An alternative Lagrangian approach

In this note, I present alternative expressions for the standard Lagrangian and action, and I discuss the corresponding Hamiltonian and canonical transformation. The new expressions are equivalent to the traditional versions, but written in a way that reveals a useful first-order differential equation for the motion of the system. The conceptual picture of this approach is a phasor diagram, which serves as an alternative to traditional energy bar graph diagrams. This approach works for particles in any one-dimensional potential and for particles in a specific set of separable higher-dimensional potentials.