I am trying to prove that kinetic energy is conserved in a collision given that it is time reversible by putting this problem in terms of functional equations.
Consider a vector-valued function which must satisfy the following properties
$$f(m_1,m_2,\mathbf{v}_1,\mathbf{v}_2) = \mathbf{v}_1'$$ $$f(m_2,m_1,\mathbf{v}_2,\mathbf{v}_1) = \mathbf{v}_2'$$ $$f(m_1,m_2,-\mathbf{v}_1',-\mathbf{v}_2') = -\mathbf{v}_1$$ $$f(m_2,m_1,-\mathbf{v}_2',-\mathbf{v}_1') = -\mathbf{v}_2$$ $$m_1\mathbf{v}_1+m_2\mathbf{v}_2=m_1\mathbf{v}_1'+m_2\mathbf{v}_2'$$ Then how does one prove that it must also satisfy the following relation? $$m_1\mathbf{v}_1^2+m_2\mathbf{v}_2^2=m_1\mathbf{v}_1'^2+m_2\mathbf{v}_2'^2$$