View source for Hamilton equations

Ordinary canonical first-order differential equations describing the motion of holonomic mechanical systems acted upon by external forces, as well as describing the extremals of problems of the classical calculus of variations.

Hamilton's equations, established by W. Hamilton [[#References|[1]]], are equivalent to the second-order [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]] (or to the [[Euler equation|Euler equation]] in the classical calculus of variations), in which the unknown magnitudes are the generalized coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462001.png" /> as well as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462002.png" />. Hamilton replaced, in his considerations, the generalized velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462003.png" /> by the generalized momenta

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462004.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462005.png" /> is the [[Lagrange function|Lagrange function]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462006.png" /> is the number of degrees of freedom of the system, and he defined the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

which has since received the name of [[Hamilton function|Hamilton function]] (or Hamiltonian). In the right-hand side of (2) the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462008.png" /> are replaced by the expressions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h0462009.png" /></td> </tr></table>

obtained by solving the equations (1). For dynamical systems, in which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620010.png" /></td> </tr></table>

such a solution always exists.

Hamilton's equations have the canonical form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620012.png" /> denotes the non-potential generalized forces if these are acting on the system. The number of equations (3) is equal to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620013.png" /> of unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620014.png" />.

The order of the system (3) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620015.png" />, which is equal to that of the system of second-order Lagrange equations.

The transition from the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620016.png" /> and the Lagrange function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620017.png" /> to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620018.png" /> and the Hamilton function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046200/h04620019.png" /> by the use of formulas (1) and (2) is given by the [[Legendre transform|Legendre transform]]. The Hamilton equations have certain advantages over the Lagrange equations; hence the important role they play in analytical mechanics. See also [[Hamiltonian system|Hamiltonian system]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.R. Hamilton,   ''Philos. Transact. Roy. Soc. London Ser. A'' , '''1'''  (1835)  pp. 95–144</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR></table>