- #1
Super_Leunam
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Hi!
I've been studying Dirac's programme for some time and I realized that there's something missing:
Actually this is missing in every standard book on classical mechanics concerning how constraints are implemented in the lagrangian.
They are usually inserted with some unknown variables called "Lagrange's Multipliers". In Dirac's theory of constrained systems these are derived as a consequence of a "theorem" appearing in Henneaux and teitelboim's "Quantization of gauge systems", page 8. Dirac, in his "Lectures on Quantum Mechanics" didn't even say a word as to how it should be derived, he simply said that it's part of the standard methods of the calculus of variations with constraints and I assume he was referring to the well known method of Lagrange's multipliers. This theorem is proved in the appendix for the first chapter of Henneaux's but I kinda feel like there's something missing.
<br /> \text{If } \lambda_n\delta q^n + \mu^n \delta p_n = 0 for arbitrary variations of the coordinates of the phase (q and p) space tangent to the constraints surface then:
<br /> \lambda_n = u^m\frac{\partial \phi_m}{\partial q^n}
<br /> \mu^n = u^m\frac{\partial \phi_m}{\partial p_n} <br />
Where the u's are the multipliers mentioned and \phi_m are the constraints derived from the Lagrangian because of it being "singular" as explained below. By the way, the proof of this theorem in Henneaux's book is almost a copy of the proof for the method of Lagrange's multipliers found in any Analysis book I think the problem stems from the definition of the Hamiltonian. I know that the Hamiltonian is defined as a Legendre Transform
<br /> {\cal H}(q, p) = \textit{Ext}_{v} H (q, v, p) = \textit{Ext}_{v} \{p\cdot v - {\cal L}(q, v)\}<br />
Where "Ext" denotes an extremum, I still don't worry about its uniqueness ,though. It follows, according to the rules of calculus and under some technical conditions, that we should have for an extremum ( if they exist, that is) that it satisfies
<br /> p = \frac{\partial {\cal L}}{\partial v} <br />
Then we can replace all the velocities back to the function H
The conditions for this to work out is that the lagrangian be not singular, in order to solve the velocities in a unique way, that is,
<br /> det\left(\frac{{\partial}^2 {\cal L}}{\partial v_i \partial v_j}\right) \neq 0<br />
But what if that doesn't happen? (in certain region of the tangent bundle)
This is precisely the case when we have a constrained system and this is where that little theorem I mentioned is to be used.
My Problem is the following: Lagrange multipliers are to be used when we want to find critical points for a function restricted to some conditions or "constraints" \phi_j. I thought that the function to be optimized was the H(q, v, p) but so far I just don't seem to make it work since Lagrange's method requires that the gradient of H with respect to the velocities arguments be proportional to the gradient of those restrictions with respect to the same velocities.
The constraints that one derives when having a singular Lagrangian are functions of p and q but not of the velocities. Moreover, if we want to reconcile this assumption with that theorem, the problem is that the gradients considered should be with respect to q and p but not with respect to v . If I want to consider that kind of gradient ( involving derivatives with respect to q and p) that would mean that I'm trying to optimize H considering p and q as variables too, which I don't think is correct.
The final result I want to obtain directly using the method of Lagrange's multipliers as stated in any Analysis book is
<br /> v = \frac{\partial {\cal H}}{\partial p_n} + u^m \frac{\partial \phi_m}{\partial p_n}<br />
<br /> - \left.\frac{\partial {\cal L}}{\partial q^n}\right|_{v}= \left.\frac{\partial {\cal H}}{\partial q^n}\right|_p + u^m \frac{\partial \phi_m}{\partial q^n}<br />
with v = \dot{q}
which are derived from standard considerations about the hamiltonian using that little theorem. I'll just copy the steps Henneaux considered on page 8 :
Consider arbitrary independent variations of the positions and velocities \delta q^n, \hspace{5pt}\delta p_n which should be "tangent" to the surface generated by the \phi_m in order to preserve these constraints in first order.(This argument of tangency isn't included in the book, I had to think about it). Then the change induced on H is
<br /> \delta H = \dot{q}_n\delta p_n +\delta\dot{q}_np_n - \delta\dot{q}\frac{\partial L}{\partial \dot{q}_n} - \delta q_n\frac{\partial L}{\partial q_n}<br /> = \dot{q}_n\delta p_n - \frac{\partial L}{\partial q_n}\delta q_n<br />
But it should be also that
<br /> \delta H = \frac{\partial H}{\partial q_n}\delta q_n + \frac{\partial H}{\partial p_n}\delta p_n = \dot{q}_n\delta p_n - \frac{\partial L}{\partial q_n}\delta q_n<br />
<br /> \left(\frac{\partial H}{\partial q_n} + \frac{\partial L}{\partial q_n}\right)\delta q_n + \left(\frac{\partial H}{\partial p_n} - \dot{q}_n\right)\delta p_n = 0 <br />
where \delta p_n are not independent variations but are regarded as linear combinations of \delta q^n \text{and } \delta \dot{q^n} (Here I'm using the notation from Henneaux's book where \dot{q} = v).
And, using that little theorem, Henneaux is able to obtain the desired result for the Hamilton's equations of motion for constrained systems. But I still can't obtain this result from the considerations I've made. ( using directly Lagrange's method so I can justify where those multipliers come from)
What do you think is the origin of those multipliers? In other words, what am I optimizing?
