Difference in "momentum" names in Lagrangian mechanics In the context of Lagrangian formulation of classical mechanics, the following names keep occurring in most textbooks, which confuse me a lot, are they different in any way?


*

*Momentum

*Generalized momentum

*Conjugate momentum 

*Canonical momentum
 A: Usually there's a great deal of overlap between the definitions of the momenta you've listed, so your confusion is understandable, but nonetheless there are cases (that I know of at least) where the distinction is more clearly enunciated:

*

*Momentum as known in Newtonian mechanics: The momentum is a vector quantity (its vectorial superposition for many particles in system possible) defined as the product of its mass $m$ times its velocity $ \vec{v}$, of a particle.
$$\vec{p}=m\vec{v}$$
If the system is isolated, then the momentum of the system is a first integral (constant of the motion), which calls for the principle of conservation of momentum, our particle's momentum is conserved (total momentum, considering all possible interactions of the system with its environment, is  always conserved). (same definition of momentum is extended and adapted in relativistic physics). All remaining details/elaborations can well covered here.



*

*Generalized momentum: In most problems, the dynamic variables ($x_1$,$x_2$,... & $\dot{x}_1$,$\dot{x}_2$,...) can be reduced to smaller number of independent generalized coordinates (usually denoted as $q_i$) and generalized velocities (denoted as $\dot{q}_i$), the latter usually adopted in the mechanics done via lagrangian. E.g. for the simple harmonic pendulum, the two coordinates $x$ and $y$ describing the position of the mass, are reduced to the generalized coordinate $\theta$, being the angle between the position vector of the mass and the vertical. In contrast to the generalized coordinates and velocities of Lagrange, we can attribute a generalized momentum $p_i$ using the Lagrange function $L(q_i,\dot{q}_i,t)$, by $$p_i=\frac{\partial L}{\partial \dot{q}_i}$$
Or equivalently, in Hamilton's description of mechanics: $$\dot{p}_i=-\frac{\partial H}{\partial q_i}$$


*

*Conjugate momentum can be slightly misleading, as one may draw different interpretations from it.  Probably the one you've seen when studying on the Lagrangian formalism, is defined as the derivative of the action with respect to the corresponding position, as: $$p=\frac{\partial S}{\partial q}$$ where the classical action is defined as
$$S=\int_{t_1}^{t_2} L\left(q(t),\dot{q}(t)\right)dt$$
Source of confusion to watch out for: Conjugate variables in general are variable pairs mathematically defined as Fourier-transform-duals of one another, e.g. time $t$ and frequency $\nu$. In contrast to such definition, in Quantum Mechanics, the wavefunction in momentum space, $\tilde{\psi}(p)$, is obtained from $\psi(x)$ by a Fourier transform(omitting the details and constants): $$\tilde{\psi}(p)\propto \int \psi(x)\exp(ixp/\hbar)dx$$


*

*Canonical momentum: I believe for this one wikipedia does a pretty good job of summarizing:


In classical mechanics, canonical coordinates are coordinates $q_i\,$
and $p_i\,$ in phase space that are used in the Hamiltonian formalism.
The canonical coordinates satisfy the fundamental Poisson bracket
relations: $$\{q_i,q_j\}=0, \, \{p_i,p_j\}=0, \, \{q_i,p_j\}=\delta_{ij}$$
Canonical coordinates can be obtained from the generalized coordinates
of the Lagrangian formalism by a Legendre transformation, or from
another set of canonical coordinates by a canonical transformation.

Finally one also talks about canonical momentum of a charged particle, $$\mathbf{p}_{\rm canonical}=\mathbf{p}+q_{\rm charge}\mathbf{A}$$ with $\mathbf{p}$ the momentum as defined in case 1 e.g., and $\mathbf{A}$ can be a magnetic vector potential. For more clarifications on this, check out this post.
