot\{q\}\_j,\; t)$

with the subscripted variables understood to represent all "N" variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as "conjugate momenta". By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics, that would otherwise be even more complicated.

For each generalized velocity, there is one corresponding

conjugate momentum , defined as::$p\_j\; =\; \{partial\; mathcal\{L\}\; over\; partial\; dot\{q\}\_j\}$

In

Cartesian coordinates , the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physicalangular momentum . For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinatizations of the same

symplectic manifold .The "Hamiltonian" is the Legendre transform of the

Lagrangian ::$mathcal\{H\}left(q\_j,p\_j,t\; ight)\; =\; sum\_i\; dot\{q\}\_i\; p\_i\; -\; mathcal\{L\}(q\_j,dot\{q\}\_j,t)$

If the transformation equations defining the generalized coordinates are independent of "t", and the Lagrangian is a sum of products of functions (in the generalised coordinates) which are homogeneous of order 0, 1 or 2, then it can be shown that "H" is equal to the total energy "E" = "T" + "V".

Each side in the definition of "$mathcal\{H\}$" produces a differential:

:$egin\{align\}mathrm\{d\}mathcal\{H\}\; =\; sum\_i\; left\; [\; left(\{partial\; mathcal\{H\}\; over\; partial\; q\_i\}\; ight)\; mathrm\{d\}q\_i\; +\; left(\{partial\; mathcal\{H\}\; over\; partial\; p\_i\}\; ight)\; mathrm\{d\}p\_i\; ight]\; +\; left(\{partial\; mathcal\{H\}\; over\; partial\; t\}\; ight)\; mathrm\{d\}tqquadqquadquadquad\; \backslash \; \backslash \; =\; sum\_i\; left\; [\; dot\{q\}\_i,\; mathrm\{d\}p\_i\; +\; p\_i,\; mathrm\{d\}dot\{q\}\_i\; -\; left(\{partial\; mathcal\{L\}\; over\; partial\; q\_i\}\; ight)\; mathrm\{d\}q\_i\; -\; left(\{partial\; mathcal\{L\}\; over\; partial\; dot\{q\}\_i\}\; ight)\; mathrm\{d\}dot\{q\}\_i\; ight]\; -\; left(\{partial\; mathcal\{L\}\; over\; partial\; t\}\; ight)\; mathrm\{d\}tend\{align\}$

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:

:$frac\{partial\; mathcal\{H\{partial\; q\_j\}\; =\; -\; dot\{p\}\_j,\; qquadfrac\{partial\; mathcal\{H\{partial\; p\_j\}\; =\; dot\{q\}\_j,\; qquadfrac\{partial\; mathcal\{H\{partial\; t\; \}\; =\; -\; \{partial\; mathcal\{L\}\; over\; partial\; t\}$

Hamilton's equations are first-order

differential equation s, and thus easier to solve than Lagrange's equations, which are second-order. However, the steps leading to the equations of motion are more onerous than in Lagrangian mechanics - beginning with the generalized coordinates and the Lagrangian, we must calculate the Hamiltonian, express each generalized velocity in terms of the conjugate momenta, and replace the generalized velocities in the Hamiltonian with the conjugate momenta. All in all, there is little labor saved from solving a problem with Hamiltonian mechanics rather than Lagrangian mechanics. Ultimately, it will produce the same solution as Lagrangian mechanics andNewton's laws of motion .The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical mechanics.

**Geometry of Hamiltonian systems**A Hamiltonian system may be understood as a

fiber bundle "E" overtime "R", with the fibers "E"_{"t"}, "t" ∈ "R" being the position space. The Lagrangian is thus a function on thejet bundle "J" over "E"; taking the fiberwiseLegendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at "t" is thecotangent space "T"^{*}"E"_{"t"}, which comes equipped with a naturalsymplectic form , and this latter function is the Hamiltonian.**Generalization to quantum mechanics through Poisson bracket**The Hamilton's equations above work well for

classical mechanics , but not forquantum mechanics , since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to apply to quantum mechanics as well as to classical mechanics, through the deformation of thePoisson algebra over "p" and "q" to the algebra ofMoyal bracket s. In this case, the more general form of the Hamilton's equation reads:$frac\{mathrm\{d\}f\}\{mathrm\{d\}t\}\; =\; \{f,\; mathcal\{H\}\}\; +\; frac\{partial\; f\}\{partial\; t\}$

where "f" is some function of "p" and "q", and "H" is the Hamiltonian. To find out the rules for evaluating a

