13.4.4 Hamiltonian Mechanics

The Lagrangian formulation of mechanics is the most convenient for
determining a state transition equation for a collection of bodies.
Once the kinetic and potential energies are determined, the remaining
efforts are straightforward computation of derivatives and algebraic
manipulation. Hamiltonian mechanics provides an alternative
formulation that is closely related to the Lagrangian. Instead of
expressing second-order differential constraints on an -dimensional
C-space, it expresses first-order constraints on a -dimensional
phase space. This idea should be familiar from Section
13.2. The new phase space considered here is an example of
a *symplectic manifold*, which has many important properties, such
as being orientable and having an even number of dimensions
[39]. The standard phase vector is defined as
; however, instead of , variables will be
introduced and denoted as . Thus, a transformation exists between
and . The variables are related to the
configuration variables through a special function over the phase
space called the *Hamiltonian*.
Although the Hamiltonian formulation usually does not help in the
determination of
, it is covered here because its
generalization to optimal control problems is quite powerful. This
generalization is called Pontryagin's minimum principle and is
covered in Section 15.2.3. In the context of mechanics, it
provides a general expression of energy conservation laws, which aids
in proving many theoretical results [39,397].

The relationship between
and can be obtained by
using the *Legendre transformation* [39,397]. Consider
a real-valued function of two variables,
. Its
*total differential* [508] is

in which

Consider constructing a total differential that depends on and , instead of and . Let be a function of and defined as

The total differential of is

(13.188) |

Using (13.185) to express , this simplifies to

(13.189) |

The and variables are now interpreted as

which appear to be a kind of inversion of (13.186). This idea will be extended to vector form to arrive the Hamiltonian formulation.

Assume that the dynamics do not depend on the particular time (the
extension to time-varying dynamics is not difficult; see
[39,397]). Let
be the Lagrangian
function defined (13.129). Let
represent a
*generalized momentum* vector (or *adjoint variables*), which
serves the same purpose as in (13.185). Each is
defined as

In some literature, is instead denoted as because it can also be interpreted as a vector of Lagrange multipliers. The

and can be interpreted as the total energy of a conservative system [397]. This is a vector-based extension of (13.187) in which and replace and , respectively. Also, and are the vector versions of and , respectively.

Considered as a function of and only, the total differential of is

Using (13.192), can be expressed as

(13.194) |

The terms all cancel by using (13.191), to obtain

(13.195) |

Using (13.118),

(13.196) |

This implies that

Equating (13.197) and (13.193) yields equations called

for each from to . These equations are analogous to (13.190).

Hamilton's equations are equivalent to the Euler-Lagrange equation. Extremals in both cases yield equivalent differential constraints. The difference is that the Lagrangian formulation uses and the Hamiltonian uses . The Hamiltonian results in first-order partial differential equations. It was assumed here that the dynamics are time-invariant and the motions occur in a conservative field. In this case, , which corresponds to conservation of total energy. In the time-varying case, the additional equation appears along with Hamilton's equations. As stated previously, Hamilton's equations are primarily of interest in establishing basic results in theoretical mechanics, as opposed to determining the motions of particular systems. For example, the Hamiltonian is used to establish Louisville's theorem, which states that phase flows preserve volume, implying that a Hamiltonian system cannot be asymptotically stable [39]. Asymptotic stability is covered in Section 15.1.1. Pontryagin's minimum principle, an extension of Hamilton's equations to optimal control theory, is covered in 15.2.3.

Steven M LaValle 2020-08-14