Solve parametric equation with parameters

Gaetano Pavone

2026 4 13

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Hi, I have the following problem, in which a and b are constants (geometric parameters), phi is the variable (angle). Moreover, h and lbda are geometrical indicators depending on phi. U is the total elastic energy of the system, in which lbdan is the natural length (lbda>lbdan).

eqn1 represents the sationarity condition for U. It leads to lbda=lbdan or lbda'=0 (its derivative). By considering the last case, we have that tan(phi)=-(sqrt(3))/3.

clc; clear all; close all

syms lb a b phi k lbda_n

h = (lb^2 - a^2 - b^2 + 2*a*b*cos(phi))^1/2;

lbda = (lb^2 + a*b*(3*cos(phi) - sqrt(3)*sin(phi)))^(1/2);

dlbda_dphi = diff(lbda, phi);

% disp(dlbda_dphi)

d2_dlbda_dphi = diff(lbda, phi,2);

% disp(d2_dlbda_dphi)

assume (lbda > lbda_n)

U = (3*k*(lbda - lbda_n)^2)/2;

dU_dphi = diff(U, phi);

d2_dU_dphi = diff(U, phi,2);

% disp(dU_dphi)

eqn1=solve(dlbda_dphi==0,phi,'ReturnConditions',true);

I know analitically the solution of eqn1, how can I obtain the same result in matlab?

Is my script correct?

Thanks

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Ritam 2026-4-16，6:12

Could you please let me know what result you are expecting? It would also be helpful if you could briefly explain what each variable represents, and whether it is intended to be a constant or a variable. This additional context will help me verify whether the script is behaving as intended.