Lagrangian Mechanics

Conservative vs. Non-conservative

The Lagrangian

\begin{equation} L = T - U \end{equation}

\begin{equation} \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial \dot{x}} = 0 \end{equation} leads to: \begin{equation} \frac{\partial L}{\partial x} = - \frac{\partial U}{\partial x} = F_x \end{equation} and \begin{equation} \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = \frac{d}{dt}m \dot{x} = m\ddot{x} \end{equation} so: \begin{equation} F_x = m\ddot{x} \end{equation}

Do it for all three dimensions and you have \begin{equation} -\nabla U = m \mathbf{a} \Rightarrow \mathbf{F} = m \mathbf{a} \end{equation}

Hamilton's Principle / Principle of Least Action

\begin{equation} S[q_k(t)] = \int_{t_1}^{t_b} dt L(t,q_1,q_2,\ldots,\dot{q}_1,\dot{q}_2,\ldots) = \int_{t_a}^{t_b} L(t,q_k,\dot{q}_k) \end{equation} When $S$ is stationary, i.e. \begin{equation} \delta S = \delta \int_{t_a}^{t_b} L(t,q_k,\dot{q}_k) = 0 \end{equation} then the $q_k(t)$s will satisfy the equations of motions for the system between the boundary conditions.

Generalized Coordinates

Polar Coordinates →

The basic polar coordinate system

Find the Lagrange Equations for a particle moving in two dimensions under the influence of a conservative force using polar coordinates

Force of Constraint

From 3 to 2 dimensions

By reducing the number of coordinates, we can imply a 'force of constraint'

Examples of Constraints

A mass on a ramp

In this classic case, a naive application of Newton's Laws would suggest two coordinates. However, since the block is constrained to the surface of the ramp, there really is only one independent variable.

The Atwood Machine has 1 coordinate

Likewise, with the Atwood Machine, there really is only one coordinate.

A regular pendulum and a double pendulum.

Pendulums are also great examples of systems with a constrain: the length of the string doesn't change.

Examples of Mechanical Systems:

Simple Pendulum a la Lagrange →

Pendulum on a stationary support

Pendulum with oscillating support →

Pendulum on Oscillating Support