In physics and mathematics, Hamilton's equations is the set of differential equations A Superconductor demonstrating the Meissner Effect. ... Euclid, detail from The School of Athens by Raphael. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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 Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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. Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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: Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Kinetic energy (also called vis viva, or living force) is energy possessed by a body by virtue of its motion. ... Potential energy is stored energy. ...

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

Generalization through Poisson bracket

The Hamilton's equations above work perfectly for classical mechanics, but not for the quantum mechanics, since the differential equations assume that we can find out the position and momentum of the particle simultaneously at any point in time. The equations can be further generalized to apply to quantum mechanics as well as to classical mechanics through the use of the Poisson algebra over p and q. In this case, the more general form of the Hamilton's equation reads It has been suggested that this article or section be merged with Newtonian mechanics. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ...

where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating Poisson bracket without resorting to differential equations, see Lie algebra, as Poisson bracket is just a different name for the Lie bracket in a Poisson algebra. In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ...

In fact, this more algebraic approach not only allows us to use probability distributions and wavefunctions for q and p, but also provides more power the classical setting, in particular by helping to find the conserved quantities. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |ψ(x)|2 is the... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

Further reading

Hamilton's equations are appealing in view of their beautiful simplicity and (slightly broken) symmetry. Broken symmetry is a concept used in mathematics and physics when an object breaks either rotational symmetry or translational symmetry. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

They have been analyzed under any imaginable angle of view, from basic physics up to symplectic geometry. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

A lot is known about solutions of these equations, yet the exact general case solution of the equations of motion cannot be given explicitly for a system of more than two massive point particles. In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ...

The finding of conserved quantities plays an important role in the search for solutions or information about their nature. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

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. The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...

See also

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

Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Field theory (mathematics), the theory of the algebraic concept of field. ... In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ...

References

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