@Hullzie16
I wasn't sure if you'd be able to read this so I highlighted the bits I thought were important. Or if theres a way to attache the worksheet I'll do that.
restart;
with(LinearAlgebra);
L := Vector[column](3, [1, 1, 1]);
m := Vector[column](3, [1, 1, 1]);
g = 9.81;
[1]
[ ]
L := [1]
[ ]
[1]
[1]
[ ]
m := [1]
[ ]
[1]
g = 9.81
;
phi := (j, k) -> piecewise(j <> k, 1, j = k, 0);
phi := proc (j, k) options operator, arrow, function_assign;
piecewise(j <> k, 1, j = k, 0) end proc
phi(1, 2);
Theta(t)[1];
diff(Theta(t), t)[2];
m[1];
L[1];
diff(Theta(t), t, t)[2];
1
Theta(t)[1]
/ d \
|--- Theta(t)|[2]
\ dt /
1
1
/ 2 \
| d |
|---- Theta(t)|[2]
| 2 |
\ dt /
for q to 3 do
ode(q) := add(g*m[k]*L[q]*sin(Theta(t)[q]) + m[k]*add(L[i]*L[q]*diff(Theta(t), t)[i]*diff(Theta(t), t)[q]*sin(Theta(t)[q] - Theta(t)[i]), i = 1 .. k) + m[k]*L[q]^2*diff(Theta(t), t, t)[q] + m[k]*add(L[i]*L[q]*diff(Theta(t), t)[i]*diff(Theta(t), t)[q]*sin(Theta(t)[q] - Theta(t)[i])*(diff(Theta(t), t)[q] - diff(Theta(t), t)[i]), i = 1 .. k) + m[k]*add(phi(i, q)*L[i]*L[q]*diff(Theta(t), t)[i]*diff(Theta(t), t)[q]*sin(Theta(t)[i] - Theta(t)[q])*diff(Theta(t), t, t)[i], i = 1 .. k), k = q .. 3) = 0;
end do;
/ 2 \
| d | / d
ode(1) := 3 g sin(Theta(t)[1]) + 3 |---- Theta(t)|[1] + 2 |---
| 2 | \ dt
\ dt /
\ / d \
Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[2]) + 2
/ \ dt /
/ d \ / d \
|--- Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1]
\ dt / \ dt /
// d \ / d \ \ /
- Theta(t)[2]) ||--- Theta(t)|[1] - |--- Theta(t)|[2]| - 2 |
\\ dt / \ dt / / \
d \ / d \
--- Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1]
dt / \ dt /
/ 2 \
| d | / d \ / d
- Theta(t)[2]) |---- Theta(t)|[2] + |--- Theta(t)|[3] |---
| 2 | \ dt / \ dt
\ dt /
\ / d \
Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[3]) + |--- Theta(t)|[3]
/ \ dt /
/ d \ // d
|--- Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[3]) ||--- Theta(t)
\ dt / \\ dt
\ / d \ \ / d \ / d \
|[1] - |--- Theta(t)|[3]| - |--- Theta(t)|[3] |--- Theta(t)|[1]
/ \ dt / / \ dt / \ dt /
/ 2 \
| d |
sin(Theta(t)[1] - Theta(t)[3]) |---- Theta(t)|[3] = 0
| 2 |
\ dt /
/ d \ / d
ode(2) := 2 g sin(Theta(t)[2]) - 2 |--- Theta(t)|[2] |--- Theta(t)
\ dt / \ dt
/ 2 \
\ | d |
|[1] sin(Theta(t)[1] - Theta(t)[2]) + 2 |---- Theta(t)|[2] - 2
/ | 2 |
\ dt /
/ d \ / d \
|--- Theta(t)|[1] |--- Theta(t)|[2] sin(Theta(t)[1]
