13.4.2 General Lagrangian Expressions

As more complicated mechanics problems are considered, it is convenient to express the differential constraints in a general form. For example, evaluating (13.130) for a kinematic chain of bodies leads to very complicated expressions. The terms of these expressions, however, can be organized into standard forms that appear simpler and give some intuitive meanings to the components.

Suppose that the kinetic energy is expressed using (13.126), and let $m_{ij}(q)$ denote an entry of $M(q)$. Suppose that the potential energy is $V(q)$. By performing the derivatives expressed in (13.136), the Euler-Lagrange equation can be expressed as $n$ scalar equations of the form [856]

$$\sum_{j=1}^n m_{ij}(q) {\ddot q}_j + \sum_{j=1}^n \sum_{k=1}^n h_{ijk}(q) {\dot q}_j {\dot q}_k + g_i(q) = u_i$$ (13.140)

in which

$$h_{ijk} = \frac{\partial m_{ij}}{\partial q_k} - \frac{1}{2} \frac{\partial m_{jk}}{\partial q_i} .$$ (13.141)

There is one equation for each $i$ from $1$ to $n$. The components of (13.140) have physical interpretations. The $m_{ii}$ coefficients represent the inertia with respect to $q_i$. The $m_{ij}$ represent the affect on $q_j$ of accelerating $q_i$. The $h_{ijj}{\dot q}_j^2$ terms represent the centrifugal effect induced on $q_i$ by the velocity of $q_j$. The $h_{ijk} {\dot q}_j{\dot q}_k$ terms represent the Coriolis effect induced on $q_i$ by the velocities of $q_j$ and $q_k$. The $g_i$ term usually arises from gravity.

An alternative to (13.140) is often given in terms of matrices. It can be shown that the Euler-Lagrange equation reduces to

$$M(q) {\ddot q}+ C(q,{\dot q}){\dot q}+ g(q) = u ,$$ (13.142)

which represents $n$ scalar equations. This introduces $C(q,{\dot q})$, which is an $n \times n$ Coriolis matrix. It turns out that many possible Coriolis matrices may produce equivalent different constraints. With respect to (13.140), the Coriolis matrix must be chosen so that

$$\sum_{j=1}^n c_{ij} {\dot q}_j = \sum_{j=1}^n \sum_{k=1}^n h_{ijk} {\dot q}_j {\dot q}_k .$$ (13.143)

Using (13.141),

$$\sum_{j=1}^n c_{ij} {\dot q}_j = \sum_{j=1}^n \sum_{k=1}^n \left(... ...frac{1}{2} \frac{\partial m_{jk}}{\partial q_i} \right) {\dot q}_j {\dot q}_k .$$ (13.144)

A standard way to determine $C(q,{\dot q})$ is by computing Christoffel symbols. By subtracting $\frac{1}{2} \frac{\partial m_{jk}}{\partial q_i}$ from the inside of the nested sums in (13.144), the equation can be rewritten as

$$\sum_{j=1}^n c_{ij} {\dot q}_j = \frac{1}{2} \sum_{j=1}^n \sum_{k... ...ial q_k} - \frac{\partial m_{jk}}{\partial q_i} \right) {\dot q}_j {\dot q}_k .$$ (13.145)

This enables an element of $C(q,{\dot q})$ to be written as

$$c_{ij} = \sum_{k=1}^n c_{ijk} {\dot q}_k ,$$ (13.146)

in which

$$c_{ijk} = \frac{1}{2} \left( \frac{\partial m_{ij}}{\partial q_k}... ...\partial m_{ik}}{\partial q_j} - \frac{\partial m_{jk}}{\partial q_i} \right) .$$ (13.147)

This is called a Christoffel symbol, and it is obtained from (13.145). Note that $c_{ijk} = c_{ikj}$. Christoffel symbols arise in the study of affine connections in differential geometry and are usually denoted as $\Gamma^i_{jk}$. Affine connections provide a way to express acceleration without coordinates, in the same way that the tangent space was expressed without coordinates in Section 8.3.2. For affine connections in differential geometry, see [133]; for their application to mechanics, see [156].