analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...

, 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 In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...

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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

, 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 Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...

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.

References