Let
C
=
{
q
(
t
)
|
t
1
≤
t
≤
t
2
}
{\displaystyle C=\{q(t)\,|\,t_{1}\leq t\leq t_{2}\}}
be a trajectory for a particle. Then, the action is given by:
Φ
[
C
]
=
Action of trajectory C
=
∫
t
1
t
2
L
(
q
(
t
)
,
q
˙
(
t
)
,
t
)
d
t
{\displaystyle \Phi [C]={\text{Action of trajectory C}}=\int _{t_{1}}^{t_{2}}L(q(t),{\dot {q}}(t),t)\,dt}
Let a variation of the trajectory be given as:
C
′
=
{
q
′
(
t
)
=
q
(
t
)
+
δ
q
(
t
)
|
t
1
+
δ
t
1
≤
t
≤
t
2
+
δ
t
2
}
{\displaystyle C'=\{q'(t)=q(t)+\delta q(t)\,|\,t_{1}+\delta t_{1}\leq t\leq t_{2}+\delta t_{2}\}}
, then the change in action is given by:
δ
Φ
[
C
]
=
Φ
[
C
′
]
−
Φ
[
C
]
=
∫
t
1
+
δ
t
1
t
2
+
δ
t
2
d
t
L
(
q
(
t
)
+
δ
q
(
t
)
,
q
˙
(
t
)
+
δ
q
˙
(
t
)
)
−
∫
t
1
t
2
d
t
L
(
q
(
t
)
,
q
˙
(
t
)
,
t
)
=
∫
t
1
t
2
[
L
(
q
(
t
)
+
δ
q
(
t
)
,
q
˙
(
t
)
+
δ
q
˙
(
t
)
,
t
)
−
L
(
q
(
t
)
,
q
˙
(
t
)
,
t
)
]
d
t
+
∫
t
2
t
2
L
(
q
s
′
q
˙
s
′
t
)
d
t
−
∫
t
1
t
1
L
(
q
s
′
q
˙
s
′
,
t
)
d
t
=
∫
t
1
t
2
(
∂
L
∂
q
δ
q
+
∂
L
∂
q
˙
δ
q
˙
)
d
t
+
∫
t
2
t
2
+
δ
t
2
L
(
q
′
,
q
˙
′
,
t
)
d
t
−
∫
t
1
t
1
+
δ
t
1
L
(
q
′
,
q
˙
′
,
t
)
d
t
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
δ
q
(
t
)
+
∂
L
∂
q
˙
d
d
t
δ
q
(
t
)
]
+
[
L
Δ
t
]
t
1
t
2
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
−
d
d
t
(
∂
L
∂
q
˙
)
]
δ
q
(
t
)
+
∂
L
∂
q
˙
δ
q
(
t
)
|
t
1
t
2
+
L
Δ
t
|
t
1
t
2
{\displaystyle {\begin{aligned}\delta \Phi [C]&=\Phi [C^{\prime }]-\Phi [C]\\&=\int _{t_{1}+\delta t_{1}}^{t_{2}+\delta t_{2}}dtL(q(t)+\delta q(t),{\dot {q}}(t)+\delta {\dot {q}}(t))-\int _{t_{1}}^{t_{2}}dtL(q(t),{\dot {q}}(t),t)\\&=\int _{t_{1}}^{t_{2}}[L(q(t)+\delta q(t),{\dot {q}}(t)+\delta {\dot {q}}(t),t)-L(q(t),{\dot {q}}(t),t)]\,dt+\,\int _{t_{2}}^{t_{2}}L(q_{s}^{\prime }\,{\dot {q}}_{s}^{\prime }\,t)\,dt\,-\,\int _{t_{1}}^{t_{1}}L(q_{s}^{\prime }\,{\dot {q}}_{s}^{\prime },t)\,dt\\&=\int _{t_{1}}^{t_{2}}\left({\frac {\partial L}{\partial q}}\delta q+{\frac {\partial L}{\partial {\dot {q}}}}\delta {\dot {q}}\right)\,dt+\,\int _{t_{2}}^{t_{2}+\delta t_{2}}L(q^{\prime },{\dot {q}}^{\prime },t)\,dt\,-\,\int _{t_{1}}^{t_{1}+\delta t_{1}}L(q^{\prime },{\dot {q}}^{\prime },t)\,dt\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}\,\delta q(t)+{\frac {\partial L}{\partial {\dot {q}}}}\ {\frac {d}{dt}}\,\delta q(t)\right]+{\bigg [}L\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)\right]\delta q(t)+{\frac {\partial L}{\partial {\dot {q}}}}\delta q(t){\biggr |}_{t_{1}}^{t_{2}}+L\Delta t{\biggr |}_{t_{1}}^{t_{2}}\end{aligned}}}
Defining the total variation of path as
Δ
q
=
q
′
(
t
′
)
−
q
(
t
)
=
δ
q
(
t
)
+
q
˙
(
t
)
δ
t
{\displaystyle \Delta q=q'(t')-q(t)=\delta q(t)+{\dot {q}}(t)\delta t}
and momentum as
∂
L
∂
q
˙
=
p
{\displaystyle {\frac {\partial L}{\partial {\dot {q}}}}=p}
, we get:
Δ
Φ
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
−
d
d
t
∂
L
∂
q
˙
]
δ
q
(
t
)
+
[
p
Δ
q
−
(
p
q
˙
−
L
)
Δ
t
]
t
1
t
2
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
−
d
d
t
∂
L
∂
q
˙
]
δ
q
(
t
)
+
[
p
Δ
q
−
H
Δ
t
]
t
1
t
2
{\displaystyle {\begin{aligned}\Delta \Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-(p{\dot {q}}-L)\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\\end{aligned}}}
Hamilton's action principle
edit
Considering a special class of variation of path that leaves the end-points and terminal times unchanged, ie.
Δ
t
i
=
Δ
q
(
t
i
)
=
0.
{\displaystyle \Delta t_{i}=\Delta q(t_{i})=0.}
For such actions, the change in action functional is given by:
Δ
σ
Φ
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
−
d
d
t
∂
L
∂
q
˙
]
δ
q
(
t
)
{\displaystyle {\begin{aligned}\Delta _{\sigma }\Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)\end{aligned}}}
From Lagrange's equation of motion, it follows that the infinitesimal change in action functional vanishes if the given trajectory is a solution for trajectory of the particle.
Weiss action principle
edit
Using Lagrange's equations of motion, we have the following value for the change in action functional:
Δ
Φ
=
∫
t
1
t
2
d
t
[
∂
L
∂
q
−
d
d
t
∂
L
∂
q
˙
]
δ
q
(
t
)
+
[
p
Δ
q
−
H
Δ
t
]
t
1
t
2
=
[
p
Δ
q
−
H
Δ
t
]
t
1
t
2
{\displaystyle {\begin{aligned}\Delta \Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&={\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\\end{aligned}}}
Hence, Hamilton's action principle can be extended to Weiss action principle as the dynamical trajectory in configuration space is that which only provides end-point contributions to
Δ
Φ
{\displaystyle \Delta \Phi }
.[ 1]