Effect (physics)

Physical size
Surname	effect
Formula symbol	${\displaystyle S}$
Size and unit system unit dimension SI Js = kg · m 2 · s -1 M · L 2 · T −1

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 . ${\displaystyle S}$

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 ${\displaystyle S}$ ${\displaystyle \Gamma \colon t\mapsto x(t)\,}$ ${\displaystyle t}$ ${\displaystyle {\underline {x}}=x(t_{1})}$ ${\displaystyle {\overline {x}}=x(t_{2})}$

${\displaystyle S[\Gamma ]=\int _{t_{1}}^{t_{2}}L\!\left(t,x(t),{\frac {\mathrm {d} x}{\mathrm {d} t}}(t)\right)\,\mathrm {d} t}$

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 , ${\displaystyle L(t,x,v)}$ ${\displaystyle m}$ ${\displaystyle V(t,x)}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle v}$

${\displaystyle L(t,x,v)={\frac {1}{2}}\,m\,v^{2}-V(t,x)\ .}$

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 . ${\displaystyle S[\Gamma ]}$ ${\displaystyle x}$ ${\displaystyle x(t)}$ ${\displaystyle t}$ ${\displaystyle v}$ ${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}(t)}$ ${\displaystyle \Gamma \colon t\mapsto x(t)}$

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 ${\displaystyle {\underline {x}}}$ ${\displaystyle {\overline {x}}}$

${\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+\partial _{x}V(t,x)=0}$

is the Euler-Lagrange equation of the effect . ${\displaystyle S}$

Example: harmonic oscillator

For example is

${\displaystyle L(t,x,v)={\frac {1}{2}}mv^{2}-{\frac {1}{2}}m\omega ^{2}x^{2}}$

the Lagrangian function of a harmonic oscillator with mass and the spring constant . ${\displaystyle m}$ ${\displaystyle \kappa =m\omega ^{2}}$

The physical orbits satisfy the Euler-Lagrange equation, according to which the Euler derivative is at all times ${\displaystyle t}$

${\displaystyle {\frac {\partial L}{\partial x}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial v}}=-m\left(\omega ^{2}x+{\frac {\mathrm {d} }{\mathrm {d} t}}v\right)}$

disappears if one intervenes for the location that is currently being traversed and for the time derivative of the orbit . ${\displaystyle x}$ ${\displaystyle x(t)}$ ${\displaystyle t}$ ${\displaystyle v}$ ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}x(t)}$

The associated physical orbits thus fulfill ${\displaystyle L}$ ${\displaystyle t\mapsto x(t)}$

${\displaystyle -m\left({\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}x(t)+\omega ^{2}x(t)\right)=0}$.

Any solution to this equation is of form

${\displaystyle \Gamma _{A,\alpha }\colon t\mapsto x(t)=A\cos(\omega t-\alpha )}$,

where is the amplitude of the oscillation and its phase shift. ${\displaystyle A}$ ${\displaystyle \alpha }$

At the moment she is passing through the place and now the place . ${\displaystyle t_{1}}$ ${\displaystyle {\underline {x}}=A\cos(\omega t_{1}-\alpha )}$ ${\displaystyle t_{2}}$ ${\displaystyle {\overline {x}}=A\cos(\omega t_{2}-\alpha )}$

Their effect is the integral

${\displaystyle S[\Gamma _{A,\alpha }]=\int _{t_{1}}^{t_{2}}\mathrm {d} t\,{\frac {1}{2}}m\,A^{2}\,\omega ^{2}{\bigl (}\sin ^{2}(\omega t-\alpha )-\cos ^{2}(\omega t-\alpha ){\bigr )}}$.

The integral can with the addition theorem

${\displaystyle \cos ^{2}\beta -\sin ^{2}\beta =\cos(2\beta )}$

can be easily evaluated, but that is irrelevant for our considerations,

${\displaystyle S[\Gamma _{A,\alpha }]=-\int _{t_{1}}^{t_{2}}\mathrm {d} t\,{\frac {1}{2}}m\,A^{2}\,\omega ^{2}\cos 2(\omega t-\alpha )={\frac {1}{4}}m\,A^{2}\omega {\bigl (}\sin 2(\omega t_{2}-\alpha )-\sin 2(\omega t_{1}-\alpha ){\bigr )}}$.

On any other track

${\displaystyle \Gamma _{A,\alpha }+\delta \colon t\mapsto A\cos(\omega t-\alpha )+\delta (t)}$,

which in the meantime deviates a little from ,, the effect differs primarily in um ${\displaystyle \delta (t)}$ ${\displaystyle \Gamma _{A,\alpha }}$ ${\displaystyle \delta (t_{1})=\delta (t_{2})=0}$ ${\displaystyle \delta }$

${\displaystyle \delta S[\Gamma _{A,\alpha },\delta ]=S[\Gamma _{A,\alpha }+\delta ]-S[\Gamma _{A,\alpha }]=\int _{t_{1}}^{t_{2}}\mathrm {d} t\,A\,m\,\omega {\bigl (}-\sin(\omega t-\alpha ){\dot {\delta }}(t)-\omega \cos(\omega t-\alpha )\delta (t){\bigr )}\ .}$

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 ${\displaystyle {\dot {\delta }}}$ ${\displaystyle \delta }$ ${\displaystyle \sin(\omega t-\alpha )}$ ${\displaystyle \delta (t)}$

${\displaystyle \delta S=\int _{t_{1}}^{t_{2}}\mathrm {d} t\,A\,m\,\omega ^{2}{\bigl (}\cos(\omega t-\alpha )\delta (t)-\cos(\omega t-\alpha )\delta (t){\bigr )}=0\ .}$

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 978-3-8171-1326-2
• Florian Scheck : Theoretical Physics 1: Mechanics , Springer, 2007, ISBN 978-3-540-71377-7

Web links

• Norbert Dragon, keywords and additions to mathematical methods of physics (PDF; 1.9 MB) Chapter 13