Given a canonical transformation phi which takes a vector Âfield to equilibrium and is generated by a function f, there existsanother function that differs from f only by a function of the time, and which also generates phi, and satisfiÂes this equation,according to a theorem bearing this equation's name. Singularities in the solution to this equation correspond to the pointson the Lagrangian manifold where the projection to the coordinate space is not a local diffeomorphism, called caustic points.When this equation describes a system with n degrees of freedom, a complete integral of this equation will have n + 1 
Given a canonical transformation phi which takes a vector Âfield to equilibrium and is generated by a function f, there existsanother function that differs from f only by a function of the time, and which also generates phi, and satisfiÂes this equation,according to a theorem bearing this equation's name. Singularities in the solution to this equation correspond to the pointson the Lagrangian manifold where the projection to the coordinate space is not a local diffeomorphism, called caustic points.When this equation describes a system with n degrees of freedom, a complete integral of this equation will have n + 1 parametersand will provide a solution to the equations of motion for positions and momenta. For 10 points, identify this equationwhich sets an operator representing total energy, plus the time derivative of the action, equal to zero, a fundamental equationof mechanics.
Given a canonical transformation phi which takes a vector Âfield to equilibrium and is generated by a function f, there existsanother function that differs from f only by a function of the time, and which also generates phi, and satisfiÂes this equation,according to a theorem bearing this equation's name.
Hamilton-Jacobi equation