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 [1], are equivalent to the second-order Lagrange equations (in mechanics) (or to the Euler equation in the classical calculus of variations), in which the unknown magnitudes are the generalized coordinates $ q _ {i} $ as well as the $ \dot{q} _ {i} = d q _ {i} / d t $. Hamilton replaced, in his considerations, the generalized velocities $ \dot{q} _ {i} $ by the generalized momenta

$$ \tag{1 } p _ {i} = \ \frac{\partial L }{\partial \dot{q} _ {i} } ,\ \ i = 1 \dots n, $$

where $ L ( q _ {i} , \dot{q} _ {i} , t) $ is the Lagrange function, $ n $ is the number of degrees of freedom of the system, and he defined the function

$$ \tag{2 } H ( q _ {i} , p _ {i} , t) = \ \sum _ {i = 1 } ^ { n } p _ {i} \dot{q} _ {i} - L, $$

which has since received the name of Hamilton function (or Hamiltonian). In the right-hand side of (2) the variables $ \dot{q} _ {i} $ are replaced by the expressions

$$ \dot{q} _ {i} = \ \phi _ {i} ( q _ {s} , p _ {s} , t), $$

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

$$ \mathop{\rm det} \left \| \frac{\partial ^ {2} L }{\partial \dot{q} _ {i} \partial \dot{q} _ {j} } \right \| \neq 0, $$

such a solution always exists.

Hamilton's equations have the canonical form

$$ \tag{3 } \frac{dq _ {i} }{dt} = \ \frac{\partial H }{\partial p _ {i} } ,\ \ \frac{dp _ {i} }{dt} = \ - \frac{\partial H }{\partial q _ {i} } + Q _ {i} ^ {*} ,\ \ i = 1 \dots n. $$

Here $ Q _ {i} ^ {*} $ denotes the non-potential generalized forces if these are acting on the system. The number of equations (3) is equal to the number $ 2n $ of unknowns $ q _ {i} , p _ {i} $.

The order of the system (3) is $ 2n $, which is equal to that of the system of second-order Lagrange equations.

The transition from the variables $ q _ {i} , \dot{q} _ {i} , t $ and the Lagrange function $ L $ to the variables $ q _ {i} , p _ {i} , t $ and the Hamilton function $ H $ by the use of formulas (1) and (2) is given by the 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.

References

[1]	W.R. Hamilton, Philos. Transact. Roy. Soc. London Ser. A , 1 (1835) pp. 95–144

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)