Maths Theory/Mechanics
Basic Mechanics
Constraints
When there are constraints, the coordinates are no longer independent - they are connected by the equations of constraint.
The procedure is to introduce generalized coordinates.
For an action
the condition for a stationary value is equivalent to Lagrange's Equation
It contains all the mechanics of holonomic systems with forces derivable from potentials.
Further it involves only physical quantities that can be defined without reference to a set of generalized coordinates.
It is therefore automatically invariant with respect to whatever coordinates are used.
Lagrange's Equation
Hamilton's Principle
Calculus of Variations
The problem is to find the curve for which some given line integral has a stationary value.
Advantages of Variational Principles
At its most useful when a Lagrangian of independent coordinates can be set up.
Conservation Theorems from Symmetry Properties