Physics 105A
Analytical Mechanics
Constraints
Generalized Coordinates
Cyclic Coordinates and
Conserved Quantities
28 June 2016
Manuel Calderón de la Barca Sánchez
Forces of Constraint
 Revisit particle sliding off frictionless sphere.
 Solve using Lagrangian, leaving r=R from beginning.
 Solve using Lagrangian and find Normal force
Force of Constraint
Lagrange Multipliers
m
Rq
28 June 2016
MCBS
Coordinate Transformations
 Cartesian to Spherical
r  x2  y 2  z 2
  atan2  y, x 

z
q  cos 
 x2  y 2  z 2

1
 Cartesian to Cylindrical




  x2  y 2
  atan2( y, x)
zz
 Generalized coordinates
28 June 2016
qi  qi ( x1 , x2 , x3 , x4 ,..., xN , t )
xi  xi (q1 , q2 , q3 , q4 ,..., qN , t )
MCBS
E-L Equations in Generalized Coordinates
 If the E-L equations hold for the xi coordinates,
d  L  L
 0, for 1  i  N


dt  xi  xi
and the qi coordinates are related to the xi coordinates by
qi  qi ( x1 , x2 , x3 , x4 ,..., xN , t )
xi  xi (q1 , q2 , q3 , q4 ,..., qN , t )
then the E-L equations hold for the qi coordinates.
28 June 2016
d  L  L
 0, for 1  j  N

 
dt  q j  q j
MCBS
Analogy with functions




Function has stationary points at f’(xmin)=0.
Change coordinates x=g(y)
Functions has stationary points at f’(ymin)=0, where ymin=g-1(xmin)
E-L Equations: Variational derivative = 0. Streching coordinates does not change the existence of
stationary values.
28 June 2016
MCBS
From Stationary Action to F=ma

Hamilton’s Principle of Stationary Action is equivalent to Newton’s Laws
Stationary Action
E-L equations
Newton’s 2nd Law F=ma
We showed it in cartesian coordinates
We showed that if it E-L eqs. hold in one coordinate system, they are true in all other
(sensible) coordinates.

Comments:
Deals only with scalars, makes problems easier.
– Recall inclined plane, problem 3.8 and problem 6.1.
d  L  L
 
dt  x  x
Local, not global.
Multiple coordinates? Multiple E-L equations.
Better suited for generalizations.
Feynman’s approach to Quantum Mechanics: Based on Hamilton’s Principle.
Maxwell’s Equations: Can be derived applying stationary action to Quantum
Electrodynamics.
28 June 2016
MCBS