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
\[ \{f,p_i\} = \frac{\partial f}{\partial q_i} \] \[ \{f,q_i\} = -\frac{\partial f}{\partial p_i} \]