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Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates

and matching generalized velocities

Abusing the notation, we write the Lagrangian as

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. By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics that would otherwise be even more complicated.

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.

One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinatizations of the same symplectic manifold.

The Hamiltonian is the Legendre transformIn mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: : f and g are then said to be related by a Legendre transformation . Legendre transformat of the Lagrangian:

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:

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 equationsIn mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. In practice the "unknown function" is usua, 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 motionNewton's laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. These laws are fundamental to classical mechanics. Newton first published these laws in Philosophiae Naturalis Principia Mathema.

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

A more geometric way of seeing this is to note we have a fiber bundleIn mathematics, in particular in topology, a fiber bundle is a continuous surjective map π from a topological space E to another topological space B satisfying a further condition making it locally of a particularly simple form. Putting it in intuitive E over timeFor alternate uses of "time", see Time (disambiguation). Time quantifies or measures the interval between events, or the duration of events. Time has long been perceived as a dimension in which each event has a definite (but not necessarily unique) positi, R with the fiberFiber ( American English) or fibre ( International English) is elongated stringy natural, man-made or manufactured material. In the case of natural fibers, they often tie together the parts of natural creatures. Natural vegetable fibers, generally consists E_t, being the position space and the Lagrangian is a function over the jet bundleIn differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Historica JE. Taking the fiberwise Legendre transform of the Lagrangian, we get a function called the Hamiltonian over the dual bundle over time whose fiber at t is the cotangent space T^*E_t, which comes equipped with a natural symplectic form.