I've been studying Dirac's programme for some time and I realized that there's something missing:
Actually this is missing in every standard book on classical mechanics concerning how constraints are implemented in the lagrangian.
They are usually inserted with some unknown variables called "Lagrange's Multipliers". In Dirac's theory of constrained systems these are derived as a consequence of a "theorem" appearing in Henneaux and teitelboim's "Quantization of gauge systems", page 8. Dirac, in his "Lectures on Quantum Mechanics" didn't even say a word as to how it should be derived, he simply said that it's part of the standard methods of the calculus of variations with constraints and I assume he was referring to the well known method of Lagrange's multipliers. This theorem is proved in the appendix for the first chapter of Henneaux's but I kinda feel like there's something missing.
<br /> \text{If } \lambda_n\delta q^n + \mu^n \delta p_n = 0 for arbitrary variations of the coordinates of the phase (q and p) space tangent to the constraints surface then:
<br /> \lambda_n = u^m\frac{\partial \phi_m}{\partial q^n}
<br /> \mu^n = u^m\frac{\partial \phi_m}{\partial p_n} <br />
Where the u's are the multipliers mentioned and \phi_m are the constraints derived from the Lagrangian because of it being "singular" as explained below. By the way, the proof of this theorem in Henneaux's book is almost a copy of the proof for the method of Lagrange's multipliers found in any Analysis book I think the problem stems from the definition of the Hamiltonian. I know that the Hamiltonian is defined as a Legendre Transform
<br /> {\cal H}(q, p) = \textit{Ext}_{v} H (q, v, p) = \textit{Ext}_{v} \{p\cdot v - {\cal L}(q, v)\}<br />
Where "Ext" denotes an extremum, I still don't worry about its uniqueness ,though. It follows, according to the rules of calculus and under some technical conditions, that we should have for an extremum ( if they exist, that is) that it satisfies
<br /> p = \frac{\partial {\cal L}}{\partial v} <br />
Then we can replace all the velocities back to the function H
The conditions for this to work out is that the lagrangian be not singular, in order to solve the velocities in a unique way, that is,
<br /> det\left(\frac{{\partial}^2 {\cal L}}{\partial v_i \partial v_j}\right) \neq 0<br />
But what if that doesn't happen? (in certain region of the tangent bundle)
This is precisely the case when we have a constrained system and this is where that little theorem I mentioned is to be used.
My Problem is the following: Lagrange multipliers are to be used when we want to find critical points for a function restricted to some conditions or "constraints" \phi_j. I thought that the function to be optimized was the H(q, v, p) but so far I just don't seem to make it work since Lagrange's method requires that the gradient of H with respect to the velocities arguments be proportional to the gradient of those restrictions with respect to the same velocities.
The constraints that one derives when having a singular Lagrangian are functions of p and q but not of the velocities. Moreover, if we want to reconcile this assumption with that theorem, the problem is that the gradients considered should be with respect to q and p but not with respect to v . If I want to consider that kind of gradient ( involving derivatives with respect to q and p) that would mean that I'm trying to optimize H considering p and q as variables too, which I don't think is correct.
The final result I want to obtain directly using the method of Lagrange's multipliers as stated in any Analysis book is
<br /> v = \frac{\partial {\cal H}}{\partial p_n} + u^m \frac{\partial \phi_m}{\partial p_n}<br />
<br /> - \left.\frac{\partial {\cal L}}{\partial q^n}\right|_{v}= \left.\frac{\partial {\cal H}}{\partial q^n}\right|_p + u^m \frac{\partial \phi_m}{\partial q^n}<br />
with v = \dot{q}
which are derived from standard considerations about the hamiltonian using that little theorem. I'll just copy the steps Henneaux considered on page 8 :
Consider arbitrary independent variations of the positions and velocities \delta q^n, \hspace{5pt}\delta p_n which should be "tangent" to the surface generated by the \phi_m in order to preserve these constraints in first order.(This argument of tangency isn't included in the book, I had to think about it). Then the change induced on H is
<br /> \delta H = \dot{q}_n\delta p_n +\delta\dot{q}_np_n - \delta\dot{q}\frac{\partial L}{\partial \dot{q}_n} - \delta q_n\frac{\partial L}{\partial q_n}<br /> = \dot{q}_n\delta p_n - \frac{\partial L}{\partial q_n}\delta q_n<br />
But it should be also that
<br /> \delta H = \frac{\partial H}{\partial q_n}\delta q_n + \frac{\partial H}{\partial p_n}\delta p_n = \dot{q}_n\delta p_n - \frac{\partial L}{\partial q_n}\delta q_n<br />
<br /> \left(\frac{\partial H}{\partial q_n} + \frac{\partial L}{\partial q_n}\right)\delta q_n + \left(\frac{\partial H}{\partial p_n} - \dot{q}_n\right)\delta p_n = 0 <br />
where \delta p_n are not independent variations but are regarded as linear combinations of \delta q^n \text{and } \delta \dot{q^n} (Here I'm using the notation from Henneaux's book where \dot{q} = v).
And, using that little theorem, Henneaux is able to obtain the desired result for the Hamilton's equations of motion for constrained systems. But I still can't obtain this result from the considerations I've made. ( using directly Lagrange's method so I can justify where those multipliers come from)
What do you think is the origin of those multipliers? In other words, what am I optimizing?
Last edited:
. I just remebered that. so..The Hamiltonian isn't always a Legendre Transform, properly 
... 
I don't like this kind of Hamiltonians