# Mathematical Glossary

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#### 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

#### Ellipse

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

#### 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.

where

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.

#### 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

then

If there is no force field

which gives

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