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''Hamiltonian''
 
''Hamiltonian''
  
A function introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classical calculus of variations to represent the [[Euler equation|Euler equation]] in canonical form. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462101.png" /> be the [[Lagrange function|Lagrange function]] of a mechanical system or the integrand in the problem of minimization of the functional
+
A function introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classical calculus of variations to represent the [[Euler equation|Euler equation]] in canonical form. Let $  L ( t, x, \dot{x} ) $
 +
be the [[Lagrange function|Lagrange function]] of a mechanical system or the integrand in the problem of minimization of the functional
  
<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/h046210/h0462102.png" /></td> </tr></table>
+
$$
 +
J ( x)  = \int\limits L ( t, x , \dot{x} )  dt
 +
$$
  
of the classical calculus of variations, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462104.png" />. The Hamilton function is the [[Legendre transform|Legendre transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462105.png" /> with respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462106.png" /> or, in other words,
+
of the classical calculus of variations, where $  x= ( x _ {1} \dots x _ {n} ) $,  
 +
$  \mathop{\rm det}  \| L _ {\dot{x} \dot{x} }  \| \neq 0 $.  
 +
The Hamilton function is the [[Legendre transform|Legendre transform]] of $  L $
 +
with respect to the variables $  \dot{x} $
 +
or, in other words,
  
<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/h046210/h0462107.png" /></td> </tr></table>
+
$$
 +
H ( t, x, p)  = ( p \mid  \dot{x} ) - L ( t, x, \dot{x} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462108.png" /> is expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h0462109.png" /> by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621011.png" /> is the scalar product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621013.png" />. If the Hamilton function is used, the Euler equation
+
where $  \dot{x} $
 +
is expressed in terms of $  p $
 +
by the relation $  p = L _ {\dot{x} }  $
 +
and $  ( p \mid  \dot{x} ) $
 +
is the scalar product of the vectors $  p = ( p _ {1} \dots p _ {n} ) $
 +
and $  \dot{x} $.  
 +
If the Hamilton function is used, the Euler equation
  
<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/h046210/h04621014.png" /></td> </tr></table>
+
$$
 +
-  
 +
\frac{d L _ {x} dot }{dt }
 +
+ L _ {x}  = 0
 +
$$
  
 
(also known as the Lagrange equation (cf. [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]) in problems of classical mechanics) is written in the form of a system of first-order equations:
 
(also known as the Lagrange equation (cf. [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]) in problems of classical mechanics) is written in the form of a system of first-order equations:
  
<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/h046210/h04621015.png" /></td> </tr></table>
+
$$
 +
- \dot{p}  = \
 +
 
 +
\frac{\partial  H }{\partial  x }
 +
,\ \
 +
\dot{x}  = \
 +
 
 +
\frac{\partial  H }{\partial  p }
 +
.
 +
$$
  
 
These equations are called the [[Hamilton equations|Hamilton equations]], the [[Hamiltonian system|Hamiltonian system]] and also the canonical system. The Hamilton–Jacobi equations for the action function (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) can be written in terms of a Hamilton function.
 
These equations are called the [[Hamilton equations|Hamilton equations]], the [[Hamiltonian system|Hamiltonian system]] and also the canonical system. The Hamilton–Jacobi equations for the action function (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) can be written in terms of a Hamilton function.
Line 21: Line 61:
 
In problems of optimal control a Hamilton function is determined as follows. One has to find a minimum of the functional
 
In problems of optimal control a Hamilton function is determined as follows. One has to find a minimum of the functional
  
<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/h046210/h04621016.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { t _ {0} } ^ { {t _ 1 } }
 +
f ^ { 0 } ( t, x, u)  dt
 +
$$
  
 
under the differential constraints
 
under the differential constraints
  
<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/h046210/h04621017.png" /></td> </tr></table>
+
$$
 +
\dot{x}  ^ {i}  = \
 +
f ^ { i } ( t, x, u),
 +
$$
  
