A Parametric Lagrangian is a Homogeneous Form in Parametric Velocities?

AI Thread Summary
The discussion centers on the characterization of the Lagrangian ##L_1## as a homogeneous form of the first order in parametric velocities, as stated by Cornelius Lanczos. Initially, there is confusion regarding this classification, particularly in relation to the canonical integral and its variations. An example of a Lagrangian for planetary motion is presented, illustrating that it does not appear to be a homogeneous form in the specified variables. However, the realization that negative powers are permissible leads to the conclusion that ##L_1## indeed qualifies as a homogeneous form of the first order, aligning with Euler's theorem on homogeneous functions. This clarification resolves the initial misunderstanding about the nature of ##L_1##.
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Confusion about a book "The Variational Principles of Mechanics" by Cornelius Lanczos, equation (610.4) where Lanczos uses the fact that "The function ##L_1## is a homogeneous form of the first order..."
In the book "The Variational Principles of Mechanics" by Cornelius Lanczos, the following statement is made about a lagrangian ##L_1## where time is given as an dependent parameter, and a new parameter ##\tau## is introduced as the independent variable, see (610.3) and (610.4) pg. 186,187 Dover Fourth edition:
The function ##L_1## is a homogeneous form of the first order in the ##n+1## variables ##q_1', \dots , q_{n+1}'##.
where ##L_1## was defined as
$$L_1 = L\left(q_1, \dots q_{n+1}; \frac{q_1'}{q_{n+1}'} , \dots , \frac{q_n'}{q_{n+1}'}\right) q_{n+1}' \text{ where } q_i' = \frac{\mathrm d q_i}{\mathrm d \tau} \text{ and } q_{n+1} = t \text{ (time)}.$$
This fact is used to show that the Hamiltonian, defined in the usual way for this new parametrisation, is identically zero.

I don't see how ##L_1## is a homogeneous form. Can someone explain this? I could understand if Lanczos was referring to the canonical integral, which is indeed a homogeneous form in the ##\dot{q_i}##. The canonical integral in the Hamiltonian formulation is:
$$A = \int_{t_0}^{t^1} \sum_i p_i \dot{q_i} - H((q_i)_i, (p_i)_i) \, \mathrm d t \, .$$
As I understand, the canonical integral for action has exactly the same variation as ##\int_{t_0}^{t_1} L \, \mathrm d t## under variations ##\delta q_i##. This is not an obvious fact to me but it seems this is the case, as shown in the text.

In general the function ##L_1## defined above is certainly not going to be a homogeneous form... i.e. take the lagrangian for planetary motion confined to the plane:
$$L(r,\theta;\dot{r},\dot{\theta}) = \frac{m}{2} \left( \dot{r}^2 + r^2 \dot{\theta}^2 \right) + \frac{GMm}{r}$$
where ##M## is the mass of the massive gravitational body (e.g. the Sun). Here, we have
$$L_1\left(r,\theta;\frac{\mathrm d r}{\mathrm d \tau}, \frac{\mathrm d \theta}{\mathrm d \tau}\right) = L\left(r,\theta;\frac{\mathrm d r}{\mathrm d \tau}\left(\frac{\mathrm d t}{\mathrm d \tau}\right)^{-1}, \frac{\mathrm d \theta}{\mathrm d \tau}\left(\frac{\mathrm d t}{\mathrm d \tau}\right)^{-1}\right)\frac{\mathrm d t}{\mathrm d \tau} = \frac{m}{2} \left( \left(\frac{\mathrm d r}{\mathrm d \tau}\right)^2 + r^2 \left(\frac{\mathrm d \theta}{\mathrm d \tau}\right)^2 \right) \left(\frac{\mathrm d t}{\mathrm d \tau}\right)^{-1} + \frac{GMm}{r} \frac{\mathrm d t}{\mathrm d \tau}$$
which is clearly not a homogeneous form in ##\frac{\mathrm d r}{\mathrm d \tau}, \frac{\mathrm d \theta}{\mathrm d \tau}, \frac{\mathrm d t}{\mathrm d \tau}##.
 
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I see my mistake now. The last equation actually is a homogeneous form of the first order. I had not realized that negative powers were permitted. I was not aware of Euler's theorem on homogeneous functions. This is mentioned in the text viz. (610.4) but I had not realized it would apply even to this kind of form. Problem solved
 
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