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

f_{m}(**r**_{1},**r**_{2},**r**_{3},...,**r**_{n},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 d**r**_{i},
consistent with the constraints imposed on the system at a given instant t. The
work done by the force in the virtual displacement d**r**_{i
}is called the virtual work.
For holonomic constraints, the forces of constraints are perpendicular to the
virtual displacements and do no virtual work.

Any set of
independent quantities q_{1}, q_{2}, … , q_{s}, 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
^{2 }+ y^{2 }- r^{2 }= 0. The constraints are holonomic. Possible generalized coordinates for the system with 1 degree of freedom are φ ; φ^{3}, … .

The
generalized coordinates q_{1}, … , q_{s} can be expressed in
terms of the Cartesian coordinates the system.

q_{1} = q_{1}(**r**_{1, }**r**_{2, … , }**r**_{n }),
… , q_{n} = q_{n}(**r**_{1, }**r**_{2, … , }**r**_{n }).

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

Assume a system has n independent generalized coordinates {q_{i}}.
Assume that the **generalized applied forces
**{Q

are given by

Q

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

d/dt(∂L/∂(dq_{i}/dt)) -
∂L/∂q_{i} = 0,

where **L
= T - U** is
the** Lagrangian **of the system. **L is a function of the
coordinates **q_{i} **and the velocities **
dq_{i}/dt.

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

d/dt(∂L/∂(dq_{i}/dt)) -
∂L/∂q_{i} = Q_{j},

where L contains the potential of the conservative forces and Q_{j}
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

∂L/∂(dq_{i}/dt) = p_{j}.

If the Lagrangian does not contain a given coordinate q_{j}
then the coordinate is said to be **cyclic** and
the corresponding conjugate momentum p_{j} is conserved.

The **Hamiltonian** H of a system is given by

H(q, p, t) = ∑_{i}(dq_{i}/dt)p_{i} – L.

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

dq_{i}/dt = ∂H/∂p_{i},
dp_{i}/dt = -∂H/∂q_{i}.

Assume now that the generalized forces are given by
Q_{j} = -∂U/dq_{j}.

- 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.

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

Σ_{k} a_{lk
}dq_{k
}+ a_{lt }dt = 0, l = 1, ..., m.

Then the equations of motion can be obtained from

d/dt(∂L/∂(dq_{k}/dt)) - ∂L/∂q_{k} = ∑_{l}λ_{l}a_{lk}, (n
equations), Σ_{k} a_{lk
}dq_{k
}+ a_{lt }dt = 0 (m equations).

We have m + n equations and m + n unknowns, the n coordinates and the m
λ’s. T

he λ_{l}
are called the **undetermined Lagrange multipliers**,
∑_{l}λ_{l}a_{lk} is the generalized force of
constraint associated with the coordinate q_{k}.