Lagrangian Mechanics

Constraints

Imposing constraints on a system is simply another way of stating that there are forces present in the problem that cannot be specified directly, but are known in term of their effect on the motion of the system.  Holonomic constraints are constraints of the form

fm(r1,r2,r3,...,rn,t) = 0,   m = 1,2,3, … , k.

They reduce the number of degrees of freedom of the system;  k equations of constraints reduce the number of the degree of freedom of an n-particle system from 3n to 3n - k, if the constraints are holonomic.

If the constraints are holonomic, then the forces of constraints do no virtual work.

Consider a virtual displacement of the system, i.e. an infinitesimal change in the coordinates of the system, denoted by dri, consistent with the constraints imposed on the system at a given instant t.  The work done by the force in the virtual displacement dri is called the virtual work.  For holonomic constraints, the forces of constraints are perpendicular to the virtual displacements and do no virtual work.

Generalized coordinates

Any set of independent quantities q1, q2, … , qs, which completely define the position of the system with s degrees of freedom, are called generalized coordinates of the system, and the derivatives  are called generalized velocities.
Examples:

A particle is constraint to move in the x-y plane, the equation of constraint is z = 0, the constraint is holonomic.  Possible generalized coordinates for the system with two degrees of freedom are x, y;  r, φ;  … .
A particle is constraint to move on a circle in the x-y plane, the equations of constraints are z = 0,  x+ y2 - r= 0.  The constraints are holonomic.  Possible generalized coordinates for the system with 1 degree of freedom are φ ; φ3, … .

The generalized coordinates q1, … , qs can be expressed in terms of the Cartesian coordinates the system.

q1 = q1(r1, r2, … , rn ), … , qn = qn(r1, r2, … , rn ).

These equations, together with the equation of constraints, can be inverted to find the r’s in terms of the q’s.

Lagrangian Mechanics

Assume a system has n independent generalized coordinates {qi}.  Assume that the generalized applied forces

are given by

,

with U some scalar function, i.e. the generalized applied forces are derivable from a potential.  Then the equations of motion may be obtained from Lagrange’s equations,

,

where  L = T - U  is the Lagrangian of the system.  L is a function of the coordinates and the velocities.

If not all the forces acting on the system are derivable from a potential, then Lagrange's equations can be written in the form

,

where L contains the potential of the conservative forces and Qj represents the generalized forces not arising from a potential.

Link:  Derive Lagrange's equations from D'Alembert's Principle

Define the generalized momentum or conjugate momentum or canonical momentum through

 .

If the Lagrangian does not contain a given coordinate qj then the coordinate is said to be cyclic and the corresponding conjugate momentum pj is conserved.

The Hamiltonian H of a system is given by

.

H is a function of the generalized coordinates and momenta of the system.  The equations of motion can be obtained from Hamilton’s equations,

.

If the Lagrangian does not explicitly depend on time, then the Hamiltonian does not explicitly depend on time and H is a constant of motion.  [If H does explicitly depend on time, H = H(t), then H is not a constant of motion.]
If the generalized coordinates do not explicitly depend on time, then  H = T + U = E, the total energy of the system.  [If the generalized coordinates do explicitly depend on time, then  H is not the total energy of the system.]
So only if Lagrangian does not explicitly depend on time and the generalized coordinates do not explicitly depend on time, then H = T + U = E and the energy is a constant of motion.

Lagrange multipliers

Assume you have chosen coordinate for a system that are not independent, but are connected by m equations of constraints of the form

.

Then the equations of motion can be obtained from

(n equations), and  (m equations).

We have m + n equations and m + n unknowns, the n coordinates and the m λ’s.  The λl are called the undetermined Lagrange multipliers is the generalized force of constraint associated with the coordinate qk.