Mass matrix

In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative $\mathbf\dot q$ of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation
:$T = \frac \mathbf^\textsf \mathbf \mathbf$

where $\mathbf^\textsf$ denotes the transpose of the vector $\mathbf$. This equation is analogous to the formula for the kinetic energy of a particle with mass and velocity , namely
:$T = \frac m, \mathbf, ^2 = \frac \mathbf \cdot m\mathbf$

and can be derived from it, by expressing the position of each particle of the system in terms of .

In general, the mass matrix depends on the state , and therefore varies with time.

Lagrangian mechanics yields an ordinary differential equation (actually, a system of coupled differential equations) that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system. The kinetic energy formula above is one term of that equation, that represents the total kinetic energy of all the particles.

Examples

Two-body unidimensional system

For example, consider a system consisting of two point-like masses confined to a straight track. The state of that systems can be described by a vector of two generalized coordinates, namely the positions of the two particles along the track.
:$\mathbf q = \begin x_1 & x_2 \end^\textsf$

Supposing the particles have masses , the kinetic energy of the system is
:$T = \sum_^ \frac m_i \dot ^2$

This formula can also be written as
:$T = \frac \dot \mathbf^\textsf \mathbf M \dot \mathbf$

where
:$\mathbf M = \begin m_1 & 0 \\ 0 & m_2 \end$

N-body system

More generally, consider a system of particles labelled by an index , where the position of particle number is defined by free Cartesian coordinates (where ). Let be the column vector comprising all those coordinates. The mass matrix is the diagonal block matrix where in each block the diagonal elements are the mass of the corresponding particle:

:\mathbf M = \operatorname\left m_1 \mathbf_,\, m_2 \mathbf_,\, \ldots,\, m_N \mathbf_ \right/math>

where is the identity matrix, or more fully:
: $\mathbf M = \begin m_1 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\ 0 & \cdots & m_1 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ 0 & \cdots & 0 & m_2 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\ 0 & \cdots & 0 & 0 & \cdots & m_2 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & m_N & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & m_N\\ \end$

Rotating dumbbell

For a less trivial example, consider two point-like objects with masses , attached to the ends of a rigid massless bar with length , the assembly being free to rotate and slide over a fixed plane. The state of the system can be described by the generalized coordinate vector
:$\mathbf q = \begin x & y & \alpha \end$

where are the Cartesian coordinates of the bar's midpoint and is the angle of the bar from some arbitrary reference direction. The positions and velocities of the two particles are
:$\begin x_1 &= (x, y) + R(\cos\alpha, \sin\alpha) & v_1 &= \left(\dot x, \dot y\right) + R\dot \alpha(-\sin\alpha, \cos\alpha) \\ x_2 &= (x, y) - R(\cos\alpha, \sin\alpha) & v_2 &= \left(\dot x, \dot y\right) - R\dot \alpha(-\sin\alpha, \cos\alpha) \end$

and their total kinetic energy is
:$2T = m\dot x^2 + m\dot y^2 + mR^2\dot\alpha^2 - 2Rd\sin(\alpha) \dot x \dot\alpha + 2Rd\cos(\alpha) \dot y \dot\alpha$

where $m = m_1 + m_2$ and $d = m_1 - m_2$. This formula can be written in matrix form as
:$T = \frac \dot \mathbf^\textsf \mathbf M \dot \mathbf q$

where
:$\mathbf M = \begin m & 0 & -Rd\sin\alpha \\ 0 & m & Rd\cos\alpha \\ -Rd\sin\alpha & Rd\cos\alpha & R^2 m \end$

Note that the matrix depends on the current angle of the bar.

Continuum mechanics

For discrete approximations of continuum mechanics as in the finite element method, there may be more than one way to construct the mass matrix, depending on desired computational accuracy and performance. For example, a lumped-mass method, in which the deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element.

See also

* Moment of inertia
* Stress–energy tensor
* Stiffness matrix
* Scleronomous

References

Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,

Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, {{ISBN, 978 0 521 57572 0

Computational science