Effect (physics)
Physical size  

Surname  effect  
Formula symbol  

In theoretical physics, the effect is a physical quantity with the dimension energy times time or length times momentum . So it has the same dimension as the angular momentum .
The effect is a functional that distinguishes the physically traversed paths in the number of conceivable paths. The equations of motion of the physically traversed paths state that if the start and end point in phase space are fixed, the effect of the physical path assumes a local extreme value among all conceivable paths . This condition is called Hamilton's principle or principle of least action .
Effect of a point particle
In classical mechanics, the effect of every twofold differentiable path that a point particle traverses from a starting point to an end point over time , orders the value of the integral
to. In Newton's mechanics, the Lagrangian function of a particle of mass moving in potential is the difference between kinetic and potential energy as a function of time , place and speed ,
In the integrand of the effect , the location of the orbit at the time and its time derivative are used. The integral of this linked function of time is the effect of the orbit .
Compared to the effect of all other twofold differentiable paths, which initially run through and finally through , the effect of the physical path is minimal because its equation of motion
is the EulerLagrange equation of the effect .
Example: harmonic oscillator
For example is
the Lagrangian function of a harmonic oscillator with mass and the spring constant .
The physical orbits satisfy the EulerLagrange equation, according to which the Euler derivative is at all times
disappears if one intervenes for the location that is currently being traversed and for the time derivative of the orbit .
The associated physical orbits thus fulfill
 .
Any solution to this equation is of form
 ,
where is the amplitude of the oscillation and its phase shift.
At the moment she is passing through the place and now the place .
Their effect is the integral
 .
The integral can with the addition theorem
can be easily evaluated, but that is irrelevant for our considerations,
 .
On any other track
 ,
which in the meantime deviates a little from ,, the effect differs primarily in um
In the first term, partial integration rolls the derivation of without boundary terms (because it disappears there) upwards with a minus sign and results in the negative of the second term for all changes in the meantime
So the effect of every physical path is stationary under all interim changes.
Significance in theoretical physics
The effect as a functional of tracks or fields is also fundamental for
 the relativistic mechanics
 the Maxwell equations of electrodynamics
 the Einstein equations of general relativity
 the standard model of elementary interactions .
literature
 L. Landau / JM Lifschitz : Textbook of theoretical physics (Volume 1): Mechanik , Verlag Harri Deutsch, reprint of the 14th, corrected edition 1997 (2007), ISBN 9783817113262
 Florian Scheck : Theoretical Physics 1: Mechanics , Springer, 2007, ISBN 9783540713777
Web links
 Norbert Dragon, keywords and additions to mathematical methods of physics (PDF; 1.9 MB) Chapter 13