### 5.3 Hamiltonian Formalism

While the Lagrangian formalism already provides extensive capabilities to handle elaborate problems,
further reﬁnement is possible resulting in the so called phase space, which also makes concepts
available, which can even be carried beyond the conﬁnes of classical mechanics.

Motivated by the aim to ﬁnd a simpler, more symmetric expression of the equation of motion as
given in Equation 5.14, it is fruitful to utilize the fact from the settings of the Lagrangian in
Equation 5.10 and Equation 5.12, from which follows that the momentum is expressible
as

Adopting this momentum to express the degrees of freedom previously expressed by using a
Legendre map (Deﬁnition 53) results in
thus
deﬁning a Hamiltonian corresponding to a given Lagrangian. Contrary to the Lagrangian, which is a
function on the tangent bundle , the Hamiltonian is a function on the co-tangent bundle ,
which is called the phase space.
This
is the deﬁning expression in the Hamiltonian formalism. Furthermore, the cotangent bundle is
equipped with a symplectic structure (see Deﬁnition 81), which turns the cotangent bundle
of an dimensional base manifold into a symplectic manifold (Deﬁnition 82)
of dimension . Among the great strengths of the Hamilton formalism is the fact that
Hamilton’s equations can be expressed using this geometric structure inherent to the phase
space.
The symplectic nature of the manifold used ensures that it is always possible, as asserted by
Deﬁnition 84, to ﬁnd coordinates such that

called canonical coordinates.

The equations of motion as observed in the Lagrangian case take on the simpler, almost symmetric
form

called
Hamilton’s equations. Together they deﬁne a vector ﬁeld (Deﬁnition 62) along with
associated integral curves (Deﬁnition 63), which all taken together deﬁne the phase ﬂow
(Deﬁnition 67). This phase ﬂow is a symplectomorphism (Deﬁnition 83), thus leaving the
symplectic structure (Deﬁnition 81) of the manifold invariant.
The vector ﬁeld due to the Hamiltonian can be expressed utilizing the geometrical structure (compare
Equation 4.162) inherent to the phase space.

This
vector ﬁeld provides the opportunity to express the evolution of the system under investigation as the
parameter changes. In case of a function , which measures a quantity of interest in the phase
space
the
evolution with regard to the parameter due to the Hamiltonian and its own dependence on the
parameter is then simply given by
recalling that a vector ﬁeld is a mapping of the form (see Equation 4.109),
which is usually given explicitly using the Poisson bracket (Deﬁnition 86) in the form
which indicates how the geometry of the system, deﬁned by the Hamiltonian, is responsible for its
evolution.