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# Hamiltonian mechanics

Hamiltonian mechanics was invented in 1833 by Hamilton. Like Lagrangian mechanics, it is a re-formulation of classical mechanics.

Hamiltonian mechanics can be formulated on its own, using symplectic spaces, and not refer to any prior concepts of force or Lagrangian mechanics. See the section on its mathematical formulation for this. For the first part of this article, we will show how it has arisen historically from the study of Lagrangian mechanics.

In Lagrangian mechanics, the equations of motion are dependent on generalized coordinates {qj | j=1,...N} and matching generalized velocities {qj′ | j=1,...N}. Abusing the notation, we write the Lagrangian as L(qj, qj′, t), with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta. For each generalized velocity, there is one corresponding conjugate momentum, defined as:

.

In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.

The Hamiltonian is the Legendre transform of the Lagrangian:

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If the transformation equations defining the generalized coordinates are independent of t, it can be shown that H is equal to the total energy E = T + V.

Each side in the definition of H produces a differential:

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Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:

Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order. However, the steps leading to the equations of motion are more onerous than in Lagrangian mechanics - beginning with the generalized coordinates and the Lagrangian, we must calculate the Hamiltonian, express each generalized velocity in terms of the conjugate momenta, and replace the generalized velocities in the Hamiltonian with the conjugate momenta. All in all, there is little labor saved from solving a problem with Hamiltonian mechanics rather than Lagrangian mechanics. Ultimately, it will produce the same solution as Lagrangian mechanics and Newton's laws of motion.

The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical mechanics.

## Mathematical formalism

If we have a symplectic space, which comes naturally equipped with a Poisson bracket and a smooth function H over it, then H defines a one-parameter family of transformations with respect to time and this is called Hamiltonian mechanics. In particular, . So, if we have a probability distribution, ρ, then . This is called Liouville's theorem. Every smooth function, G, over the symplectic manifold generates a one-parameter family of symplectomorphisms and if {G,H}=0, then G is conserved and the symplectomorphisms are symmetry transformations.

### Poisson algebras

There's a further generalization we can make. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element of the algebra, A, A^2 maps to a nonnegative real number.

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