7.4 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Glossary of aerospace engineering | 19/27 | https://en.wikipedia.org/wiki/Glossary_of_aerospace_engineering | reference | science, encyclopedia | 2026-05-05T07:50:15.292303+00:00 | kb-cron |
== L == Lander – spacecraft designed to soft-land intact or almost undamaged on the surface of a celestial body and eventually take-off from it Landing – is the last part of a flight, where an aircraft, or spacecraft returns to the ground. When the flying object returns to water, the process is called alighting, although it is commonly called "landing", "touchdown"a or "splashdown" as well. A normal aircraft flight would include several parts of flight including taxi, takeoff, climb, cruise, descent and landing. Landing gear – is the undercarriage of an aircraft or spacecraft and may be used for either takeoff or landing. For aircraft it is generally needed for both. Also, for aircraft, the landing gear supports the craft when it is not flying, allowing it to take off, land, and taxi without damage. Wheeled landing gear is the most common, with skis or floats needed to operate from snow/ice/water and skids for vertical operation on land. Faster aircraft have retractable undercarriages, which fold away during flight to reduce drag. Lagrangian mechanics – Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle. Lagrangian mechanics defines a mechanical system to be a pair
(
M
,
L
)
{\displaystyle (M,L)}
of a configuration space
M
{\displaystyle M}
and a smooth function
L
=
L
(
q
,
v
,
t
)
{\displaystyle L=L(q,v,t)}
called Lagrangian. By convention,
L
=
T
−
V
,
{\displaystyle L=T-V,}
where
T
{\displaystyle T}
and
V
{\displaystyle V}
are the kinetic and potential energy of the system, respectively. Here
q
∈
M
,
{\displaystyle q\in M,}
and
v
{\displaystyle v}
is the velocity vector at
q
{\displaystyle q}
(
v
{\displaystyle (v}
is tangential to
M
)
.
{\displaystyle M).}
(For those familiar with tangent bundles,
L
:
T
M
×
R
t
→
R
,
{\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}
and
v
∈
T
q
M
)
.
{\displaystyle v\in T_{q}M).}
Given the time instants
t
1
{\displaystyle t_{1}}
and
t
2
,
{\displaystyle t_{2},}
Lagrangian mechanics postulates that a smooth path
x
0
:
[
t
1
,
t
2
]
→
M
{\displaystyle x_{0}:[t_{1},t_{2}]\to M}
describes the time evolution of the given system if and only if
x
0
{\displaystyle x_{0}}
is a stationary point of the action functional
S
[
x
]
=
def
∫
t
1
t
2
L
(
x
(
t
)
,
x
˙
(
t
)
,
t
)
d
t
.
{\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}
If
M
{\displaystyle M}
is an open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
and
t
1
,
{\displaystyle t_{1},}
t
2
{\displaystyle t_{2}}
are finite, then the smooth path
x
0
{\displaystyle x_{0}}
is a stationary point of
S
{\displaystyle {\cal {S}}}
if all its directional derivatives at
x
0
{\displaystyle x_{0}}
vanish, i.e., for every smooth
δ
:
[
t
1
,
t
2
]
→
R
n
,
{\displaystyle \delta :[t_{1},t_{2}]\to \mathbb {R} ^{n},}
δ
S
=
def
d
d
ε
|
ε
=
0
S
[
x
0
+
ε
δ
]
=
0.
{\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}