How to obtain differential equations of motion using Lagrangian dynamics?

Question

I'm confused to find a set of differential equations of motion of a pair of masses, m1 and m2 joined by a spring of constant k.

The unstretched length of the spring is L, and the initial conditions are; x1[0]=0, x2[0]=L+a, x1'[t]=0, x2'[t]=0. x1[t] and x2[t] are the positions of the masses m1 and m2 at any time t.

I used Lagrangian dynamics to find the differential equations of motion, but I'm not sure my codes are correct. And I have no idea where I need to put the initial conditions to find the differential equations. Can someone help me?

Needs["VariationalMethods`"];
T = 0.5*m1*x1'[t]^2 + 0.5*m2*x2'[t]^2;
V = 0.5*k*(x2[t] - x1[t] - L);
lag = T - V;
eqn[n_] = EulerEquations[lag, {x1[t], x2[t]}, t];

Answer 1

Be Carful: Work and define your function correctly, Check physics laws and kinetic and potential correctly.

Here's my try for you: Go ahead and review and work more all the best

Needs["VariationalMethods`"];
m1 = m2 = 2;
k = 3;

T[t_, x1_, x2_] := 0.5*m1*(x1'[t])^2 + 0.5*m2*(x2'[t])^2;

V[t_, x1_, x2_] := 0.5*k*(x2[t] - x1[t]);

lag[t_, x1_, x2_] := T[t, x1, x2] - V[t, x1, x2];

EulerEquations[lag[t, x1, x2], {x1[t], x2[t]}, t]