Mathematical Glossary

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Angular Momentum

Angular Momentum

Configuration Space

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

Conic

The general polar form (apart from a circle) is

General Conic Formula in Polar Form

See Conics

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

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

Formula for Ellipse in Polar Form

Euler-Lagrange Equation

Euler-Lagrange Equation

see Lagrange Equation

Generalized Coordinates

Hamiltonian

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

Generalizable Scalar Potential

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

The Action
where
Lagrangian

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

Calculus of Variations - Ansatz

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
Action

then the Euler-Lagrange Equations become Lagrange's Equation

Lagrange Equation

If the Lagrangian is not a function of x

Lagrangian independent of x

If the Lagrangian is independent of t

Lagrangian independent of t

where

Definition of V

Since

Generalized momentum defined
then
Rejigged U

If there is no force field

Lagrangian when no Force Field present
which gives
mrDash

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

Lagrangian
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

Lagrangian when no Force Field present

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

Inverse R Potential

then the orbit is an ellipse

Torque

Torque