Brief Review of Undergraduate Classical Mechanics (2) - From Action extremization to Lagrangian Mechanics

이 글은 2020.10.14에 작성됨.

This post was written on Oct 14, 2020.

 

Which path $x(t)$ minimize the action $S[x(t)]$? Use variational calculus to answer this question.

$$S[x(t)]=\int_{t_0}^{t_1} L(x^i(t), \dot{x}^i(t)) dt $$

Here $i=1,2,3,\cdots,3N$, The reason for not including higher order terms more than 1st is that physical state is completely determined by coordinates and speeds. Suppose there also are k constraints.

$$f_j(x^i)=0$$

Also $j=1,2,3, \cdots, k$. Simply varying $S[x(t)]$, we get

$$\delta S =0 \:\:\:\:\: \Rightarrow \:\:\:\:\: \int dt \left( \frac{d}{dt} \frac{\partial L}{\partial \dot{x}^i}-\frac{\partial L}{\partial x^i} \delta x^i \right)= 0$$

Since all $\delta x^i$s are not independent each other, we need to vary the condition of constraints.

$$\delta f_j =0 \:\:\:\:\: \Rightarrow \:\:\:\:\: \frac{\partial f_j}{\partial x^i} \delta x^j = 0$$

Each term for index i  is the functional of only $x^i(t)$, so we can introduce some Lagrange multiplier $\lambda(t)$ to solve this equation.

$$\frac{d}{dt} \frac{\partial L}{\partial \dot{x}^i}-\frac{\partial L}{\partial x^i} = \lambda_j(t) \frac{\partial f_j}{\partial \dot{x}^i}$$

We want to know k unknown functions $\lambda_j(t)$ and 3N unknown functions $x^i(t)$. Also we have k unknown equations $f_j(x^1, x^2, \cdots, x^{3N})=0$ and 3N Lagrange equations. So we can find all unknown functions with appropriate initial conditions. In addition, the right side of the above equations means the $x^i$ directional constraint force.