ClearAll["Global`*"];y[t_] := Y[t] + R Cos[θ[t]];
x[t_] := R Sin[θ[t]];
T = m (x'[t]^2 + y'[t]^2) +(1/2) M Y'[t]^2;
U  = 2 m g y[t] + M g Y[t];
lag = T-U

FullSimplify[lag]

lagWithConstraint = lag + λ Y[t];
FullSimplify[
D[D[lagWithConstraint, Y'[t]],t] == D[lagWithConstraint, Y[t]]
]

FullSimplify[
D[D[lagWithConstraint, θ'[t]], t] == D[lagWithConstraint, θ[t]]
]

ClearAll["Global`*"];
ϕ = 0;ωt = 1; 
Manipulate[
r = 1; xc = r Cos[ωt]; yc = r Sin[ωt];
c = {xc, yc}; a = {0, 0}; b = 2 c;
xyrelative = r {Cos[ωt + ϕ], Sin[ωt + ϕ]};
Graphics[{
Circle[c, r],
Line[{a, b}],
Disk[c+xyrelative, 0.05 r],
Dashed,
Line[{c, c + xyrelative}]
},
PlotRange->{{-2.1,2.1}, {-2.1,2.1}},
PlotRangeClipping->True, Frame->True],
{ϕ, 0, 2 π}, {ωt, 0, 2π}, 
LabelStyle->Large]