Mechanical similarity

The mechanical similarity is a concept in classical theoretical mechanics of Lagrangian mechanics . With the help of the mechanical similarity, without having to solve the equations of motion , the basic mechanical quantities of different trajectories can be related to one another in a conservative force field .

The prerequisite for the applicability of the concept is a scale-invariant potential from which the force field emerges.

statement

Let be a scale-invariant potential of degree ; d. H. for whatever applies . ${\displaystyle V(x_{1},\dots ,x_{n})}$ ${\displaystyle k}$ ${\displaystyle \alpha }$ ${\displaystyle \textstyle V(\alpha x_{1},\dots ,\alpha x_{n})=\alpha ^{k}V(x_{1},\dots ,x_{n})}$

Then it follows for two trajectories in this potential for each size of the dimension of the coordinates or and each size of the dimension time or : ${\displaystyle l}$ ${\displaystyle l'}$ ${\displaystyle t}$ ${\displaystyle t'}$

${\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {k}{2}}}}$

The corresponding relations for the speeds , the energies and the angular momentum are derived from this : ${\displaystyle v}$ ${\displaystyle E}$ ${\displaystyle L}$

${\displaystyle {\frac {v'}{v}}=\left({\frac {l'}{l}}\right)^{\frac {k}{2}}\qquad {\frac {E'}{E}}=\left({\frac {l'}{l}}\right)^{k}\qquad {\frac {L'}{L}}=\left({\frac {l'}{l}}\right)^{1+{\frac {k}{2}}}}$

Derivation

The Lagrangian equations are invariant under a scaling , where is the Lagrangian of the system. In classical mechanics, the following applies ${\displaystyle {\mathcal {L}}\to {\mathcal {L}}'=\xi {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$

${\displaystyle {\mathcal {L}}={\frac {1}{2}}m\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}-V(x)}$

If you scale all coordinates with the factor and all times with the factor , then with the assumption of a scale-invariant potential applies ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle {\mathcal {L}}'={\frac {1}{2}}m{\frac {\alpha ^{2}}{\beta ^{2}}}\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}-\alpha ^{k}V(x)}$

and thus the invariance of the equations of motion for

${\displaystyle \beta =\alpha ^{1-{\frac {k}{2}}}}$.

Scaling the coordinate-like quantities by a factor therefore requires scaling the time-like quantities by a factor in order to obtain “the same physics”. ${\displaystyle \alpha }$ ${\displaystyle \textstyle \alpha ^{1-{\frac {k}{2}}}}$

Applications

In classical physics , there are three well-known applications of mechanical similarity:

• For free fall it is . The law of fall follows from this : The fall time of a body is proportional to the square root of the height of fall. ${\displaystyle V=mgz\sim z^{1}}$ ${\displaystyle t'/t=(l'/l)^{1/2}\Rightarrow t\propto {\sqrt {l}}}$
• For the harmonic oscillator is . From this follows the pendulum law : The period of the pendulum does not depend on its deflection . ${\displaystyle V=mgl\phi ^{2}\sim \phi ^{2}}$ ${\displaystyle t'/t=1}$
• For the law of gravitation is . From this follows Kepler's third law : The squares of the orbital times of the planets behave like the cubes of their major semiaxes . ${\displaystyle V=-GmMr^{-1}\sim r^{-1}}$ ${\displaystyle t'/t=(l'/l)^{3/2}\Leftrightarrow (t'/t)^{2}=(l'/l)^{3}}$

literature

• LD Landau and EM Lifshitz: Mechanics . 3. Edition. Butterworth-Heinemann, Oxford 1976, ISBN 978-0-7506-2896-9 , pp. 22-24 (English).