Maths Theory/Mechanics

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Basic Mechanics

Angular Momentum
Torque

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.

Lagrange's Equation

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.

For an action

Action

the condition for a stationary value is equivalent to Lagrange's Equation

Advantages of Variational Principles

At its most useful when a Lagrangian of independent coordinates can be set up.

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.

Conservation Theorems from Symmetry Properties