applies. The Hamiltonian of control compared to the Hamiltonian of mechanics William Rowan Hamilton defined the Hamiltonian for describing the mechanics of a system. It is a function of three variables: where is the Lagrangian, the extremizing of which determines the dynamics (not the Lagrangian defined above), is the state variable and is its time derivative. is the so-called "conjugate momentum", defined by Hamilton then formulated his equations to describe the dynamics of the system as The Hamiltonian of control theory describes not the dynamics of a system but conditions for extremizing some scalar function thereof (the Lagrangian) with respect to a control variable . As normally defined, it is a function of 4 variables where is the state variable and is the control variable with respect to that which we are extremizing. The associated conditions for a maximum are This definition agrees with that given by the article by Sussmann and Willems. (see p. 39, equation 14). Sussmann and Willems show how the control Hamiltonian can be used in dynamics e.g. for the brachistochrone problem, but do not mention the prior work of Carathéodory on this approach. Current value and present value Hamiltonian In economics, the objective function in dynamic optimization problems often depends directly on time only through exponential discounting, such that it takes the form where is referred to as the instantaneous utility function, or felicity function. This allows a redefinition of the Hamiltonian as where which is referred to as the current value Hamiltonian, in contrast to the present value Hamiltonian defined in the first section. Most notably the costate variables are redefined as , which leads to modified first-order conditions. , which follows immediately from the product rule. Economically, represent current-valued shadow prices for the capital goods . Example: Ramsey–Cass–Koopmans model In economics, the Ramsey–Cass–Koopmans model is used to determine an optimal savings behavior for an economy. The objective function is the social welfare function, to be maximized by choice of an optimal consumption path . The function indicates the utility the representative agent of consuming at any given point in time. The factor represents discounting. The maximization problem is subject to the following differential equation for capital intensity, describing the time evolution of capital per effective worker: where is period t consumption, is period t capital per worker (with ), is period t production, is the population growth rate, is the capital depreciation rate,