Hamilton's equations

In physics and mathematics, Hamilton's equations is the set of differential equations

${\displaystyle {\dot {p}}=-{\frac {\partial H}{\partial q}}}$

${\displaystyle {\dot {q}}=~~{\frac {\partial H}{\partial p}}}$

that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science.

More precisely...

In the above equations, the dot denotes the ordinary derivative of the functions p = p(t) (called momentum) and q = q(t) (called coordinates), taking values in some vector space, and H = H(p,q,t) is the so-called Hamiltonian, or (scalar valued) Hamiltonian function. Thus, a little bit more explicitly, one should write

${\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} t}}(t)=-{\frac {\partial H}{\partial q}}(p(t),q(t),t)}$ 

${\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}(t)=~~{\frac {\partial H}{\partial p}}(p(t),q(t),t)}$ 

and precise the ___domain of values the parameter t (the "time") varies in.

For a quite detailed derivation of these equations from Lagrangian mechanics, see the article on Hamiltonian mechanics.

Basic physical interpretation, mnemotechnics

The most simple interpretation of the equations is as follows: The Hamiltonian H represents the energy of the physical system, which is the sum of kinetic and potential energy, traditionally denoted T resp. V:

H = T + V , T = p²/2m , V = V(q) = V(x)

i.e. T is a quadratic function of the momenta p, while V depends only on the coordinates q which we identify here with the positions x of the particle(s).

The second equation then just gives the link between the velocities v = dx/dt and the momenta:

v = p / m , or: p = m v ,

while the first equation is Newton's law,

dp/dt = - grad V(x) , or: m a = F

where a = dv/dt is the acceleration and F = - grad V the force deriving from potential energy.

All of this is not only a superficial analogy, but applies in a very general and profound sense.

Further reading

Hamilton's equations are appealing in view of their beautiful simplicity and (slightly broken) symmetry.

They have been analyzed under any imaginable angle of view, from basic physics up to symplectic geometry.

A lot is known about solutions of these equations, yet not even for a system describing only three massive point particles, the exact solution of the equations of motion can be given explicitly for the general case.

The finding of conserved quantities plays an important rôle 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 of integrable systems, where an infinite number of independent conserved quantities can be constructed.

See also

• Hamiltonian mechanics
• Lagrangian mechanics
• Classical mechanics
• Dynamical systems
• Quantum mechanics
• Maxwell's equations
• Field theory

References

• L. Landau, L. D. Lifshitz: Theoretical physics, vol.1: Mechanics.
• Goldstein, H. Classical Mechanics, second edition, pp.16 (Addison-Wesley, 1980)