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

p = -∂˙q .                                 (5.22)
Adopting this momentum to express the degrees of freedom previously expressed by q˙ using a Legendre map (Definition 53) results in
H (q,p, t) = ⟨p, ˙q⟩ − L(q,q˙, t),                        (5 .23)
thus defining a Hamiltonian corresponding to a given Lagrangian. Contrary to the Lagrangian, which is a function on the tangent bundle T M , the Hamiltonian is a function on the co-tangent bundle  ∗
T M , which is called the phase space.
H (q,p ) : T∗M →  ℝ                             (5.24)
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 T∗M of an n dimensional base manifold into a symplectic manifold (Definition 82) of dimension 2n . 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

 [p,p ] = 0,                                (5 .25a)

[ [q,q ]] = 0,                                (5.25b)
 pi,qj = δji,                                (5 .25c)

called canonical coordinates.

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

˙p = − ----                                 (5.26)
˙q =  ∂p                                    (5.27)
called Hamilton’s equations. Together they define a vector field vH   (Definition 62) along with associated integral curves γH (t)   (Definition 63), which all taken together define the phase flow Φt  (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 H can be expressed utilizing the geometrical structure (compare Equation 4.162) inherent to the phase space.

vH =  IdH  = γ˙H                               (5.28)
This vector field provides the opportunity to express the evolution of the system under investigation as the parameter t changes. In case of a function f , which measures a quantity of interest in the phase space
f(q,p ) ∈ 𝔽 (T∗M ) : T ∗M → ℝ,                        (5 .29)
the evolution with regard to the parameter t due to the Hamiltonian and its own dependence on the parameter is then simply given by
˙f = vH (f) + ∂f-,                              (5 .30)
recalling that a vector field is a mapping of the form 𝔽(T∗M  ) → 𝔽 (T ∗M )  (see Equation 4.109), which is usually given explicitly using the Poisson bracket (Definition 86) in the form
f˙=  df-=  {f,H } + ∂f-,                           (5 .31)
     dt             ∂t
which indicates how the geometry of the system, defined by the Hamiltonian, is responsible for its evolution.