classical mechanics - Derivation of Euler-Lagrange equations from Hamilton's and D'Alembert's principle

Goldstein's book of Classical Mechanics derive the Euler-Lagrange equations from two different principles:

1. Hamilton's principle states that

$$\delta S = \delta\int_{t_1}^{t_2}L(q^{i},\dot{q}^{i},t)dt=0,$$

where no conditions on $L(q^{i},\dot{q}^{i},t)$ seem to be required, and from here one can directly obtain Euler-Lagrange equations.

2. D'Alembert's principle (in terms of generalized coordinates) states that

$$\sum_{j}\left\lbrace\bigg[\frac{d}{dt}\bigg(\frac{\partial T}{\partial \dot{q}^{j}}\bigg)-\frac{\partial T}{\partial q^{j}}\bigg]-Q_{j}\right\rbrace\delta q^{j}=0,$$

where one needs to consider that the kinetic energy $T$ is a quadratic function of the velocities $\dot{q}^{j}$.

In order to obtain the Euler-Lagrange equations from this principle, one also needs to consider that the generalized force $Q_j$ is derivable from the potential energy function $V$, which must be a function dependent only on the coordinates $q^{j}$. The form of the Lagrangian must be $L=T-V$ in this case.

1. 
   

   Is the Hamilton's principle more powerful in the sense that puts no restrictions on the form of the Lagrangian? or there is some missing restriction?

   
   
   


2. 
   

   Is there another way of deriving Euler-Lagrange equations from D'Alembert's principle without restricting the forms of $T$, $V$ and $L$?

   
   

Answer

I) Actually, it's the other way around. Within the context of Newtonian mechanics, the hierarchy is the following from most to least applicable:

1. 
   

   Newton's laws are always applicable.

   
   
   


2. 
   

   D'Alembert's principle or Lagrange equations. E.g. sliding friction typically violates D'Alembert's principle.

   
   


3. 
   

   The stationary action principle $S=\int\! dt~L$, with Lagrangian $L=T-U$, and its Euler-Lagrange equations. E.g. a generalized force might not have a generalized potential $U$.

   
   

II) In point 3 we have tacitly assumed that the Lagrangian is of the form $$L~=~T-U,\tag{1}$$ as is customary. $T$ and $U$ in eq. (1) may be viewed as representing the kinematic and the dynamical side of Newton's 2nd law, cf. e.g. this Phys.SE post. The linear structure of eq. (1) also reflects a categorical-like composition rules for how to build physical models out of physical subsystems.

There exist strictly speaking exceptions to the form (1), cf. e.g. this Phys.SE post, but these exceptions often lacks categorical-like composition rules, which make them unsuitable for useful model building.

III) For further details and discussions, see e.g. my related Phys.SE answers here, here, and links therein.