# Why can a partial derivative be added to a hamiltonian in canonical transformations?

In canonical transformations, how come we allow hamiltonian to change by a partial derivative of time? $$H'(P, Q, t) = H(p, q, t) + \frac{\partial F}{\partial t}.$$ Here $$F$$ is the generating function.

I mean geometrically that is not how a function should be transformed when there is a change of variables. Geometrically it should be $$H'(P, Q, t) = H(p, q, t).$$ In Lagrangian mechanics it is indeed so $$L'(Q, \dot{Q}, T) = L(q, \dot{q}, t).$$

In a canonical transformation, the new hamiltonian could have nothing to do with the initial hamiltonian, it just have to preserve Hamilton's equations. So in the new variables $$(Q,P,t)$$ you have to have that

$$\dot{Q} = \frac{\partial K}{\partial P} \qquad \dot{P} = -\frac{\partial K}{\partial Q}$$

where $$K$$ is the new Hamiltonian. Whenever this happens

$$K(Q,P,t) = H(q(Q,P),p(Q,P),t)$$

we call the transformation completely canonical (with the added bonus that the transformation is time independent), and it's a particular type of canonical transformation.

A more geometric approach is to consider the $$(2n+1)$$-dimensional contact manifold $${\cal M}$$ with coordinates $$(q^i,p_j,t)$$. The Hamiltonian action functional is $$S_H[\gamma]~=~\int_I \gamma^{\ast} \Theta, \qquad \Theta~=~p_j \mathrm{d}q^j -H \mathrm{d}t, \tag{1}$$ where $$\gamma:I\to {\cal M}$$ is a curve. This action formulation (1) is world-line (WL) reparametrization invariant. Let us for simplicity work in the static gauge $$\gamma^0(t)=t$$. The Euler-Lagrange (EL) equations (i.e Hamilton's equations) remain the same if we change the contact 1-form $$\Theta$$ by an exact 1-form $$P_j \mathrm{d}Q^j -K\mathrm{d}t ~=~ \Theta^{\prime}~=~\Theta- \mathrm{d}F.\tag{2}$$ From this geometric perspective, the transform law $$K~=~H + \frac{\partial F}{\partial t} \tag{3}$$ is just the standard way how the $$t$$-component $$\Theta_t=-H$$ of the contact 1-form $$\Theta$$ transforms under a change by an exact 1-form (given various other restrictions on the transformation).

References:

1. S. G. Rajeev, A Hamilton-Jacobi Formalism for Thermodynamics, Annals. Phys. 323 (2008) 2265, arXiv:0711.4319.

2. H. Geiges, An introduction to contact topology, 2008. (A pdf file of lecture notes from 2004 can be found on the author's webpage.)

• Where could I find more informations about this formulation? I've never seen this more formal approach Commented Apr 17, 2020 at 8:01
• Yeah, it's rare that textbooks give it much space. Here is a related MO post for a start. Commented Apr 17, 2020 at 8:27
• Thank you very much! Commented Apr 17, 2020 at 8:48