1. D’Alembert’s Principle the sum of the work done by
MA4248 Weeks 8-9.
Topics Generalized Forces, Velocities, and Momenta;
Lagrange’s, Lagrange-Euler’s, and Hamilton’s Equations
D’Alembert’s Principle the sum of the work done by
the applied and inertial forces in a virtual disp. = 0 1

2. GENERALIZED FORCES 2 If the 3N particle coordinates are expressed
in terms of a set of f generalized coordinates
then
where
are called generalized forces. 2

3. GENERALIZED VELOCITIES
Chain-rule gives derivatives of the particle velocities
and the kinetic energy
with respect to the generalized velocities 3

4. GENERALIZED MOMENTA
We define generalized momenta
Example (linear momenta) for unconstrained motion
of a single particle with generalized coordinates x,y,z
and the generalized momenta
are components of the linear momentum vector 4

5. GENERALIZED MOMENTA
Example (angular momenta) for motion of a particle in
a circle with radius with generalized coordinate
the generalized momentum
is the angular momentum 5

6. TIME DERIVATIVES OF GENERALIZED MOMENTA
Example (linear momenta)
Example (angular momentum)
These consequences of Newton’s laws motivate us to
considerer the quantities 6

7. TIME DERIVATIVES OF GEN. MOMENTA
The Eq. on page 3 and Leibniz’s formula yield 7

8. LAGRANGE’S EQUATIONS
Combining these equations yields
Therefore 8

9. EULER-LAGRANGE EQUATIONS
If the applied forces are conservative, then
where
is the potential energy
Therefore
where
is the Lagrangian (it may or may not depend on t) 9

10. EXAMPLE: PLANE PENDULUM
kinetic energy
potential energy
Lagrangian 10

11. EXAMPLE: SPHERICAL PENDULUM
Lagrangian 11

12. MOMENT OF INERTIA
For a particle with mass m and
position vector that rotates
with angular velocity about
a line through the origin
hence with respect to orthonormal coordinates x, y, z 12

13. MOMENT OF INERTIA
Therefore the kinetic energy of the particle equals
where I is the 3 x 3 inertia matrix for the particle 13

14. MOMENT OF INERTIA
For a rigid body with density (mass per volume) If
then
where (see Calkin, page 38) 14

15. CENTER OF MASS
The center of mass of a system of N particles with
masses and positions is
For a rigid body with density (mass per volume) 15

16. HAMILTONIAN
A function of
is said to be “conserved” if its time derivative is zero
satisfies
The Hamiltonian
hence is conserved iff L does depend explicitly on t 16

17. HAMILTONIAN AND ENERGY
For scleronomic (time independent) holonomic constraints
T is a quadratic form in the generalized velocities 17

18. HAMILTONIAN AND ENERGY
Therefore, for scleronomic holonomic constraints
and the hamiltonian
is the total energy 18

19. HAMILTONIAN AND ENERGY
For a bead with mass m on a horizontal wire rotating with
angular speed and distance q from the center
therefore
and
is conserved but
This is an example of a rheonomic (time dependent)
constraint in which the constraint (wire) can do work 19

20. Then for any smooth and small
EULER’S EQUATION
Let F be a function of 3 variables and for any smooth function
define
Then for any smooth and small 20