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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Dimensional Analysis

Dimensional Analysis is a technique based on two simple axioms about nature:

Although these laws are quite simple, they are very important and serve as powerful instrument for correct mathematical/theoretical modelling. The dimension of a physical quantity (such as length, time, or mass) is given once and for all and will never change. When the unit changes, the numerical value of the quantity also changes.

...
Quantity Symbol SI unit
Mass m Kilogram (kg)
Length ℓ meter (m)
Time t second (s)
Electric current I ampere (A)
Absolute temperature Θ kelvin (K)

A physical relation or equation is dimensionally correct if each side of an equality does not have different dimensions. For instance, the well-known formula F = m 𝑎 is dimensionally correct because

[F]     newton (N)
[m]      kilogram (kg)
[𝑎]     m/s²

Let us consider a relationship between physical quantities R₁, R₂, … , Rn. A relationship between these physical quantities is a relation or a formula, that can be written in the form

\[ \Phi \left( R_1 , R_2 , \ldots , R_n \right) = 0 , \]
where Φ is a certain function in n variables. According to the implicit function theorem, we could also write
\[ R_1 = \Psi \left( R_2 , R_3 , \ldots , R_n \right) . \]
If we choose a set of fundamental units, and use these in a consistent way for all the involved variables, we should now have
\[ \Phi \left( v(R_1 ), v(R_2 ) , \ldots , v(R_n ) \right) = 0 . \]
However, it may happen that there is no valid physical relation between v(R₁), v(R₂), … , v(Rn).

To investigate this further, it is smart to create a so-called dimension matrix containing the exponents of the fundamental units in the units for the quantities we have. If [S] = m, [t] = s and for acceleration due to gravity [g] = ms−2, the dimension matrix will be

S t g
m 1 0 1
s 0 1 −2

Let us denote the fundamenttal units as u₁, u₂, &ldots , uN. The units of any physical quantity may be expressed by means of these units, for example, the. unit for energy is ”kg m² /s²”. Generally speaking, we may thus write

\begin{align*} \left[ R_1 \right] &= u_1^{m_{1 1}} u_2^{m_{2 1}} \cdots u_N^{m_{N 1}} , \\ \ \vdots & \qquad \vdots \\ \left[ R_n \right] &= u_1^{m_1 n} u_2^{m_2 n} \cdots u_N^{m_N n} , \end{align*}
This yieldss the dimension matrix A:
\[ \begin{array}{|c|ccc} & R_1 & R_2 & \cdots & R_n \\ \hline u_1 & a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots &&& \mathbf{A} & \vdots \\ u_N & a_{N1} && \cdots & a_{Nn} \end{array} \]

We say that R₁, R₂, &ldots , Rr have independent dimension if it is impossible to make a (non- trivial) dimensionless combination of R₁, R₂, &ldots , Rr of the form

\[ R_1^{\lajgea_1} R_2^{\lambda_2} \cdots R_r^{\lambda_r} \]
Using units, we get
\begin{align*} \left[ R_1^{\lambda_1} \cdots R_r^{\lambda_r} \right] &= u_1^{a_{11} \lambda_1 + a_{12}\lambda_2 + \cdots + a_{1r}\lambda_r} \times \\ &\quad u_2^{a_{21} \lambda_1 + a_{22} \lambda_2 + \cdots + a_{2r}\lambda_r} \\ & \quad \cdots u_N^{a_{N1}\lambda_1 + a_{N2} \lambda_2 + \cdots + a_{Nr} \lambda_r} . \end{align*}
Then R₁, R₂, &ldots , Rr have independent dimensions if and only if the system of equations
\begin{align*} a_{11} \lambda_1 + a_{12} \lambda_2 + \cdots + a_{1r} \lambda_r &= 0 , \\ a_{21} \lambda_1 + a_{22} \lambda_2 + \cdots + a_{2r} \lambda_r &= 0 , \\ \qquad \vdots \\ a_{N1} \lambda_1 + a_{N2} \lambda_2 + \cdots + a_{Nr} \lambda_r &= 0 \end{align*}
has only the trivial (all zeroes) solution:
\[ \lambda_1 = \lambda_2 = \cdots = \lambda_r = 0 . \]
Recall from Linear Algebra that this may happen only when matrix A has rank r, so its column vectors are linear independent subject that r ≤ N.

 

Hamilton Principle

\begin{equation} \label{EqH.2} \frac{{\text d}q_k}{{\text d}t} = \left\{ q_k ,H \right\} , \qquad \frac{{\text d}p_k}{{\text d}t} = \left\{ p_k ,H \right\} . \end{equation}

 

Canonical transformations

The 2n-dimensional space of points specified by the canonical coordinates and momenta is called phase space.

 

Hamiltonian evolution

When the dynamics is described by Hamilton's equations, the evolution in time is made by canonical transformations.

 

  1. Jordan, T., Steppingstones in Hamiltonian dynamics, The American Journal of Physics, 2004, 72, No. 8, pp. 1095--1099. doi: 10.1119/1.1737394

 

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