# Hamilton's equations of motion

## Hamilton's equations of motion

The motion of a mechanical system may be described by a set of first-order ordinary differential equations known as Hamilton's equations. Because of their remarkably symmetrical form, they are often referred to as the canonical equations of motion of a system. They are equivalent to Lagrange's equations, but the fact that they are of first order and highly symmetrical makes them advantageous for general discussions of the motion of systems. *See* Lagrange's equations

Hamilton's equations can be derived from Lagrange's equations. Let the coordinates of the system be *q*_{j} (*j* = 1, 2, . . . , *f*), and let the dynamical description of the system be given by the lagrangian *L*(*q*, …, *t*), where *q* denotes all the coordinates and a dot denotes total time derivative. Lagrange's equations are then given by Eq. (1). The momentum *p*_{j} canonically conjugate to *q*_{j} is defined by Eq. (2).

The hamiltonian *H* is defined by Eq. (3). Then Hamilton's canonical equations are Eqs. (4).

As they stand, Hamilton's equations are no easier to integrate directly than Lagrange's. Hamilton's equations are of great advantage in more general discussions, and they permit the making of canonical transformations which can lead to simplifications. *See* Canonical transformations

The hamiltonian function *H* of classical mechanics is used to form the quantum-mechanical hamiltonian operator.

## Hamilton's equations of motion

[′ham·əl·tənz i¦kwā·zhənz əv ′mō·shən]*q̇*

_{j }= ∂

*H*/∂

*p*

_{j },

*ṗ*

_{j }= -∂

*H*/∂

*q*

_{j }; here

*q*

_{j }(

*j*= 1, 2,…) are generalized coordinates of the system,

*p*

_{j }is the momentum conjugate to

*q*

_{j }, and

*H*is the Hamiltonian. Also known as canonical equations of motion.