we can write the differencial of the hamilton using its equations :
$$ d{H(q,p)} = -\dot{p}d{q} + \dot{q}d{p} $$
However we can write :
$$ d(\dot{p}q)=\dot{p}dq+qd\dot{p}$$ and we can replace
$$ -\dot{p}dq = -d(\dot{p}q)+qd{\dot{p}}$$
We therefore get
$$ d{(H+\dot{p}q)} = \dot{q}dp+qd\dot{p}$$
We can now define a new quantities $$ W(p,\dot{p}) = H+\dot{p}q$$ which satisfies
$$dW(p,\dot{p}) = \frac{\partial W}{\partial p} dp+\frac{\partial W}{\partial \dot{p}} d\dot{p}$$ with
$$ \frac{\partial W}{\partial p} = \dot{q}$$ $$ \frac{\partial W}{\partial \dot{p}} = q$$
Now when we replace the Hamiltonian by it canonical form:
$$ H(q,p) = \frac{p^{2}}{2m} + V(q)$$
and we replace it it the W, we get
$$ W(p,\dot{p}) = \frac{p^{2}}{2m} + V(q) + \dot{p}q$$
Now we can use the fact that $$ \dot{p} = -\frac{dV}{dq}$$ and we can recognize the legendre transformation of the potentiel V which is define in term of the conjugate variable $ \dot{p}$, we can call it $A(\dot{p})$, which satisfies the equation
$$ q = \frac{dA(\dot{p})}{d \dot{p}}$$.
So my question is that i am trying to understand the physical meaning of the last equation which gives us back the fundamental principle of dynamics, and particularly the meaning of the $A(\dot{p})$ which is the legendre transformation of the potential.
Finally could you tell me if it is usefull to introduce a new such quantity $W$.
Vincent Woiline
