Generalised Coordinates

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[edit] Definition

Any $s$ variables $q_1, q_2, \dots, q_s$ completely describing the $s$ degrees of positional freedom of the system are called generalised coordinates, and their time derivatives $\dot{q}_1, \dot{q}_2, \dots, \dot{q}_s$ are known as generalised velocities.

[edit] Constraints

It is occasionally the case that elements within a system are constrained in one or more dimensions. It is often convenient to describe such systems in terms of a reduced number of generalised coordinates, matching the number of degrees of the system. More complicated constraints must be dealt with alternatively (perhaps using Lagrange multipliers).

[edit] The Metric Tensor

The metric tensor is a tensor which, roughly (but accurately, for most purposes in physics), describes the distance from one point in a generalised coordinate system to another. For the purposes of classical mechanics, it is used most often to cast the generalised coordinates to cartesian coordinates, for example, in order to write the Lagrangian.

[edit] The Lagrangian

The Lagrangian is of the form $L = T - U$ , where $T$ is the kinetic energy. For free particles, $T = \frac{1}{2}m \mathbf{v}^2$ where $\mathbf{v}^2 = \frac{dl^2}{dt^2}$ is the velocity in Cartesian space, and $d l 2$ is the line element.

To obtain the Lagrangian for a system described by generialised coordinates, we require the following transformations:

$x_a = f_a(q_1, q_2, \dots, q_s), \dot{x}_a = \sum_k \frac{\partial f_a}{\partial q_k} \dot{q}_k$ , etc. $L = \frac{1}{2} \sum_{particles} m_a (\dot{x}_a^2 + \dot{y}_a^2 + \dot{z}_a^2) = \frac{1}{2} \sum_{i,k,particles} a_{ik}(q) \dot{q}_i \dot{q}_k - U(q)$ , where $a_{ik}(q) = m_a \frac{\partial f_a}{\partial q_i}\frac{\partial f_a}{\partial q_k}$

[edit] Scale Factors

Orthogonal and Curvilinear coordinates have the interesting property that their metric tensor is diagonal, that is, $g i j = g i i δ i j$ where $δ i j$ is the Kronecker delta. As a result, the line element is