such a way that, it will make the new Hamiltonian . Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial so the new generalized coordinates and momenta are constants of motion. As they are constants, in this context the new generalized momenta are usually denoted , i.e. and the new generalized coordinates are typically denoted as , so . Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant : the HJE automatically arises When solved for , these also give us the useful equations or written in components for clarity Ideally, these N equations can be inverted to find the original generalized coordinates as a function of the constants and , thus solving the original problem. Action and Hamilton's functions Hamilton's principal function S and classical function H are both closely related to action. The total differential of is: so the time derivative of S is Therefore, so S is actually the classical action plus an undetermined constant. When H does not explicitly depend on time, in this case W is the same as abbreviated action. Separation of variables The HJE is most useful when it can be solved via additive separation of variables, which directly identifies constants of motion. For example, the time t can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative in the HJE must be a constant, usually denoted (), giving the separated solution where the time-independent function is sometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be written To illustrate separability for other variables, a certain generalized coordinate and its derivative are assumed to appear together as a single function in the Hamiltonian In that case, the function S can be partitioned into two functions, one that depends only on qk and another that depends only on the remaining generalized coordinates Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as ), yielding a first-order ordinary differential equation for In fortunate cases, the function can be separated completely into functions In such a case, the problem devolves to ordinary differential equations. The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta,