Simple code to write differential equations in a matrix formulation

Question

Is there a simple code to transform the following differential system equations :

SysDiff = {Subscript[F, x] == m Subscript[x, G]''[t],
    -g m + Subscript[F, y] == m Subscript[y, G]''[t], 
    1/2 l Sin[γ[t]] Subscript[F, x] - 1/2 l Cos[γ[t]] Subscript[F, y] == JG γ''[t]}

In a matrix formulation such as:

$$M q''(t) + \phi_q^{\mathsf{T}}*Lambda = F $$

where $q = \{γ[t], x_G[t], y_G[t]\}$ is a vector with 3 components,

and $Lambda = \{F_x, F_y\} $ is a vector with 2 components.

P.S: For a better understand, these equations are those of a simple pendulum with absolute coordinates and the constraints equations are:

SysCon = {-(1/2) l Cos[\[Gamma][t]] + Subscript[x, G][t] == 
   0, -(1/2) l Sin[\[Gamma][t]] + Subscript[y, G][t] == 0}

In others words, for the equations given, how can I determine :

• the matrix M
• the matrix $\phi_q^{\mathsf{T}}$,
• the vector F ?

Answer 1

SysDiff = {
Subscript[F, x] == m Subscript[x, G]''[t], 
-g m + Subscript[F, y] == m Subscript[y, G]''[t], 
1/2 l Sin[γ[t]] Subscript[F, x] - 1/2 l Cos[γ[t]] Subscript[F, y] == JG γ''[t]};

eqns = SysDiff /. Equal[lhs_, rhs_] :> lhs - rhs;

q = {γ[t], Subscript[x, G][t], Subscript[y, G][t]};

c = Thread[
 Join[q2 = D[q, {t, 2}], λ ={Subscript[F, x], Subscript[F, y]}] -> 0];

The resulting matrices:

M = D[eqns, {q2}] /. c

{{0, -m, 0}, {0, 0, -m}, {-JG, 0, 0}}

Subscript[ϕ, q] = Transpose[D[eqns, {λ}] /. c]

{{1, 0, 1/2 l Sin[γ[t]]}, {0, 1, -(1/2) l Cos[γ[t]]}}

And the r.h.s vector:

(*rhs F*)f = -eqns /. c

{0, g m, 0}