\ dt / \ dt /
// d \ / d \ \ /
- Theta(t)[2]) ||--- Theta(t)|[2] - |--- Theta(t)|[1]| + 2 |
\\ dt / \ dt / / \
d \ / d \
--- Theta(t)|[1] |--- Theta(t)|[2] sin(Theta(t)[1]
dt / \ dt /
/ 2 \
| d | / d \ / d
- Theta(t)[2]) |---- Theta(t)|[1] + |--- Theta(t)|[3] |---
| 2 | \ dt / \ dt
\ dt /
\ / d \
Theta(t)|[2] sin(Theta(t)[2] - Theta(t)[3]) + |--- Theta(t)|[3]
/ \ dt /
/ d \ // d
|--- Theta(t)|[2] sin(Theta(t)[2] - Theta(t)[3]) ||--- Theta(t)
\ dt / \\ dt
\ / d \ \ / d \ / d \
|[2] - |--- Theta(t)|[3]| - |--- Theta(t)|[3] |--- Theta(t)|[2]
/ \ dt / / \ dt / \ dt /
/ 2 \
| d |
sin(Theta(t)[2] - Theta(t)[3]) |---- Theta(t)|[3] = 0
| 2 |
\ dt /
/ d \ / d \
ode(3) := g sin(Theta(t)[3]) - |--- Theta(t)|[3] |--- Theta(t)|[1
\ dt / \ dt /
/ d \ / d
] sin(Theta(t)[1] - Theta(t)[3]) - |--- Theta(t)|[3] |--- Theta(t
\ dt / \ dt
/ 2 \
\ | d | /
)|[2] sin(Theta(t)[2] - Theta(t)[3]) + |---- Theta(t)|[3] - |
/ | 2 | \
\ dt /
d \ / d \
--- Theta(t)|[1] |--- Theta(t)|[3] sin(Theta(t)[1]
dt / \ dt /
// d \ / d \ \ / d
- Theta(t)[3]) ||--- Theta(t)|[3] - |--- Theta(t)|[1]| - |---
\\ dt / \ dt / / \ dt
\ / d \ /
Theta(t)|[2] |--- Theta(t)|[3] sin(Theta(t)[2] - Theta(t)[3]) |
/ \ dt / \
/ d \ / d \ \ / d \ / d
|--- Theta(t)|[3] - |--- Theta(t)|[2]| + |--- Theta(t)|[1] |---
\ dt / \ dt / / \ dt / \ dt
/ 2 \
\ | d |
Theta(t)|[3] sin(Theta(t)[1] - Theta(t)[3]) |---- Theta(t)|[1] +
/ | 2 |
\ dt /
/ d \ / d \
|--- Theta(t)|[2] |--- Theta(t)|[3] sin(Theta(t)[2]
\ dt / \ dt /
/ 2 \
| d |
- Theta(t)[3]) |---- Theta(t)|[2] = 0
| 2 |
\ dt /
NULL;
;
odesys := ode(1), ode(2), ode(3);
ICs := Theta(0)[1] = Pi/2, Theta(0)[2] = Pi/2, Theta(0)[3] = Pi/2, diff(Theta(0), t)[1] = 0, diff(Theta(0), t)[2] = 0, diff(Theta(0), t)[3] = 0;
/ 2 \
| d | / d
odesys := 3 g sin(Theta(t)[1]) + 3 |---- Theta(t)|[1] + 2 |---
| 2 | \ dt
\ dt /
\ / d \
Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[2]) + 2
/ \ dt /
/ d \ / d \
|--- Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1]
\ dt / \ dt /
// d \ / d \ \ /
- Theta(t)[2]) ||--- Theta(t)|[1] - |--- Theta(t)|[2]| - 2 |
\\ dt / \ dt / / \
d \ / d \
--- Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t)[1]
dt / \ dt /
/ 2 \
| d | / d \ / d
- Theta(t)[2]) |---- Theta(t)|[2] + |--- Theta(t)|[3] |---
| 2 | \ dt / \ dt
\ dt /
\ / d \
Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[3]) + |--- Theta(t)|[3]
