In analytical mechanics, the mass matrix is a symmetric matrix M that expresses the connection between the time derivative q ˙ {\displaystyle \mathbf {\dot {q}} } of the generalized coordinate vector q of a system and the kinetic energy T of that system, by the equation
T = 1 2 q ˙ T M q ˙ {\displaystyle T={\frac {1}{2}}\mathbf {\dot {q}} ^{\textsf {T}}\mathbf {M} \mathbf {\dot {q}} }
where q ˙ T {\displaystyle \mathbf {\dot {q}} ^{\textsf {T}}} denotes the transpose of the vector q ˙ {\displaystyle \mathbf {\dot {q}} } . This equation is analogous to the formula for the kinetic energy of a particle with mass m and velocity v, namely
T = 1 2 m | v | 2 = 1 2 v ⋅ m v {\displaystyle T={\frac {1}{2}}m|\mathbf {v} |^{2}={\frac {1}{2}}\mathbf {v} \cdot m\mathbf {v} }
and can be derived from it, by expressing the position of each particle of the system in terms of q. In general, the mass matrix M depends on the state q, 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 system can be described by a vector q of two generalized coordinates, namely the positions of the two particles along the track.
q = [ x 1 x 2 ] T {\displaystyle \mathbf {q} ={\begin{bmatrix}x_{1}&x_{2}\end{bmatrix}}^{\textsf {T}}}
Supposing the particles have masses m1, m2, the kinetic energy of the system is
T = ∑ i = 1 2 1 2 m i x i ˙ 2 {\displaystyle T=\sum _{i=1}^{2}{\frac {1}{2}}m_{i}{\dot {x_{i}}}^{2}}
This formula can also be written as
T = 1 2 q ˙ T M q ˙ {\displaystyle T={\frac {1}{2}}{\dot {\mathbf {q} }}^{\textsf {T}}\mathbf {M} {\dot {\mathbf {q} }}}
where
M = [ m 1 0 0 m 2 ] {\displaystyle \mathbf {M} ={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}
N-body system More generally, consider a system of N particles labelled by an index i = 1, 2, …, N, where the position of particle number i is defined by ni free Cartesian coordinates (where ni = 1, 2, 3). Let q be the column vector comprising all those coordinates. The mass matrix M is the diagonal block matrix where in each block the diagonal elements are the mass of the corresponding particle:
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