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Consider a dynamical system $\dot x = f(x,u)$ where $x$ is the state and $u$ is a control input. We are interested in moving the system state from one value to another in finite time, say $[0, T]$. At the same time we want to minimise a scalar function $\int_0^T L(x,u) dt$

The Pontryagin minimum principle address this very need. First, we express the constraint optimization problem in one equation, using Lagrange multipliers: $H(x,u,\lambda) = \lambda f (x,u) + L(x,u)$.

After this point, some textbooks I have consulted and even the wikipedia article present the reader with five functions that when solved will provide the optimal trajectory ($x,u,\lambda$) for the system.

When comparing these five equations with the Hamiltonian mechanics equations it is apparent that the Lagrange multiplier $\lambda$ play the role of the momentum in Hamiltonian mechanics.

I am curious to know, how can one interpret the Lagrange multipliers, which are essentially the marginal cost of a constraint violation, as momentum?

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    What do you mean by interpretation? $\lambda$ is an auxiliary abstract parameter. The Wiki article on Lagrange multipliers visually explains how they work. – 2017-02-03
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    I try to understand theory on both a technical and an intuitional level. The later eludes me in this case. I cannot see how a Lagrange multiplier can be thought of as momentum in the context of Hamiltonian mechanics. What are the features they two share? – 2017-02-04
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    If we go through the proof of the maximum principle, it becomes apparent that the co-state equation captures, up to first order, effect of variations from the optimal trajectory and it is best interpreted that way rather than as momentum. For more information why, I would refer you to [Daniel Liberzon's book](http://liberzon.csl.illinois.edu/teaching/cvoc.pdf) Chapters 2-4 or [Sussman's lectures](http://www.math.rutgers.edu/~sussmann/weizmann-course-2000.html), handouts 2-4 primarily, for more information. – 2017-02-13
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    @ITA so the costate should not be thought of as either momentum or multiplier? – 2018-02-03

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