/ \ dt /
/ d \ // d
|--- Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[3]) ||--- Theta(t)
\ dt / \\ dt
\ / d \ \ / d \ / d \
|[1] - |--- Theta(t)|[3]| - |--- Theta(t)|[3] |--- Theta(t)|[1]
/ \ dt / / \ dt / \ dt /
/ 2 \
| d |
sin(Theta(t)[1] - Theta(t)[3]) |---- Theta(t)|[3] = 0, 2 g sin(
| 2 |
\ dt /
/ d \ / d \
Theta(t)[2]) - 2 |--- Theta(t)|[2] |--- Theta(t)|[1] sin(Theta(t
\ dt / \ dt /
/ 2 \
| d | / d \
)[1] - Theta(t)[2]) + 2 |---- Theta(t)|[2] - 2 |--- Theta(t)|[1
| 2 | \ dt /
\ dt /
/ d \ // d
] |--- Theta(t)|[2] sin(Theta(t)[1] - Theta(t)[2]) ||--- Theta(t
\ dt / \\ dt
\ / d \ \ / d \ / d \
)|[2] - |--- Theta(t)|[1]| + 2 |--- Theta(t)|[1] |--- Theta(t)|
/ \ dt / / \ dt / \ dt /
/ 2 \
| d | / d
[2] sin(Theta(t)[1] - Theta(t)[2]) |---- Theta(t)|[1] + |---
| 2 | \ dt
\ dt /
\ / d \
Theta(t)|[3] |--- Theta(t)|[2] sin(Theta(t)[2] - Theta(t)[3]) +
/ \ dt /
/ d \ / d \
|--- Theta(t)|[3] |--- Theta(t)|[2] sin(Theta(t)[2]
\ dt / \ dt /
// d \ / d \ \ / d
- Theta(t)[3]) ||--- Theta(t)|[2] - |--- Theta(t)|[3]| - |---
\\ dt / \ dt / / \ dt
/
\ / d \ |
Theta(t)|[3] |--- Theta(t)|[2] sin(Theta(t)[2] - Theta(t)[3]) |
/ \ dt / |
\
2 \
d | / d \ /
---- Theta(t)|[3] = 0, g sin(Theta(t)[3]) - |--- Theta(t)|[3] |
2 | \ dt / \
dt /
d \ / d
--- Theta(t)|[1] sin(Theta(t)[1] - Theta(t)[3]) - |--- Theta(t)
dt / \ dt
\ / d \
|[3] |--- Theta(t)|[2] sin(Theta(t)[2] - Theta(t)[3])
/ \ dt /
/ 2 \
| d | / d \ / d \
+ |---- Theta(t)|[3] - |--- Theta(t)|[1] |--- Theta(t)|[3] sin(
| 2 | \ dt / \ dt /
\ dt /
// d \
Theta(t)[1] - Theta(t)[3]) ||--- Theta(t)|[3]
\\ dt /
/ d \ \ / d \ / d \
- |--- Theta(t)|[1]| - |--- Theta(t)|[2] |--- Theta(t)|[3] sin(
\ dt / / \ dt / \ dt /
// d \
Theta(t)[2] - Theta(t)[3]) ||--- Theta(t)|[3]
\\ dt /
/ d \ \ / d \ / d \
- |--- Theta(t)|[2]| + |--- Theta(t)|[1] |--- Theta(t)|[3] sin(
\ dt / / \ dt / \ dt /
/ 2 \
| d | / d \
Theta(t)[1] - Theta(t)[3]) |---- Theta(t)|[1] + |--- Theta(t)|[
| 2 | \ dt /
\ dt /
/ 2
/ d \ | d
2] |--- Theta(t)|[3] sin(Theta(t)[2] - Theta(t)[3]) |---- Theta(t
\ dt / | 2
\ dt
\
|
)|[2] = 0
|
/
1 1
ICs := Theta(0)[1] = - Pi, Theta(0)[2] = - Pi,
2 2
1
Theta(0)[3] = - Pi, 0 = 0, 0 = 0, 0 = 0
2
sols := dsolve([odesys, ICs], {Theta(t)[1], Theta(t)[2], Theta(t)[3]});
Error, (in dsolve) required an indication of the solving variables for the given system
add(L[k], k = 1 .. 3);
3
NULL;
