Hamiltonian mechanics

The Hamiltonian mechanics, named after William Rowan Hamilton, is a branch of classical mechanics. It examines the movement in the phase space. It is the set of pairs of position and momentum values ​​that you can specify when the system under consideration initially free of particles. Then the Hamiltonian function defined by the Hamiltonian equations of motion, such as changing the positions and momenta of the particles, neglecting friction with time.

The equations of motion were given in 1834 by William Rowan Hamilton.

All equations of motion that follow from an action principle can be formulated as a Hamiltonian equations of motion. It has the equivalent Hamiltonian formulation of two decisive advantages: On the one hand, one can show that the motion in phase space is equal area (see Henri Poincaré ). It follows that there are eddies and stagnation points are at the movement in the phase space, similar to the flow of an incompressible fluid. Second, the hamiltonian equations of motion have a large group of transformations, the canonical transformations it sometimes allows to transform separable Hamiltonian equations in others.

We studied them in particular integrable and chaotic movement and uses it in statistical physics.

Particulars

The Hamiltonian of a system of particles is their energy as a function of the phase space. It depends on the ( generalized ) spatial coordinates and of the ( generalized ) coordinates momentum of the particles and may also depend upon time.

The number of coordinates and momenta is called the number of degrees of freedom. The phase space is dimensional.

The Hamiltonian determines the temporal evolution of the particle momenta and Teilchenorte by the Hamiltonian equations of motion,

This is a system of ordinary first order differential equations for the unknown functions of time,

If the Hamiltonian does not depend explicitly on, then the solution curves do not intersect and it goes through each point of the phase space, a solution curve.

With time-based one can perceive the time as an additional degree of freedom associated with pulse and the time-independent Hamiltonian. Therefore, we limit ourselves to time-independent functions. However, the function is not bounded from below and the hypersurface of constant energy is not, as assumed in some considerations, compact.

Particle in the potential

For a particle of mass, the non-relativistic moving in a potential, the Hamiltonian of kinetic and potential energy is composed

The corresponding Hamiltonian equations of motion

Are Newton's equations of motion in a conservative force field,

In particular, the potential of a one-dimensional () harmonic oscillator is the Hooke's spring force in the equation of motion

Causing the web to oscillate about the rest position,

Here, the amplitude and time when this maximum displacement is executed.

Free relativistic particle

For a relativistic, free particle with the energy - momentum relation is the Hamiltonian function

The Hamiltonian equations of motion predict how these speed related to the pulse and that the pulse does not change with time,

When the Hamiltonian is not dependent, as in the examples of the time, the system retains its initial energy of particles, it is then conserved.

Action principle

The Hamiltonian equations of motion follow from the Hamiltonian principle of stationary action. Of all the possible paths in the phase space,

Initially at the time of the start point

And finally to time by the endpoint

Run, the physically traversed path is the one on which the action

Is stationary.

If we consider a one-parameter family of curves

Initially at the time of the start point

And finally to time by the endpoint

Run, so the effect for extremal if there the derivative with respect to vanishes.

We denote this derivative as variation of the action

It is equally

The variation of the location and

The variation of the pulse.

The variation of the action is on the chain rule

We write the second term as a full time derivative and a term, which occurs without a time derivative.

The integral over the total derivative gives the start and end time and disappears, because then disappears, go for it all the curves of the family by the same start and end points. We finally summarize the terms with and together, as is the variation of the action

Thus, the effect is stationary, this integral must vanish for all and all, the initially and finally disappear. This is exactly the case when the factors disappear, with which they occur in the integral,

The effect is thus stationary when the Hamiltonian equations of motion are valid.

Related to the Lagrangian

The Hamilton function with respect to the speeds of the Legendre transform Lagrangian

In this case, the functions are on the right side with the speed meant that is obtained when the definition of the pulses

Solving for the velocities.

