Dynamic optimisation Consider a simple dynamic consumption-saving problem. A solution can be characterised using a Lagrangian approach that generates a set of first order conditions and some boundary conditions. 

An alternative approach is to set use a Bellman equation. What I don't understand is how come a Bellman equation approach does not seem to use information regarding the boundary conditions in the solution methodology? That is, a Bellman equation does have a tradeoff between time period t and t+1 but not some overall budget constraint does not seem to be represented.
The Dynamic Programming ("Bellman' Equation") formulation incorporates the terminal boundary condition ("transversality conditions") needed in case we use the Lagrangian/Euler equation formulation. The initial conditions are still needed in both approaches. 

To see why, consider the problem

$$\text{max}_{\{x_t\}} \sum_{t=0}^{\infty} \beta^t U(c_t)$$

$$s.t.\;\;\; a_{t+1}= (1-r)a_t + w - c_t, \;\;\; a_0 \; \text{given}$$

Solving the budget constraint for $c_t$ and inserting into the objective function, we have compactly

$$\text{max}_{\{a_{t}\}} \sum_{t=0}^{\infty} \beta^t U(a_t, a_{t+1}),\;\;\;\; a_0 \; \text{given}$$

where we treat the stock variable next period as our decision variable.
Now for the optimal sequence, $a^*_{t+1}$ must solve the following problem:

$$\max_y \big[U(a^*_t, y) + \beta U(y, a^*_{t+2})\big],\;\;\; y\;\; \text{feasible given}\;\; a^*_t$$

In the words of Stokey and Lucas "a feasible variation on the sequence $\{a^*_t\}$ cannot lead to an improvement on an optimal policy".  

Then we get the first order conditions (differentiating with respect to $y$ essentially)

$$U_2(a^*_t, a^*_{t+1}) + \beta U_1(a^*_{t+1}, a^*_{t+2}) =0$$

where the subscripts $1,2$ denote partial derivatives. This is a second-order difference equation, and so it requires two boundary conditions. We have already one of them, the given initial value of wealth $a_0$. We need one more, hence the terminal, "transversality" condition. 

In contrast, using the Dynamic Programming approach we arrive at a first-order difference equation, and we do not need an additional boundary condition.

For the full exposition, see Stokey &amp; Lucas p. 97-100.