Mathematical formulations for physics

Personal reference on the various ways to mathematically formulate physics.

In all of the following, $$v_i = \frac{x_i}{dt}$$, and $$x_i$$ is not necessarily a Cartesian coordinate except in the Newtonian formulation.

Newtonian

$\frac{\mathbf{p}}{m} = \frac{d\mathbf{x}}{dt}$ $\mathbf{F} = \frac{d\mathbf{p}}{dt}$ $\mathbf{F} = - \nabla V$

Lagrangian

For a Lagrangian $$\mathcal{L}$$, the action $$\mathcal{A}$$ is

$\mathcal{A} = \int_{t_0}^{t_1} \mathcal{L\left(\mathbf{x}(t), \mathbf{v}(t)\right)} \, dt$

The trajectory satisfies the principle of least action or stationary action.

$\delta \mathcal{A} = 0$

Euler-Lagrange equation

$\frac{d}{dt} \frac{\partial \mathcal{L}}{\partial v_i} = \frac{\partial \mathcal{L}}{\partial x_i}$

Generalized momentum

$p_i = \frac{\partial \mathcal{L}}{\partial v_i}$

Classical Lagrangian

$\mathcal{L} = T - V = \frac{1}{2} m v^2 - V$

Symmetry and conservation

For a symmetry which leaves the Lagrangian invariant, there is a conserved quantity.

Hamiltonian

$H = \sum p_i v_i - \mathcal{L} = T + V = E$

Equations of motion

$\frac{dp_i}{dt} = - \frac{\partial H}{\partial x_i}$ $\frac{dx_i}{dt} = \frac{\partial H}{\partial p_i}$

Gibbs-Liouville theorem

For $$\mathbf{v} = (x_1, \ldots, x_n, p_1, \ldots, p_n)$$ in phase space,

$\nabla \cdot \mathbf{v} = 0$

Poisson bracket

$\{F,G\} = \sum_i \left( \frac{\partial F}{\partial q_i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q_i} \right)$

With the Hamiltonian $$H$$,

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

The equations of motions for a Hamiltonian can be expressed with the Poisson bracket.

$\frac{d q_k}{dt} = \{q_k,H\}$ $\frac{d p_k}{dt} = \{p_k,H\}$

Basic properties

$\{A,C\} = -\{C,A\}$ $\{cA,C\} = c\{A,C\}$ $\{A+B,C\} = \{A,C\} + \{B,C\}$ $\{AB,C\} = A\{B,C\} + B\{A,C\}$ $\{q_i,q_j\} = 0$ $\{p_i,p_j\} = 0$ $\{q_i,p_j\} = \delta_{ij}$

Other properties

$\{f,p_i\} = \frac{\partial f}{\partial q_i}$ $\{f,q_i\} = -\frac{\partial f}{\partial p_i}$

References

[1]
Jim Baggott. 2020. The quantum cookbook. Oxford University Press.
[2]
Leonard Susskind and George Hrabovsky. 2013. The theoretical minimum. Basic Books.