If you can invert the definition of momentum and solve for the velocities, the Hamiltonian equations of motion are valid if and only if the Euler -Lagrange equations of the action

Are fulfilled. For the partial derivative of the pulses according to results according to the chain rule and the definition of the pulses

Similarly, the derivation yields according to the local coordinates

The Euler-Lagrange equation states

So the Hamiltonian equations of motion are valid when the Euler -Lagrange equation. Conversely, the Euler -Lagrange equation when the Hamiltonian equations of motion are valid.

For example, hangs at free relativistic particle with Lagrangian

The pulse according to

From the starting speed. Conversely, the speed is therefore the function

Of the pulse. In the above equation for used there is the already mentioned Hamiltonian of the free relativistic particle.

Depends on the Lagrangian does not explicitly on the time, then the Noether theorem states that the energy

To the physical paths retains its initial value. The comparison with the Legendre transformation is that it is the Hamiltonian of the energy is, at which the speeds are to be understood as a function of the pulses,

Poisson bracket

The value of a phase space function changes to paths with time in that it explicitly depends on and the fact that changes the path point

The physically traversed paths satisfy the Hamiltonian equations of motion. Applies to physical paths

With the introduction of Siméon Denis Poisson Poisson bracket of two phase space functions and

Therefore applies

Written with Poisson brackets resembles the formula image of the Hamiltonian equations of motion the heisenberg between equations of motion of quantum mechanics.

Interpreted as coordinate functions, the phase space coordinates have the Poisson brackets

Meet you in quantum mechanics by canonical quantization, the canonical commutation relations.

The Poisson bracket is antisymmetric, linear and satisfies the product rule and the Jacobi identity. For all figures and and all phase space functions

The phase space differentiable functions form a Lie algebra with the Poisson bracket as a Lie product.

Hamiltonian flow

At any ( time-independent ) phase space function is one of the vector field

The phase space functions along the curves derived, which solve the Hamiltonian equations.

The mapping of the initial values ​​of the solution curves to the corresponding Hamiltonian flow.

Symplectic structure

The phase space with its Poisson bracket is a symplectic manifold with the symplectic form

Applied to the to and associated vector fields results in these two form the Poisson bracket of the two functions,

The symplectic form is invariant under every Hamiltonian flow. This implies the following: If initially a two-dimensional surface, where in phase space, then it is mapped over time by the Hamiltonian flow of a phase space function on the surface. The measured with the symplectic form size of the initial surface is consistent with the size at any later time. Hamiltonian flow is equal area,

Since the planar element is invariant, the volume element is invariant under hamiltonschem flow. This finding is Liouville's theorem. The volume of a region of the phase space does not change with Hamiltonian time evolution,

In particular, the range within which the system is initially due to the measurement error remains the same size. From this one can not conclude that initial lack of knowledge does not increase, however. In chaotic motion can initial values ​​, which initially differed only by small measurement errors are distributed over a large area with many small holes like whipped cream. Also whipping cream does not increase their microscopically -determined volume.

Canonical transformation

The Hamilton equations are simplified if the Hamilton function of a variable, for example, not dependent. Then there is a symmetry in front: the Hamiltonian function is invariant under the shift Conversely, if there is a symmetry ( in a neighborhood of a point which is not a fixed point ), the position and momentum variables are chosen such that the Hamiltonian of a variables does not depend. Then simply

Integrable motion

The equations of motion are integrable if the Hamiltonian function depends only on the pulses. Then, the pulses are constant and the derivatives of the Hamiltonian function after the pulses are the time-constant speeds at which increase linearly the coordinates

If, in addition, the phase space surface of constant energy compact, then it is at the coordinates by the angles on a torus, the name the same point to increases again,

The phase space of such an integrable system consists of n-dimensional tori, around which the solution curves of the Hamiltonian equations squirm.

Related to quantum mechanics

Thus, as in mechanics, the Hamiltonian function determines the time evolution, the Hamiltonian determines the time evolution in quantum mechanics. You can get it for many quantum mechanical systems from the Hamiltonian of the corresponding classical system by so-called canonical quantization, by reading the algebraic expression for as a function of operators and which satisfy the canonical commutation relations.

Swell

  • VI Arnol'd, Mathematical Methods of Classical Mechanics, Springer- Verlag ( 1989), ISBN 0-387-96890-3
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