3C5 Lagrangian & Hamiltonian

How many generalised coordinates are required to fully describe a system with N degrees of freedom?

N

Energy method for single DOF?

T + V = const

Which systems require Lagrange & Hamilton?

And what do these methods lead to?

Many DOFs

Multiple coupled differential equations

All 3 Lagrangian relationship?

p = dL/dq dot

F = dL/ dq

d/dt (p) = F

Differentiate e_r and e_theta (taking note of direction)

Holonomic vs Non-holonomic?

Holonomic: current state of system fully described by instantaneous displacements and velocities

Non-holonomic: system depends on current displacements & velocities AND path taken by system to reach current state (won’t come up on exams)

Work done by applied generalised, external forces equation?

δW = ^∑_j Q_j δq_j

Steps to find generalised forces?

not made FC yet :(

What to remember to include in T and V?

T = ½ m v²

T = ½ I ω² (include whenever there’s rotation NOT accounted for by generalised coordinates)

V = mgh

V = ½ k x²

What form should you get T into to find mass matrix?

T = ½ m [ α x² + β y² + (2 γ xy) ]

M = m |α γ|

|γ β|

Invert 2×2 matrix?

When is Kamiltonian equal to Hamiltonian?

When G has no explicit time-dependence