Poisson bracket without resorting to differential equations, seeLie algebra ; a Poisson bracket is the name for the Lie bracket in aPoisson algebra .In fact, this more algebraic approach not only allows us to ultimately extend

probability distribution s inphase space toWigner quasi-probability distribution s, but also provides more power in the classical setting, in particular by helping to analyze the relevant conserved quantities in a system.**Mathematical formalism**Any smooth real-valued function "H" on a

symplectic manifold can be used to define a Hamiltonian system. The function "H" is known as the**Hamiltonian**or the**energy function**. The symplectic manifold is then called thephase space . The Hamiltonian induces a specialvector field on the symplectic manifold, known as thesymplectic vector field .The symplectic vector field, also called the Hamiltonian vector field, induces a

Hamiltonian flow on the manifold. Theintegral curve s of the vector field are a one-parameter family of transformations of the manifold; the parameter of the curves is commonly called the**time**. The time evolution is given bysymplectomorphism s. By Liouville's theorem, each symplectomorphism preserves thevolume form on thephase space . The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called the**Hamiltonian mechanics**of the Hamiltonian system.The Hamiltonian vector field also induces a special operation, the

Poisson bracket . The Poisson bracket acts on functions on the symplectic manifold, thus giving the space of functions on the manifold the structure of aLie algebra .In particular, given a function "f"

:$frac\{mathrm\{d\{mathrm\{d\}t\}\; f=frac\{partial\; \}\{partial\; t\}\; f\; +\; \{,f,mathcal\{H\},\}.$

If we have a

probability distribution , ρ, then (since the phase space velocity ($\{dot\; p\_i\}\; ,\; \{dot\; q\; \_i\}$) has zero divergence, and probability is conserved) its convective derivative can be shown to be zero and so:$frac\{partial\}\{partial\; t\}\; ho\; =\; -\; \{,\; ho\; ,mathcal\{H\},\}.$

This is called Liouville's theorem. Every

smooth function "G" over thesymplectic manifold generates a one-parameter family ofsymplectomorphism s and if { "G", "H" } = 0, then "G" is conserved and the symplectomorphisms aresymmetry transformation s.A Hamiltonian may have multiple conserved quantities "G"

_{"i"}. If the symplectic manifold has dimension 2"n" and there are "n" functionally independent conserved quantities "G"_{"i"}which are in involution (i.e., { "G"_{"i"}, "G"_{"j"}} = 0), then the Hamiltonian is Liouville integrable. TheLiouville–Arnol'd theorem says that locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism in a new Hamiltonian with the conserved quantities "G"_{"i"}as coordinates; the new coordinates are called "action-angle coordinates". The transformed Hamiltonian depends only on the "G"_{"i"}, and hence the equations of motion have the simple form:$dot\{G\}\_i\; =\; 0,\; qquad\; dot\{varphi\}\_i\; =\; F(G),$for some function "F" (Arnol'd et al., 1988). There is an entire field focusing on small deviations from integrable systems governed by theKAM theorem .The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined. At this time, the study of

dynamical systems is primarily qualitative, and not a quantitative science.**Riemannian manifolds**An important special case consists of those Hamiltonians that are

quadratic form s, that is, Hamiltonians that can be written as:$mathcal\{H\}(q,p)=\; frac\{1\}\{2\}\; langle\; p,p\; angle\_q$

where $langlecdot,cdot\; angle\_q$ is a

cometric on the fiber $T\_q^*Q$, thecotangent space to the point "q" in theconfiguration space . This Hamiltonian consists entirely of thekinetic term .If one considers a

Riemannian manifold or apseudo-Riemannian manifold , so that one has an invertible, non-degenerate metric, then the cometric is given simply as the inverse of the metric. The solutions to theHamilton–Jacobi equation s for this Hamiltonian are then the same as thegeodesic s on the manifold. In particular, theHamiltonian flow in this case is the same thing as thegeodesic flow . The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article ongeodesic s. See alsoGeodesics as Hamiltonian flows .**ub-Riemannian manifolds**When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point "q" of the configuration space manifold "Q", so that the rank of the cometric is less than the dimension of the manifold "Q", one has a

sub-Riemannian manifold .The Hamiltonian in this case is known as a

**sub-Riemannian Hamiltonian**. Every such Hamiltonian uniquely determines the cometric, and vice-versa. This implies that everysub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by theChow-Rashevskii theorem .The continuous, real-valued

Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by:$mathcal\{H\}(x,y,z,p\_x,p\_y,p\_z)=frac\{1\}\{2\}left(\; p\_x^2\; +\; p\_y^2\; ight)$.

