Hamiltonian Mechanics Article Index for
Hamiltonian 		Website Links For
Hamiltonian Mechanics 		 

Information About

Hamiltonian Mechanics
  CATEGORIES ABOUT HAMILTONIAN MECHANICS
   
fundamental physics concepts
   
classical mechanics
   
hamiltonian mechanics
   
dynamical systems
   
symplectic geometryfundamental physics concepts
   
classical mechanics
   
hamiltonian mechanics
   
dynamical systems
   
symplectic geometry
   
classical mechanics
   
theoretical physics
   
dynamical systems
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at classical mechanics. Generally, these equations do not provide a more convenient way of solving a particular problem. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.


SIMPLIFIED OVERVIEW OF USES


For a closed system the sum of the kinetic and potential energy in the system is represented by a set of Differential Equation s known as the ''Hamilton equations'' for that system. Hamiltonians can be used to describe such simple systems as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time. Hamiltonians can also be employed to model the energy of other more complex dynamic systems such as planetary orbits and in quantum mechanics. The Hamiltonian MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007

:\dot p = - rac{\partial H}{\partial q}
:\dot q =~~ rac{\partial H}{\partial p}

In the above equations, the dot denotes the ordinary derivative with respect to time of the functions
''p = p(t)'' (called momenta) and ''q = q(t)'' (called coordinates), taking values in some vector space,
and ''H = H(p,q,t)'' is the so-called Hamiltonian , or (scalar valued) Hamiltonian function. Thus, a little bit more explicitly, one should write

: rac{\mathrm d}{\mathrm dt}p(t) = - rac{\partial}{\partial q}H(p(t),q(t),t)
: rac{\mathrm d}{\mathrm dt}q(t) =~~ rac{\partial}{\partial p}H(p(t),q(t),t)

and specify the domain of values in which the parameter ''t'' ("''time''") varies.

For a quite detailed derivation of these equations from Lagrangian Mechanics , see below.


Basic physical interpretation, mnemotechnics


The simplest interpretation of the equations is as follows:
The Hamiltonian ''H'' represents the Energy of the physical system,
which is the sum of Kinetic and Potential Energy , traditionally denoted ''T'' & ''V'' respectively:

: H = T + V , \quad T = rac{p^2}{2m} , \quad V = V(q) = V(x).


Using Hamilton's equations


#First write out L = T - V. Express T and V as though you were going to use Lagrange's equation.
#Calculate the momenta by differentiating the Lagrangian with respect to velocity.
#Express the velocities in terms of the momenta by inverting the expressions in step (2).
#Calculate the Hamiltonian using the usual definition, H = \sum_i p_i {\dot q_i} - L. Substitute for the velocities using the results in step (3).
#Apply Hamilton's equations.


Notes


Hamilton's equations are appealing in view of their beautiful simplicity and (slightly '' Broken '') Symmetry . They have been analyzed under any imaginable angle of view, from basic physics up to Symplectic Geometry . A lot is known about solutions of these equations, yet the exact general case solution of the Equations Of Motion cannot be given explicitly for a system of more than two massive point particles. The finding of Conserved Quantities plays an important role in the search for solutions or information about their nature. In models with an infinite number of Degrees Of Freedom , this is of course even more complicated.
An interesting and promising area of research is the study of Integrable System s, where an infinite number of independent conserved quantities can be constructed.


DERIVING HAMILTON'S EQUATIONS


We can derive Hamilton's equations by looking at how the Lagrangian changes as you change the time and the positions and velocities of particles.


\mathrm{d} L = \sum_i \left ( rac{\partial L}{\partial q_i} \mathrm{d} q_i + rac{\partial L}{\partial {\dot q_i}} \mathrm{d} {\dot q_i} ight ) + rac{\partial L}{\partial t} \mathrm{d}t


Now the generalized momenta were defined as p_i = rac{\partial L}{\partial {\dot q_i}} and Lagrange's equations tell us that

rac{\mathrm{d}}{\mathrm{d}t} rac{\partial L}{\partial {\dot q_i}} - rac{\partial L}{\partial q_i} = F_i

where F_i is the generalized force. We can rearrange this to get

rac{\partial L}{\partial q_i} = {\dot p}_i - F_i

and substitute the result into the variation of the Lagrangian


\mathrm{d}L = \sum_i \left \left( {\dot p}_i - F_i ight) \mathrm{d} q_i + p_i \mathrm{d} {\dot q_i} ight + rac{\partial L}{\partial t}\mathrm{d}t


We can rewrite this as


\mathrm{d} L = \sum_i \left \left ( {\dot p}_i - F_i ight ) \mathrm{d}q_i + \mathrm{d}\left ( p_i {\dot q_i} ight ) - {\dot q_i} \mathrm{d} p_i ight + rac{\partial L}{\partial t}\mathrm{d}t


and rearrange again to get


\mathrm{d} \left ( \sum_i p_i {\dot q_i} - L ight ) = \sum_i \left \left ( F_i-{\dot p}_i ight ) \mathrm{d} q_i + {\dot q_i} \mathrm{d}p_i ight - rac{\partial L}{\partial t}\mathrm{d}t


The term on the left-hand side is just the Hamiltonian that we have defined before, so we find that


\mathrm{d} H = \sum_i \left \left ( F_i-{\dot p}_i ight ) \mathrm{d} q_i + {\dot q_i} \mathrm{d} p_i ight - rac{\partial L}{\partial t}\mathrm{d}t = \sum_i \left [ rac{\partial H}{\partial q_i} \mathrm{d} q_i +
rac{\partial H}{\partial p_i} \mathrm{d} p_i ight ] + rac{\partial H}{\partial t}\mathrm{d}t


where the second equality holds because of the definition of the partial derivatives.


