Let n 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors, and consider the functional of n dependent variables x1 rt1 L[x] = [*^* dt L(t,x,x), _x(to) = xo, to x(t1) = X1, . . ., xn: where x = (x1,...,xn) and x = (x1,...,xn), and where xk = dxk for dt k = 1,..., n. dt - n ƏL ƏL Σ It k=1 ,xn). The A system of n particles of masses m₁, ..., mn lie on a line with positions x1, ..., xn, respectively in a force field of smooth potential V(t,x1,. motion of the particles from time t = to to time t = t₁ > to is given by a stationary path of the Lagrangian functional C: L[x] = rt1 to dt L(t,x,x), x(0) =X0, x(t1)=X1, where L = T - V and T is the total kinetic energy T = = n Στις k=1
Let n 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors, and consider the functional of n dependent variables x1 rt1 L[x] = [*^* dt L(t,x,x), _x(to) = xo, to x(t1) = X1, . . ., xn: where x = (x1,...,xn) and x = (x1,...,xn), and where xk = dxk for dt k = 1,..., n. dt - n ƏL ƏL Σ It k=1 ,xn). The A system of n particles of masses m₁, ..., mn lie on a line with positions x1, ..., xn, respectively in a force field of smooth potential V(t,x1,. motion of the particles from time t = to to time t = t₁ > to is given by a stationary path of the Lagrangian functional C: L[x] = rt1 to dt L(t,x,x), x(0) =X0, x(t1)=X1, where L = T - V and T is the total kinetic energy T = = n Στις k=1
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![Let n 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors,
and consider the functional of n dependent variables x1
rt1
· · ·, xn:
<[x] = ["
dt L(t,x,x), x(to)
=
✗0,
x(t1) = x1,
to
where x = (x1,...,xn) and x = (x1,xn), and where ik
k
= 1,
n.
dt
-
n
k=1
ᎥᏞ
Ik
kJXk
=
ƏL
It
dæk for
=
dt
A system of n particles of masses m₁,
=
---7
mn lie on a line with positions x1,
xn, respectively in a force field of smooth potential V (t, x1,...,xn). The
motion of the particles from time t to to time t = t₁ > to is given by a
stationary path of the Lagrangian functional C:
L[x] =
1
dt L(t,x,x), x(0) =x0, x(t)=X1,
where LT - V and T is the total kinetic energy
T =
n 1
k=1
2
mark.
Using the above first-integral, show that, if V is independent of t, the total
energy E=T+V of the particle is a constant of the motion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2Ffd09de70-e8b6-4908-977d-7afb436d6515%2Fsevdhj_processed.png&w=3840&q=75)
Transcribed Image Text:Let n 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors,
and consider the functional of n dependent variables x1
rt1
· · ·, xn:
<[x] = ["
dt L(t,x,x), x(to)
=
✗0,
x(t1) = x1,
to
where x = (x1,...,xn) and x = (x1,xn), and where ik
k
= 1,
n.
dt
-
n
k=1
ᎥᏞ
Ik
kJXk
=
ƏL
It
dæk for
=
dt
A system of n particles of masses m₁,
=
---7
mn lie on a line with positions x1,
xn, respectively in a force field of smooth potential V (t, x1,...,xn). The
motion of the particles from time t to to time t = t₁ > to is given by a
stationary path of the Lagrangian functional C:
L[x] =
1
dt L(t,x,x), x(0) =x0, x(t)=X1,
where LT - V and T is the total kinetic energy
T =
n 1
k=1
2
mark.
Using the above first-integral, show that, if V is independent of t, the total
energy E=T+V of the particle is a constant of the motion.
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