0x110 Mechanics

Newtonian Formulation

Lagrangian Formulation

Configuration space is used to capture the state of a system in the Lagrangian formulation. The time evolution of a system is described in this space by a single path.

The main idea of Lagrangian formalism is that nature is lazy, which means the action $S$ defined as follows should be minimized

$$S = \int_{t_i}^{t_f} dt (T-V) $$

where the second term is called Lagrangian $L(q, \dot q)$

$$L = T – V$$

Intuitively, the principle of least action states that the lazy man do not want to move very actively, instead, they want to preserve their potential energy (fat?) when integrating over time.

By solving the least action formula with variational calculus, we obtain the famous Euler-Lagrange equation

$$\frac{\partial L}{\partial q} – \frac{d}{dt}\big(\frac{\partial L}{\partial \dot{q}}\big) = 0$$

This Eular Lagrange Equation can be connected with the Newtonian one

$$p = \frac{\partial L}{\partial \dot q}$$

$$F = \frac{\partial L}{\partial q} = -\frac{\partial V(q)}{\partial q}$$

where $p$ is the generalized momentum and $F$ is the generalized force

This leads to Newton second law (rate of change of the momentum equals the force)

$$\frac{d}{dt} p =F$$

Hamiltonian Formulation

Hamiltonian $H(p,q)$ is an abstract concept of total energy in closed systems. It is a Legendre transform from Lagrangian.

$$H(p,q) = p\dot q – L(q,p)$$

The Hamilton’s equations are

$$\frac{\partial{p}}{\partial t} = – \frac{\partial H}{\partial q}$$

$$\frac{\partial{q}}{\partial t} = \frac{\partial H}{\partial p}$$

Poisson bracket of two phase space functions $A(q,p), B(q,p)$, is defined as

$$\{ A, B \} = \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} – \frac{\partial A}{\partial p} \frac{\partial B}{\partial q}$$

The Hamilton’s equation of motion describes the time evolution of a general phase space function

$$\frac{d}{dt} F = \{ F, H \}$$

Note that commutator is related to poisson bracket

$$[ \hat{f}, \hat{g} ] \sim i\hbar \{ f, g \}$$

A quick derivation can show that

$$\{ p_i, p_j \} = \{ q_i, q_j \} = \{ p_i, q_j \} = 0$$

A couple of properties related to Poisson brackets are

  • $\{ A, B \} = – \{ B, A \}$
  • $\{ A + B, C \} = \{ A, C \} + \{ B, C \}$
  • $\{ \lambda A, B \} = \lambda \{ A, B \}$
  • $\{ AB, C \} = \{ A, C \} B + A \{ B, C \}$

Reference

[1] Schwichtenberg, Jakob. No-Nonsense Classical Mechanics: A Student-Friendly Introduction. No-Nonsense Books, 2019.

[2] Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynman lectures on physics, Vol. I: The new millennium edition: mainly mechanics, radiation, and heat. Vol. 1. Basic books, 2011.

[3] Goldstein, Herbert, Charles Poole, and John Safko. “Classical mechanics.” (2002): 782-783.