SICM Chapter 1 Notes
by Carson Reynolds
Chapter 1 of SICM covers Legrangian Mechanics. The Lagrangian formulation of mechanics is equivalent to the newtonian formulation along the path that is stationary. Lagrangians are coordinate system independent: they describe the action of a system whether the coordinates are provided in cartesian, radial, or other styles. In many mechanical systems, the action is the integral of the Lagrangian (L = T – V). This is the difference between the Kinetic and Potential Energy.
If we have a Lagrangian, we can find a second-order system of differential equations, the Euler-Lagrange Equations (or just Lagrange Equations). Using the principle of stationary action, we can compute the motion of the system. A bonus is that we can also easily relate this to newtonian and calculus of variations formulations.
Lagrangians are not unique. For instance, by adding the total time derivative to a Lagrangian we can recover another Lagrangian. Lagrangians also nicely apply to rigid-body problems, if we treat constraints as a special sort of coordinate transformation.
The Lagrangian is based on the negative energy gradient, as opposed to simple relationships between points.