Equation of motion through the Lagrangian with Lagrange multipliers

Question

I ask for advice, I'm a little confused. I have such a Lagrangian.

$L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-\lambda(2x^2+3y^2-1^2)$

Here $\lambda(2x^2+3y^2-1^2)$ is the constraint on the phase variables.

I need to derive the equation of motion given the constraints and solve them numerically with the help of NDSolve.

We do this in accordance with the classic formula:

$\frac{d}{dt}(\frac{dL}{d\dot{q}})-\frac{dL}{dq}=0$

Where $q=[x,y]$ are generalized coordinates. I'm not sure about the Lagrange multiplier as a generalized coordinate.

Clear["Derivative"]

ClearAll["Global`*"]

T = 1/2 m (x'[t]^2 + y'[t]^2);(*Kinetic Energy*)

f = \[Lambda] (2 x[t]^2 + 3 y[t]^2 - 1^2);(*Constraint*)

L = T - f;(*Lagrangian*)

D[D[L, x'[t]], t] - D[L, x[t]];

D[D[L, y'[t]], t] - D[L, y[t]];

D[D[L, \[Lambda]'[t]], t] - D[L, \[Lambda][t]];

Question: how are the Lagrange multipliers included in this system when compiling the ODE system and numerically solving it?

EDIT (with correct initial conditions and another constraint):

Clear["Derivative"]

ClearAll["Global`*"]

q = Function[t, {x[t], y[t]}];(*coordinates*)

zwang = x[t] + x[t] y[t] + y[t] - 1;(*constraints*)

Solve[zwang == 0, y[t]];

Q = \[Lambda][t] D[zwang, {q[t]}];(*gen.forces*)

L = 1/2 (x'[t]^2 + y'[t]^2);

eqn = D[D[L, {q'[t]}], t] - 
   D[L, {q[t]}] - \[Lambda][t] D[zwang, {q[t]}];


sol = Solve[
    Join[eqn, {D[zwang, {t, 2}]}] == 0, {x''[t], 
     y''[t], \[Lambda][t]}][[1]];

x0 = 1/2;

y0 = 1/3;

s = NDSolve[{sol[[1, 1]] == sol[[1, 2]], sol[[2, 1]] == sol[[2, 2]], 
    sol[[3, 1]] == sol[[3, 2]], x[0] == x0, y[0] == y0, x'[0] == 0, 
    y'[0] == 0}, {x, y, \[Lambda]}, {t, 500}];

Plot[Evaluate[zwang /. s], {t, 0, 500}, PlotRange -> All]

Answer 1

As far as I remember Lagrangeformalism (classic form) is only valid for generalized coordinates, which fullfill the constraints . In your example x[t],y[t] aren't generalized coordinates!

For the constraint 2 x[t]^2 + 3 y[t]^2 ==1^2 possible generalized coordinate q[t] would be x[t]->Cos[q[t]]/Sqrt[2],y[t]->Sin[q[t]]/Sqrt[3]

2 x[t]^2 + 3 y[t]^2 ==1^2 /.{x[t]->Cos[q[t]]/Sqrt[2],y[t]->Sin[q[t]]/Sqrt[3]}// Simplify
(*True*)

L= 1/2 m (3 x'[t]^2 + y'[t]^2) /. {x -> ( Cos[q[#]]/Sqrt[2] &), y  -> (Sin[q[#]]/Sqrt[3] &)} //Simplify
(*-(1/24) m (-11 + 7 Cos[2 q[t]]) Derivative[1][q][t]^2*)

The equations of motion follow to

D[D[L, q'[t]], t] - D[L, q[t]]==0
(*7/12 m Sin[2 q[t]] Derivative[1][q][t]^2 -1/12 m (-11 + 7 Cos[2 q[t]]) (q^\[Prime]\[Prime])[t]*)

Hope it helps!

addendum Lagrange with constraints (thanks @DanielHuber's comment )

q = Function[t, {x[t], \[CurlyPhi][t]}]; (*coordinates*)
zwang = x[t] - r \[CurlyPhi][t] (* constraints *)
Q = \[Lambda][t] D[zwang, {q[t]}] (* gen. forces*)

L = 1/2 m x'[t]^2 + 1/2 \[Theta] \[CurlyPhi]'[t]^2 +m g Sin[\[Alpha]]  x[t]
eqn = D[D[L, {q'[t]}], t] -D[L, {q[t]}] - \[Lambda][t] D[zwang, {q[t]}]

From equations eqn and zwang the Lagrange multiplier \[Lambda][t]is evaluated, substitution gives the equations of motion

sol = Solve[Join[eqn, {D[zwang, {t, 2}]}] ==0, {x''[t], \[CurlyPhi]''[t], \[Lambda][t]}][[1]]
(*{(x^\[Prime]\[Prime])[t] -> (g m r^2 Sin[\[Alpha]])/(m r^2 + \[Theta]), (\[CurlyPhi]^\[Prime]\[Prime])[t] -> (g m r Sin[\[Alpha]])/(m r^2 + \[Theta]), \[Lambda][t] -> -((g m \[Theta] Sin[\[Alpha]])/(m r^2 + \[Theta]))}*)

Q /. sol (* Reibkraft und -moment*)