PHY 313, Mechanics

September 16, 2026

I’ll try to get some notes typed up soon, but for now here’s a quick summary of what we covered and where you can find the examples in Taylor (I followed Taylor very closely today).

Lagrangian mechanics for unconstrained systems

An unconstrained system in Lagrangian mechanics will, in general, have some conservative force \(\vec F(\vec r)\) acting on a particle and no other rules about the particle’s motion. If we are working in a 2-dimensional space (and what comes after will generalize to 3 dimensions), then we have two spatial coordinates and two degrees of freedom to describe our system. We will write down expressions for the kinetic energy \(T\) and the potential energy \(U(r)\) and use them to write down the Lagrangian \(\mathcal{L} = T - U\). We will then put the Lagrangian into the Euler-Lagrange equation to solve for the equations of motion.

  • Typical rectangular coordinates reproduce our expected equations of motion from Newton’s laws (Taylor example 7.1)
  • Polar coordinates will reproduce the expected equations of motion for rotation, that is the equation for torque and the equation for radial acceleration that includes centripedal acceleration (Taylor example 7.2)
  • Lagrangian mechanics will hold for ANY general coordinate system
  • Lagrangian mechanics suggests the idea of a generalized momentum \(\frac{\partial \mathcal{L}}{\partial \dot{q}}\)

Lagrangian mechanics for constrained system

Notice that for the two unconstrained examples, the number of Euler-Lagrange equations we needed to solve was equal to the number of degrees of freedome which was simply equal to the number of spatial coordinates. If we introduce a constraint equation, we will reduce the number of degrees of freedom and therefore the number of necessary Euler-Lagrange equations.

  • Simple pendulum (Taylor 7.2) reveals that when we add a constraint \(x^2 + y^2 = \ell^2\) we can go down to one Euler-Lagrange equation if we first transform into the correct coordinate system (\(r=\ell, \phi\)).
  • More constraints next time