Lagrangian Mechanics

• can avoid usage of Newtonian method of solving for motion utilizing forces when they become complex
• lagrangian mechanics uses velocities to find kinetic energies and is combined with potential energies to generate the required equations of motion

Introduction

• obtain kinetic energy from velocities present within the system
• if forces present within the system are conservative then potential energy is obtained
• derive the lagrangian equations of motion for each coordinate

Generalized Coordinates and Forces

• 2-dimensions:
  • rectangular coordinates ($x, y$)
  • polar coordinates ($\rho, \theta$)
• 3-dimensions:
  • rectangular coordinates ($x, y, z$)
  • cyclindrical coordinates ($\rho, \phi, z$)
  • spherical coordinates ($r, \omega, \phi$)
• generalized coordinates: lengths, angles, combination of the two ($q_1, q_2, q_3, \dots$)
  • for one particle in 3-space, need 3 coordinates
  • if $N$ particles, need $3N$ coordinates
• generalized force: associated with each generalized coordinate
  • work required to increase coordinate $q_j$ by $\delta q_j$ is $P_j \delta q_j$
  • thereby, $P_j$ is generalized force associated with $q_j$

NOTE: generalized force may be dimensionally a force or a torque and is dependent upon whether generalized coordinate is angle or length

Holonomic Constraints

• holonomic constraint: constraint that can be described by a holonomic equation and relates the coordinates (and perhaps also time)
  • ex., known contact of point in
• complete description of $N$ unconstrained particles required $3N$ coordinates
  • state of system can be viewed as a point within $3N$-dimensional space
  • if subject to $k$ constraints, the system can be viewed as moving along a surface of dimension $3N-k$ within the $3N$-dimensional space

Lagrangian Equations of Motion

• system of $N$ particles:
  • force on $i$th particle ($i=1$ to $N$) is $F_i$
  • displacement is $\delta r_i$
  • thereby work is $\sum_i{F_i \cdot \delta r_i}$$
• can express $\vec{r}$ as generalized coordinates, thereby...

$$ \sum_i{F_i \cdot \sum_j{\frac{\partial r_i}{\partial q_j} \delta q_j}} = \sum_j{\sum_i{F_i \cdot \frac{\partial r_i}{\partial q_j} \delta q_j}} $$