Hamilton's equations

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In physics and mathematics, Hamilton's equations is the set of differential equations created by the Irish mathematician William Rowan Hamilton (1805 – 1865). 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.

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[edit] Description

For a closed system the sum of the kinetic and potential energy in the system is represented by a set of differential equations known as the Hamiltonian 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. [1]

[edit] More precisely

\dot p = -\frac{\partial H}{\partial q}
\dot q =~~\frac{\partial H}{\partial p}

In the above equations, the dot denotes the ordinary derivative of the functions p = p(t) (called momentum) 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

\frac{\mathrm dp}{\mathrm dt}(t) = -\frac{\partial H}{\partial q}(p(t),q(t),t)
\frac{\mathrm dq}{\mathrm dt}(t) =~~\frac{\partial H}{\partial p}(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 the article on Hamiltonian mechanics.

[edit] Basic physical interpretation, mnemotechnics

The most simple 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 resp. V:

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

[edit] 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 ( \frac{\partial L}{\partial q_i} \mathrm{d} q_i + \frac{\partial L}{\partial {\dot q_i}} \mathrm{d} {\dot q_i} \right ) + \frac{\partial L}{\partial t} \mathrm{d}t

Now the generalized momenta were defined as p_i = \frac{\partial L}{\partial {\dot q_i}} and Lagrange's equations tell us that \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial {\dot q_i}} - \frac{\partial L}{\partial q_i} = F_i where Fi is the generalized force. We can rearrange this to get \frac{\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 \right) \mathrm{d} q_i + p_i \mathrm{d} {\dot q_i} \right] + \frac{\partial L}{\partial t}\mathrm{d}t

We can rewrite this as

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

and rearrange again to get

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

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

[edit] Generalization through Poisson bracket

The Hamilton's equations above work perfectly for classical mechanics, but not for the quantum mechanics, since the differential equations assume that we can find out the position and momentum of the particle simultaneously at any point in time. The equations can be further generalized to apply to quantum mechanics as well as to classical mechanics through the use of the Poisson algebra over p and q. In this case, the more general form of the Hamilton's equation reads

\frac{\mathrm{d}f}{\mathrm{d}t} = \{f, H\} + \frac{\partial f}{\partial t}

where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating Poisson bracket without resorting to differential equations, see Lie algebra, as Poisson bracket is just a different name for the Lie bracket in a Poisson algebra.

In fact, this more algebraic approach not only allows us to use probability distributions and wavefunctions for q and p, but also provides more power the classical setting, in particular by helping to find the conserved quantities.

[edit] Using Hamilton's Equations

  1. First write out L = T - V. Express T and V as though you were going to use Lagrange's equation.
  2. Calculate the momenta by differentiating the Lagrangian.
  3. Express the velocities in terms of the momenta by inverting the expressions in step (2).
  4. 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).
  5. Apply Hamilton's equations.

[edit] Further reading

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 systems, where an infinite number of independent conserved quantities can be constructed.

[edit] See also

[edit] References

  • L. Landau, L. D. Lifshitz: Theoretical physics, vol.1: Mechanics.
  • H. Goldstein, Classical Mechanics, second edition, pp.16 (Addison-Wesley, 1980)
  1. ^ The Hamiltonian MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007
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