How can I show that two objective functions are the same?

Question

I am trying to understand the relationship between the constrained and unconstrained versions of a convex optimization problem. The unconstrained problem is as follows: $$\min_{X}||X-Y||_2^2 + \lambda \mbox{TV}(X)$$ I am trying to show that it is equivalent to the following constrained problem $$\min_X \mbox{TV}(X) \mbox{ s.t. } ||X-Y||_2 < \beta$$ How can I show that the two optimization problems are equivalent? My idea is to express each as a Lagrangian and show that the Lagrangians are the same. The Lagrangian for the first problem would be $$L_1(X, \lambda) = ||X-Y||_2^2 + \lambda \mbox{TV}(X)$$ The Lagrangian for the second problem would be $$L_2(X, \lambda) = \mbox{TV}(X)+\lambda(||X-Y||_2^2 - \beta)$$ Is there a way to rewrite $L_1$ such that it corresponds to $L_2$ for a different choice of $\lambda$? Or vice versa? Thanks!