This course will make you study things you already know in a different way

Classical mechanics in 1 dimensions

💼 Case: consider a point particle of mass $m$ moving in a potential $V(x)$.

• Newton would say it feels a force $f=-\partial V/\partial x$ and therefore accelerates according to $F=ma$ or

  $$ m\ddot{x}=-\frac{\partial V}{\partial x} $$

  We can also calculate the kinetic energy of the particle $T=\frac 12 m \dot x^2$ and the total energy $E=T+V$

🗒️ Note: on partial derivatives

• $\partial E /\partial t$ is the answer to the question:

  ❓ How much does the expression $E$ change as $T$ varies keeping everything else constant

  ✅ Not at all $(=0)$

• $\text{d} E/\text dt$ is the answer to the question

  ❓ How much does the energy of the particle vary as it moves along its trajectory ( so $x$ varies) with some acceleration ( so $\dot x$ varies ) because time is changing?

  ✅ The answer is

  $$ \begin{aligned} \frac{\text dE}{\text dt}&=\frac{\partial E}{\partial t}+\frac{\partial E}{\partial x}\frac{\text dx}{\text dt}+ \frac{\partial E}{\partial \dot x}\frac{\text d \dot x}{\text dt} \\ &=0+\frac{\partial V}{\partial x}\dot x + m\dot x \ddot x \\ &= \left ( m\ddot x+\frac{\partial V}{\partial x} \right )\dot x

  \end{aligned} $$

Now we can take 2 approaches

1. 🍎 Newton: $m\ddot x=-\partial V/\partial x$ hence $\text dE/\text dt=0$
2. 🍋 Lagrange approach: suppose energy is conserved $\text dE/\text dt=0$, Then either $\dot x=0$, the particle is always stationary or the particle must move in such a way that $m\ddot x$ is always equal to $-\partial V/\partial x$

🗒️ Note: Either approach is good (we take one axiom and derive one corollary from it)

• 🍎 In Newton’s approach forces are the fundamental quantity,
• 🍋 In Lagrange’s approach energy is the fundamental quantity

Lagrange being based on a symmetry of nature Lagrange’s approach is seen from the modern viewpoint as more fundamental