for given boundary conditions and with constraints on the control <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621018.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621019.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621020.png" />-dimensional vector of phase coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621021.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621022.png" />-dimensional control vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621023.png" /> is a closed set of admissible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621024.png" />. The Hamilton function in this problem has the form
+
for given boundary conditions and with constraints on the control $  u \in U $.  
 +
Here $  x = ( x  ^ {1} \dots x  ^ {n} ) $
 +
is an $  n $-
 +
dimensional vector of phase coordinates, $  u = ( u  ^ {1} \dots u  ^ {m} ) $
 +
is an $  m $-
 +
dimensional control vector and $  U $
 +
is a closed set of admissible values of $  u $.  
 +
The Hamilton function in this problem has the 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/h046210/h04621025.png" /></td> </tr></table>
+
$$
 +
H ( t, x, \psi , u )  = \
 +
\psi _ {0} f ^ { 0 } ( t, x, u ) +
 +
\sum _ {i = 1 } ^ { n }
 +
\psi _ {i} f ^ { i } ( t, x, u ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621027.png" /> are conjugate variables (Lagrange multipliers, momenta) analogous to the canonical variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621028.png" /> mentioned above. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621029.png" /> is a minimum in the above problem and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621031.png" /> may then be considered as equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621032.png" />), then
+
where $  \psi _ {0} = \textrm{ const } \leq  0 $,  
 +
and $  \psi _ {1} \dots \psi _ {n} $
 +
are conjugate variables (Lagrange multipliers, momenta) analogous to the canonical variables $  p _ {i} $
 +
mentioned above. If $  ( x _ {0} , u _ {0} ) $
 +
is a minimum in the above problem and $  \psi _ {0} \neq 0 $(
 +
$  \psi _ {0} $
 +
may then be considered as equal to $  - 1 $),  
 +
then
  
<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/h046210/h04621033.png" /></td> </tr></table>
+
$$
 +
H ( t, x _ {0} ( t), p ( t), u _ {0} ( t))  = \
 +
\left . ( p \mid  f  ) \right | _ {x _ {0,} u _ {0}  } -
 +
\left . f ^ { 0 } \right | _ {x _ {0}  , u _ {0} } ,
 +
$$
  
 
where
 
where
  
<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/h046210/h04621034.png" /></td> </tr></table>
+
$$
 +
- \dot{p}  = \
 +
\left .
 +
\frac{\partial  H }{\partial  x }
 +
 
 +
\right | _ {x _ {0}  , u _ {0} } .
 +
$$
  
 
The expression obtained for the Hamilton function has the same structure as in the classical calculus of variations. According to the [[Pontryagin maximum principle|Pontryagin maximum principle]], the Euler equations for the optimal control problem may be written using a Hamilton function as follows:
 
The expression obtained for the Hamilton function has the same structure as in the classical calculus of variations. According to the [[Pontryagin maximum principle|Pontryagin maximum principle]], the Euler equations for the optimal control problem may be written using a Hamilton function as follows:
  
<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/h046210/h04621035.png" /></td> </tr></table>
+
$$
 +
\dot{x}  ^ {i}  = \
  
The optimal control <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621036.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046210/h04621037.png" /> should constitute a maximum of the Hamiltonian:
+
\frac{\partial  H }{\partial  \psi _ {i} }
 +
,\ \
 +
\dot \psi  _ {i}  = \
 +
-
 +
\frac{\partial  H }{\partial  x  ^ {i} }
 +
,\ \
 +
i = 1 \dots n.
 +
$$
  
<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/h046210/h04621038.png" /></td> </tr></table>
+
The optimal control  $  u $
 +
for each  $  t $
 +
should constitute a maximum of the Hamiltonian:
 +
 
 +
$$
 +
H ( t, x, \psi , u)  = \
 +
\max _ {v \in U }  H ( t, x, \psi , v).
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  V.G. Boltayanskii,  R.V. Gamkrelidze,  E.F. Mishchenko,  "The mathematical theory of optimal processes" , Wiley  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  V.G. Boltayanskii,  R.V. Gamkrelidze,  E.F. Mishchenko,  "The mathematical theory of optimal processes" , Wiley  (1962)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.E. Crouch,  A.J. van der Schaft,  "Variational Hamiltonian control systems" , ''Lect. notes in control and inform. science'' , '''101''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.E. Crouch,  A.J. van der Schaft,  "Variational Hamiltonian control systems" , ''Lect. notes in control and inform. science'' , '''101''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR></table>

