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 .
V
(
x
1
,
...
,
x
n
)
{\ displaystyle V (x_ {1}, \ dots, x_ {n})}
k
{\ displaystyle k}
α
{\ displaystyle \ alpha}
V
(
α
x
1
,
...
,
α
x
n
)
=
α
k
V
(
x
1
,
...
,
x
n
)
{\ 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 :
l
{\ displaystyle l}
l
′
{\ displaystyle l '}
t
{\ displaystyle t}
t
′
{\ displaystyle t '}
t
′
t
=
(
l
′
l
)
1
-
k
2
{\ 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 :
v
{\ displaystyle v}
E.
{\ displaystyle E}
L.
{\ displaystyle L}
v
′
v
=
(
l
′
l
)
k
2
E.
′
E.
=
(
l
′
l
)
k
L.
′
L.
=
(
l
′
l
)
1
+
k
2
{\ 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
L.
→
L.
′
=
ξ
L.
{\ displaystyle {\ mathcal {L}} \ to {\ mathcal {L}} '= \ xi {\ mathcal {L}}}
L.
{\ displaystyle {\ mathcal {L}}}
L.
=
1
2
m
(
d
x
d
t
)
2
-
V
(
x
)
{\ 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
α
{\ displaystyle \ alpha}
β
{\ displaystyle \ beta}
L.
′
=
1
2
m
α
2
β
2
(
d
x
d
t
)
2
-
α
k
V
(
x
)
{\ 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
β
=
α
1
-
k
2
{\ 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”.
α
{\ displaystyle \ alpha}
α
1
-
k
2
{\ 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.
V
=
m
G
z
∼
z
1
{\ displaystyle V = mgz \ sim z ^ {1}}
t
′
/
t
=
(
l
′
/
l
)
1
/
2
⇒
t
∝
l
{\ 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 .
V
=
m
G
l
ϕ
2
∼
ϕ
2
{\ displaystyle V = mgl \ phi ^ {2} \ sim \ phi ^ {2}}
t
′
/
t
=
1
{\ 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 .
V
=
-
G
m
M.
r
-
1
∼
r
-
1
{\ displaystyle V = -GmMr ^ {- 1} \ sim r ^ {- 1}}
t
′
/
t
=
(
l
′
/
l
)
3
/
2
⇔
(
t
′
/
t
)
2
=
(
l
′
/
l
)
3
{\ 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).
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