# Mathematical Glossary

__Angular Momentum__

__Configuration Space__

Not necessarily connected with 'normal' three-dimensional space.
__Conic__

The general polar form (apart from a circle) is

__Conjugate Momentum__

In general, the conjugate momentum is not the same as 'normal momentum', but if the Lagrangian L is
not dependent on the generalized coordinates q, it is a constant.
__Coordinates__

See
Generalized Coordinates
__D'Alembert's Principle__

This can be used to derive Lagrange's Equation.

__Ellipse__

From the
general polar form, the equation for an ellipse is

__Euler-Lagrange Equation__

__Generalized Coordinates__

__Hamiltonian__

same as Conservation of Energy. The quantity which is preserved is the Hamiltonian.

__Hamilton's Principle__

Describes the motion of mechanical systems in which
all forces (except for forces of constraint) are derivable
from a generalizable scalar potential

For such a system, the motion is such that the action, i.e.

has a stationary value. Or, in other words, the variation of I is zero, i.e.

When the constraints are holonomic, Hamilton's principle is a necessary and sufficient condition for Lagrange's Equations.

This allows us to use Hamilton's Principle as a basic postulate (for the conditions stated in the first paragraph), rather than using Newton's Laws of Motion.

__Holonomic Constraint__

__Lagrange Equation__

When the action is given by
then the Euler-Lagrange Equations become Lagrange's Equation

If the Lagrangian is not a function of x

If the Lagrangian is independent of t

where

Since

If there is no force field

which is **Newton's First law of Motion**

__Lagrangian__

We can set up a Lagrangian for holonomic systems with applied forces derivable from an ordinary or
general potential, and workless constraints.
A suitable way to construct a Lagrangian for a conserved system is

*where T is the Kinetic Energy and V is the Potential Energy*(Although it should be noted that this does not provide the only Lagrangian suitable for the given system)

If there is no force field

__Orbits in a Central Field__

Orbits of particles in central fields show three features
- The energy is constant
- The angular momentum is constant
- The orbit lies in a fixed plane

If the potential is