$p\_z$ is not involved in the Hamiltonian.

**Poisson algebras**Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of

smooth function s over asymplectic manifold , Hamiltonian mechanics can be formulated on generalcommutative unital realPoisson algebra s. A state is a continuouslinear functional on the Poisson algebra (equipped with some suitable topology) such that for any element "A" of the algebra, "A"² maps to a nonnegative real number.A further generalization is given by

Nambu dynamics .**Charged particle in an electromagnetic field**A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged

particle in anelectromagnetic field. InCartesian coordinates (i.e. $q\_i\; =\; x\_i$), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (inSI Units ):: $mathcal\{L\}\; =\; sum\_i\; frac\{1\}\{2\}\; m\; dot\{x\}\_i^2\; +\; sum\_i\; e\; dot\{x\}\_i\; A\_i\; -\; e\; phi,$

where e is the

electric charge of the particle (not necessarily the electron charge), $phi$ is the electric scalar potential, and the $A\_i$ are the components of themagnetic vector potential (these may be modified through a gauge tranformations).The generalized momenta may be derived by:

: $p\_j\; =\; frac\{partial\; L\}\{\; partial\; dot\{x\}\_j\}\; =\; m\; dot\{x\}\_j\; +\; e\; A\_j.$

Rearranging, we may express the velocities in terms of the momenta, as:

: $dot\{x\}\_j\; =\; frac\{\; p\_j\; -\; e\; A\_j\; \}\{m\}.$

If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:

: $mathcal\{H\}\; =\; sum\_i\; dot\{x\}\_i\; p\_i\; -\; mathcal\{L\}\; =\; sum\_i\; frac\{\; (p\_i\; -\; e\; A\_i)^2\; \}\; \{2\; m\; \}\; +\; e\; phi.$

This equation is used frequently in

quantum mechanics .**Hamilton's Principle Applied to Deformable Bodies**Hamilton's Principle is an important variational principle in elastodynamics. As opposed to a system composed of rigid bodies, deformable bodies have an infinite number of degrees of freedom and occupy continuous regions of space; consequently, the state of the system is described by using continuous functions of space and time. The extended Hamilton Principle for such bodies is given by

: $int\_\{t1\}^\{t2\}\; left\; [\; delta\; W\_e\; +\; delta\; T\; -\; delta\; U\; ight]\; dt\; =\; 0$

where $T$ is the kinetic energy, $U$ is the elastic energy, $W\_e$ is the work done byexternal loads on the body, and $t\_1,\; t\_2$ the initial and final times. If the system is conservative, the work done by external forces may be derived from a scalar potential $V$. In this case,

: $delta\; int\_\{t1\}^\{t2\}\; left\; [\; T\; -\; (U\; +\; V)\; ight]\; dt\; =\; 0$

This is called Hamilton's Principle and it is invariant under coordinate transformations.

**References*** V.I. Arnol'd, "Mathematical Methods of Classical Mechanics", Springer-Verlag (1989), [ISBN 0-387-96890-3]

*Ralph Abraham andJerrold E. Marsden , "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X

* V.I. Arnol'd, V.V. Kozlov and A.I. Neĩshtadt, "Mathematical aspects of classical and celestial mechanics." In: "Encyclopaedia of Mathematical Sciences, Dynamical Systems III" (vol. 3), Springer-Verlag, 1988.

* A. M. Vinogradov , B. A. Kupershmidt " [*http://diffiety.ac.ru/djvu/structures.djvu The structure of Hamiltonian mechanics*] " (djvu ), London Math. Soc. Lect. Notes Ser., 60 (1981), Cambridge Univ. Press, London

* Binney, James, " [*http://www-thphys.physics.ox.ac.uk/users/JamesBinney/CMech_notes.ps Classical Mechanics*] " (PostScript ) [*http://www-thphys.physics.ox.ac.uk/users/JamesBinney/cmech.pdf lecture notes*] (PDF)

* Tong, David, [*http://www.damtp.cam.ac.uk/user/tong/dynamics.html Classical Dynamics*] (Cambridge lecture notes)**ee also***

Hamiltonian (quantum mechanics)

*Lagrangian mechanics

*Canonical transformation

*Classical mechanics

*Dynamical systems

*Quantum mechanics

*Maxwell's equations

*Field theory

*Hamilton–Jacobi equation

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