AS A REFORMULATION OF LAGRANGIAN MECHANICS


Starting with Lagrangian Mechanics , the Equations Of Motion are based on Generalized Coordinates

  :<math>\left\{\, \dot{q} J J 1, \ldots ,N \, ight\} </math>


If we have a Probability Distribution , ρ, then (since the phase space velocity ( {\dot p_i} , {\dot q _i} ) has zero divergence, and probability is conserved) its convective derivative can be shown to be zero and so

: rac{\partial}{\partial t} ho = - \{\, ho ,H\,\}.

This is called Liouville's Theorem . Every Smooth Function ''G'' over the Symplectic Manifold generates a one-parameter family of Symplectomorphism s and if { ''G'', ''H'' } = 0, then ''G'' is conserved and the symplectomorphisms are Symmetry Transformation s.

A Hamiltonian may have multiple conserved quantities ''G''''i''. If the symplectic manifold has dimension 2''n'' and there are ''n'' functionally independent conserved quantities ''G''''i'' which are in involution (i.e., { ''G''''i'', ''G''''j'' } = 0), then the Hamiltonian is Liouville Integrable . The Liouville–Arnol'd Theorem says that locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism in a new Hamiltonian with the conserved quantities ''G''''i'' as coordinates; the new coordinates are called ''action-angle coordinates''. The transformed Hamiltonian depends only on the ''G''''i'', and hence the equations of motion have the simple form
: \dot{G}_i = 0, \qquad \dot{ arphi}_i = F(G),
for some function ''F'' (Arnol'd et al., 1988). There is an entire field focusing on small deviations from integrable systems governed by the KAM Theorem .

The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are Chaotic ; concepts of measure, completeness, integrability and stability are poorly defined. At this time, the study of Dynamical Systems is primarily qualitative, and not a quantitative science.


RIEMANNIAN MANIFOLDS


An important special case consists of those Hamiltonians that are Quadratic Form s, that is, Hamiltonians that can be written as

:H(q,p)= rac{1}{2} \langle p,p angle_q


If one considers a Riemannian Manifold or a Pseudo-Riemannian Manifold , so that one has an invertible, non-degenerate Metric , then the cometric is given simply as the inverse of the metric. The solutions to the Hamilton–Jacobi Equation s for this Hamiltonian are then the same as the Geodesic s on the manifold. In particular, the Hamiltonian Flow in this case is the same thing as the Geodesic Flow . The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on Geodesic s. See also Geodesics As Hamiltonian Flows .


SUB-RIEMANNIAN MANIFOLDS


When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point ''q'' of the configuration space manifold ''Q'', so that the Rank of the cometric is less than the dimension of the manifold ''Q'', one has a Sub-Riemannian Manifold .

The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice-versa. This implies that every .

The continuous, real-valued Heisenberg Group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by

:H(x,y,z,p_x,p_y,p_z)= rac{1}{2}\left( p_x^2 + p_y^2 ight).

p_z is not involved in the Hamiltonian.


POISSON ALGEBRAS


Hamiltonian systems can be generalized in various ways. Instead of simply looking at the Algebra of Smooth Function s over a Symplectic Manifold , Hamiltonian mechanics can be formulated on general Commutative Unital Real Poisson Algebra s. A State is a Continuous Linear Functional on the Poisson algebra (equipped with some suitable Topology ) such that for any element ''A'' of the algebra, ''A''2 maps to a nonnegative real number.

A further generalization is given by Nambu Dynamics .


CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD


A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an Electromagnetic field. In Cartesian Coordinates (i.e. q_i = x_i ), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units ):

: L = \sum_i frac{1}{2} m \dot{x}_i^2 + \sum_i e \dot{x}_i A_i - e \phi,

where e is the Electric Charge of the particle (not necessarily the electron charge), \phi is the Electric Scalar Potential , and the A_i are the components of the Magnetic Vector Potential (these may be modified through a Gauge Tranformations ).

The generalized momenta may be derived by:

: p_j = rac{\partial L}{ \partial \dot{x}_j} = m \dot{x}_j + e A_j.

Rearranging, we may express the velocities in terms of the momenta, as:

: \dot{x}_j = rac{ p_j - e A_j }{m}.

If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:

: H = \sum_i \dot{x}_i p_i - L = \sum_i rac{ (p_i - e A_i)^2 } {2 m } + e \phi.

This equation is used frequently in Quantum Mechanics .


REFERENCES




SEE ALSO