Hamilton's principle
Hamilton's principle of theoretical mechanics is an extremum. Physical fields and particles take then for a specific size one extremal (ie largest or smallest ) value. This review is called effect, mathematically the effect is a functional. The effect turns out in many cases to be minimal, but only as a "stationary" (ie extremal ). Therefore, the principle is called by some textbook authors, the principle of stationary action. Some authors call the Hamilton's principle and principle of least action, which, however - as noted above - is not accurate.
One example is the Fermat's principle according to which a ray of light in a medium of all possible paths from the start point to the end point passes through the path with the shortest running time.
The Newtonian equations of motion follow at suitably chosen effect from the Hamilton 's principle. However, the Maxwell equations of electrodynamics, the Einstein equations of general relativity can be traced back to a principle the smallest effect.
History
Pierre Maupertuis in 1746 as the first language of a general principle of nature, extremal or optimal begins ( see also Occam's Razor ). Leonhard Euler and Joseph Lagrange clarified in the middle of the eighteenth century that such a principle implies the validity of the Euler -Lagrange equations. The Lagrangian formulation of mechanics is from 1788. 1834 William Hamilton formulated the principle named after him.
Mathematical Description
In the mechanism, the effect is the time integral of a function of time, location and speed, the so-called Lagrange
For example, in Newtonian mechanics, the Lagrangian of a particle of mass moving in the potential, the difference of kinetic and potential energy:
In relativistic mechanics is the Lagrangian of a free particle
Each path, which is traversed from a starting point to an end point in time, the effect assigned to the following value:
Thus, the effect has the dimension of energy times time.
The Hamilton's principle now states that those paths are traversed in the nature of all possible paths that initially, and finally run through, the smallest, more precisely is a static effect. For the physically traversed paths, the first variation of the action vanishes:
Therefore, to satisfy the Euler-Lagrange equation
For example, arise for the non-relativistic motion of a particle in the potential of the Newtonian equations of motion
For a free relativistic particle of momentum, however, is time-independent:
Hamilton's principle for fields
In the field theory, however, the behavior of fields is investigated, ie the way in which they change and interact with their environment.
If in Hamilton's principle
The Lagrangian density
A, we obtain the Hamilton's principle for fields with
With the obvious identification
, the integrand now than
Be written.
It can be seen that this formulation is particularly interesting for the theory of relativity, because is integrated here about the place and the time. Similar to the ordinary Hamilton 's principle can be determined from this modified version of the Lagrange equations for fields.
Related to quantum mechanics
If we develop the quantum mechanics starting from the path integral formalism, it is very clear quickly, so minimizing effect for the description of classical particle trajectories is so effective. This applies namely that the effect of paths that one usually encounter in daily life, very large as measured by the Planck's quantum of action, which is often the case, already been because of the large mass of macroscopic objects. Thus, the exponential function in the integral path, which contains the effect is rapidly oscillating function. The main contribution to the path integral now supply terms, for which the effect is stationary. This is very important to note that only the requirement of stationarity follows and not a requirement of a minimum value. This also provides the appropriate justification for that is usually not checked whether the extreme values that are obtained by minimizing the effects are actually minimum values , because actually you only need extreme values to a classical description to obtain.
Properties
Since the principle of action is independent of the coordinate system used, one can examine the Euler -Lagrange equations in such coordinates that are proportionate to the problem and use spherical coordinates, for example, when it comes to the movement in drehinvarianten gravitational field of the sun. This simplifies the solution of the equation.
In addition, constraints can be easily taken into account when mechanical devices, the free movement of the mass restrict points such as the suspension in a ball pendulum.
But above all, can be used in this formulation of the equations of motion of the Noether theorem proving, which states that every symmetry of the effect of a conserved quantity is one and that, conversely, every conserved quantity includes a symmetry of the action.
The conserved quantities in turn are crucial for whether it is possible to solve the equations of motion by integrals on given functions.