Question

The general Galtican transformat?on $$ t^{\prime}=t+a, \quad r^{\prime}=A r-v t+b \quad \text { where } \quad A^{\prime} A=1 $$ may be denoted by the abstrict symbol \((a, v, b, A)\). Show that the result of performing two Galilean transformations $$ G_{1}=\left(a_{1}, v_{1}, b_{1} \cdot A_{1}\right) \quad \text { and } \quad G_{2}=\left(a_{2} \cdot v_{2}, b_{2}, A_{2}\right) $$ in succession is $$ G=G_{2} G_{1}=(a, \mathbf{v}, \mathbf{b} . \mathrm{A}) $$ where $$ a=a_{1}+a_{2}, \quad v=A_{2} v_{1}+v_{2}, \quad b=b_{2}-a_{1} v_{2}+A_{2} b_{1} \quad \text { and } \quad A=A_{2} A_{1} $$ Show from thts rule of composition that the Galican transformations form a group In partucular venfy explicitly that the assoclative law holds

Short answer

The result of two Galilean transformations, \( G_{1} \) and \( G_{2} \), performed in succession is another Galilean transformation, \( G \), where \( G = G_{2} G_{1} \). The expressions for \( a \), \( v \), \( b \), and \( A \) in terms of the original transformations have been derived. The Galilean transformations form a group as they satisfy the associative law.

Step-by-step solution

Understanding the Transformed Variables

The key to answering this question is to understand what happens to the variables when a transformation is performed. According to the rules of Galilean transformations, the transformed variables \( t' \), \( r' \), \( a' \) and \( A' \) are modified in the following ways: \( t' = t + a \), \( r' = A r - vt + b \), \( A' = A \), and \( a' = a \).

Successive Transformations

Now we consider what happens when two transformations are performed in succession. Given two Galilean transformations denoted as \( G_{1} = (a_{1}, v_{1}, b_{1} \cdot A_{1}) \) and \( G_{2} = (a_{2}, v_{2}, b_{2}, A_{2}) \), the resulting transformation is given by \( G = G_{2} G_{1} \). We prove this with the provided transformation rules given in the exercise.

Apply Transformations

Using the transformation equations, we define the new relationship in terms of \( a \), \( v \), \( b \), and \( A \) which are obtained from both \( G_{1} \) and \( G_{2} \) respectively. These new relationships are \( a=a_{1}+a_{2} \), \( v=A_{2} v_{1}+v_{2} \), \( b=b_{2}-a_{1} v_{2}+A_{2}b_{1} \), and \( A=A_{2} A_{1} \). This gives us the result of two transformations in succession.

Checking Group Properties

To check if the Galilean transformations form a group, we must verify that the associative law holds. The associative property states that for three transformations \( G_{1} \), \( G_{2} \), and \( G_{3} \), the equation \( (G_{1} G_{2}) G_{3} = G_{1} (G_{2} G_{3}) \) must hold. The equality can be verified using the expressions obtained from performing transformations in Step 3.

Key concepts

Group Theory

Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is a set of elements combined with a binary operation that satisfies four key properties:

• Closure: Performing the operation on any two elements from the group results in another element within the group.
• Associativity: The grouping of operations does not change the result. This means \((a \, * \, b) \, * \, c = a \, * \, (b \, * \, c)\).
• Identity Element: There is an element in the group that, when used in the operation with any element in the group, results in the same element. For example, for addition, the identity is 0 because \(a + 0 = a\).
• Inverse Element: For every element in the group, there is another element that, when the operation is performed between the two, results in the identity element. In terms of addition, the inverse of \(a\) is \(-a\) because \(a - a = 0\).

Understanding these properties is foundational for analyzing whether transformations, such as Galilean transformations, can form a group. If they do, it implies regular, reliable outcomes from combining these transformations, which is crucial in physics.

Associative Law

The associative law is a fundamental property that characterizes many algebraic structures, particularly groups. It states that for any three elements \(a, b, \,\text{and} \, c\) in a set, the equation \((a \, * \, b) \, * \, c = a \, * \, (b \, * \, c)\) is always true. This means that the order in which operations are performed does not affect the outcome.
In the context of Galilean transformations, verifying the associative law involves showing that successive transformations can be grouped in any order without affecting the final result. Mathematically, this requires that when you perform transformations \(G_1\), \(G_2\), and \(G_3\) in sequence, the transformation \((G_1 \, G_2) \, G_3\) results in the same outcome as \((G_3 \, G_2) \, G_1\).
This property ensures consistency and predictability in how transformations combine, making it possible to create complex sequences of transformations that yield expected outcomes. It is particularly important in physics, where transformations describe how systems change under different conditions.

Matrix Multiplication

Matrix multiplication is a mathematical operation that takes a pair of matrices and produces another matrix. The process is crucial in a variety of applications, including transformations in geometry and physics. In Galilean transformations, matrices can be used to represent linear transformations of space and time.
When multiplying two matrices, the entry in the resulting matrix at position \(i, j\) is calculated by taking the dot product of the \(i\)-th row of the first matrix with the \(j\)-th column of the second matrix. Formally, if \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, the product \(C = A \times B\) is an \(m \times p\) matrix defined as:
\[ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}. \]
This formula might seem complex, but it essentially combines elements from the two matrices in a structured way to produce the new matrix. This result is especially relevant in transformations because it allows for the representation of successive transformations as a single matrix, encapsulating the transformation rules concisely and making calculations easier.

Physics Transformations

In physics, transformations are used to describe how different states of a system can change based on various parameters, like time or space shifts. Galilean transformations, in particular, relate to transformations in classical mechanics. They describe how the laws of motion are perceived in moving reference frames, highlighting the importance of relative motion.
Galilean transformations include parameters like shifts in time \(t' = t + a\), transformations of spatial coordinates \(r' = A r - vt + b\), and adjustments related to velocity and position. These transformations are used to move from one inertial frame to another, often necessary in classical mechanics problems.
Understanding these transformations is critical in solving real-world physics problems, ensuring that calculations remain consistent across different frames of reference. This consistency is a cornerstone of physics and is safeguarded by using proper mathematical structures like groups and matrices, ensuring all transformations remain coherent and predictable in nature.