Latest revision as of 19:43, 5 June 2020


Hamiltonian

A function introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classical calculus of variations to represent the Euler equation in canonical form. Let $ L ( t, x, \dot{x} ) $ be the Lagrange function of a mechanical system or the integrand in the problem of minimization of the functional

$$ J ( x) = \int\limits L ( t, x , \dot{x} ) dt $$

of the classical calculus of variations, where $ x= ( x _ {1} \dots x _ {n} ) $, $ \mathop{\rm det} \| L _ {\dot{x} \dot{x} } \| \neq 0 $. The Hamilton function is the Legendre transform of $ L $ with respect to the variables $ \dot{x} $ or, in other words,

$$ H ( t, x, p) = ( p \mid \dot{x} ) - L ( t, x, \dot{x} ), $$

where $ \dot{x} $ is expressed in terms of $ p $ by the relation $ p = L _ {\dot{x} } $ and $ ( p \mid \dot{x} ) $ is the scalar product of the vectors $ p = ( p _ {1} \dots p _ {n} ) $ and $ \dot{x} $. If the Hamilton function is used, the Euler equation

$$ - \frac{d L _ {x} dot }{dt } + L _ {x} = 0 $$

(also known as the Lagrange equation (cf. Lagrange equations (in mechanics)) in problems of classical mechanics) is written in the form of a system of first-order equations:

$$ - \dot{p} = \ \frac{\partial H }{\partial x } ,\ \ \dot{x} = \ \frac{\partial H }{\partial p } . $$

These equations are called the Hamilton equations, the Hamiltonian system and also the canonical system. The Hamilton–Jacobi equations for the action function (cf. Hamilton–Jacobi theory) can be written in terms of a Hamilton function.

In problems of optimal control a Hamilton function is determined as follows. One has to find a minimum of the functional

$$ J = \int\limits _ { t _ {0} } ^ { {t _ 1 } } f ^ { 0 } ( t, x, u) dt $$

under the differential constraints

$$ \dot{x} ^ {i} = \ f ^ { i } ( t, x, u), $$

for given boundary conditions and with constraints on the control $ u \in U $. Here $ x = ( x ^ {1} \dots x ^ {n} ) $ is an $ n $- dimensional vector of phase coordinates, $ u = ( u ^ {1} \dots u ^ {m} ) $ is an $ m $- dimensional control vector and $ U $ is a closed set of admissible values of $ u $. The Hamilton function in this problem has the form

$$ H ( t, x, \psi , u ) = \ \psi _ {0} f ^ { 0 } ( t, x, u ) + \sum _ {i = 1 } ^ { n } \psi _ {i} f ^ { i } ( t, x, u ), $$

where $ \psi _ {0} = \textrm{ const } \leq 0 $, and $ \psi _ {1} \dots \psi _ {n} $ are conjugate variables (Lagrange multipliers, momenta) analogous to the canonical variables $ p _ {i} $ mentioned above. If $ ( x _ {0} , u _ {0} ) $ is a minimum in the above problem and $ \psi _ {0} \neq 0 $( $ \psi _ {0} $ may then be considered as equal to $ - 1 $), then

$$ H ( t, x _ {0} ( t), p ( t), u _ {0} ( t)) = \ \left . ( p \mid f ) \right | _ {x _ {0,} u _ {0} } - \left . f ^ { 0 } \right | _ {x _ {0} , u _ {0} } , $$

where

$$ - \dot{p} = \ \left . \frac{\partial H }{\partial x } \right | _ {x _ {0} , u _ {0} } . $$

The expression obtained for the Hamilton function has the same structure as in the classical calculus of variations. According to the Pontryagin maximum principle, the Euler equations for the optimal control problem may be written using a Hamilton function as follows:

$$ \dot{x} ^ {i} = \ \frac{\partial H }{\partial \psi _ {i} } ,\ \ \dot \psi _ {i} = \ - \frac{\partial H }{\partial x ^ {i} } ,\ \ i = 1 \dots n. $$

The optimal control $ u $ for each $ t $ should constitute a maximum of the Hamiltonian:

$$ H ( t, x, \psi , u) = \ \max _ {v \in U } H ( t, x, \psi , v). $$

References

[1] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947)
[2] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)

Comments

References

[a1] H. Goldstein, "Classical mechanics" , Addison-Wesley (1950)
[a2] P.E. Crouch, A.J. van der Schaft, "Variational Hamiltonian control systems" , Lect. notes in control and inform. science , 101 , Springer (1987)
[a3] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
How to Cite This Entry:
Hamilton function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hamilton_function&oldid=47168
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article