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Lagrange multipliers and constraints

The additional term in the Euler Lagrange equations, in the case of Cartesian coordinate system can be interpreted as force that is perpendicular to the constraint surface i.e. parallel to gradient of the constraint equation. The method of Lagrange constraint applies in the same manner for generalized coordinates due to invariance property of the Euler Lagrange equation, summarized as the following set of equations.d$${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}+\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial q_{i}}}.$$

Invariance under point transformations

Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates Q according to a point transformation Q = Q(q, t) which is invertible as q = q(Q, t), the new Lagrangian L′ is a function of the new coordinates and similarly for the constraints $${\begin{aligned}L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),\\\phi _{j}'(\mathbf {Q} ,t)&=\phi _{j}(\mathbf {q} (\mathbf {Q} ,t),t)\end{aligned}}$$ and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation;

$${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}+\sum _{j}\lambda _{j}{\frac {\partial \phi '_{j}}{\partial Q_{i}}}.$$

Proof

For a coordinate transformation $Q_{i}=Q_{i}(\mathbf {q} ,t)$, we have $$d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,$$ which implies that ${\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)$ which implies that ${\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}$.

It also follows that: $${\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)$$ and similarly: $${\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)$$ which imply that ${\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}$. The two derived relations can be employed in the proof.

Starting from Euler Lagrange equations in initial set of generalized coordinates, we have: $${\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial q_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial q_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial Q_{k}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\end{aligned}}$$

Since the transformation from $q\rightarrow Q$ is invertible, it follows that the form of the Euler-Lagrange equation is invariant i.e., $${\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{i}}}=0.$$

Convexity of Lagrangian in generalized velocity

Since $L\rightarrow -L$ gives another valid Lagrangian satisfying Euler Lagrange equation, the condition of convexity or concavity of Lagrangian is a matter of convention. Nonetheless, given a basic form of the Lagrangian, it can be shown, for some forms of the potential function, that the Lagrangian is either convex or concave in generalized velocity. For Newtonian mechanics and also in special relativity, given that the potential function's Hessian with respect to generalized velocities vanish or is positive definite, the resulting Lagrangian can only be strictly convex or strictly concave in generalized velocities.

Proof

Assumption The potential function has a hessian, $K_{rs}=-{\frac {\partial ^{2}V}{\partial {\dot {q}}^{r}\partial {\dot {q}}^{s}}}$, with respect to generalized velocities that is either vanishing everywhere or positive (semi) definite everywhere. The calculation of hessian of the Lagrangian, $W_{rs}={\frac {\partial ^{2}L}{\partial {\dot {q}}^{r}\partial {\dot {q}}^{s}}}$, can be used to show convexity or strict convexity by showing positive semi-definiteness or positive definiteness of the hessian, respectively. Proof 1: Newtonian Mechanics Note that the assumption also applies to potentials that are upto linear order in ${\dot {\mathbf {r} }}$ such as that of electromagnetic field $V=q(\phi -{\dot {\mathbf {r} }}\cdot \mathbf {A} )$ since their hessian with respect to generalized velocities always vanish at every point. This follows from the expansion of the velocity vector in terms of generalized velocities, ${\dot {\mathbf {r} }}=\left({\frac {\partial \mathbf {r} }{\partial q^{r}}}{\dot {q}}^{r}+{\frac {\partial \mathbf {r} }{\partial t}}\right)$. Assuming the form of the Lagrangian to be $L=T-V={\frac {1}{2}}m{\dot {\mathbf {r} }}^{2}-V$ and substituting the expression for velocity vector, the hessian can be calculated to be: $$W_{rs}={\frac {\partial ^{2}L}{\partial {\dot {q}}^{r}\partial {\dot {q}}^{s}}}=m{\frac {\partial \mathbf {r} }{\partial q^{r}}}\cdot {\frac {\partial \mathbf {r} }{\partial q^{s}}}+K_{rs}=M_{rs}+K_{rs}$$To show positive definiteness of the matrix, it is sufficient to show that $x^{T}Mx=m\left({\frac {\partial \mathbf {r} }{\partial q^{s}}}x^{s}\right)^{2}>0$ for non zero $x$. This is because $\left\{{\frac {\partial \mathbf {r} }{\partial q^{s}}}\right\}$ are linearly independent vectors for invertible coordinate transformations (and are also parallel to the local basis vectors for the new coordinates) implying the quantity will only be zero for a zero vector. Hence it is proved that $W$ is a positive definite matrix implying strict convexity of Lagrangian of the given form in generalized velocity. Proof 2: Special Relativity Taking the relativistic Lagrangian, $L=-{\frac {m_{0}c^{2}}{\gamma ({\dot {\mathbf {r} }})}}-V$ and employing the same steps, the hessian of the Lagrangian is calculated to be: $$W_{rs}={\frac {\partial L}{\partial {\dot {q}}^{r}\partial {\dot {q}}^{s}}}={m_{0}c^{2}}{\gamma ({\dot {\mathbf {r} }})}{\frac {\partial ^{2}{\frac {v^{2}}{2}}}{\partial {\dot {q}}^{r}\partial {\dot {q}}^{s}}}+m_{0}c^{2}\gamma ^{3}({\dot {\mathbf {r} }}){\frac {\partial {\frac {v^{2}}{2}}}{\partial {\dot {q}}^{s}}}{\frac {\partial {\frac {v^{2}}{2}}}{\partial {\dot {q}}^{r}}}+K_{rs}=M'_{rs}+M_{rs}+K_{rs}$$ Since the first term is known to be positive definite, it is sufficient to show semi-positive definiteness of the second term which is easily shown as, $x^{T}Mx=m_{0}c^{2}\gamma ^{3}({\dot {\mathbf {r} }})\left({\frac {\partial {\frac {v^{2}}{2}}}{\partial {\dot {q}}^{s}}}x^{s}\right)^{2}\geq 0$. Hence it is proved that $W$ is positive definite matrix implying strict convexity of Lagrangian of the given form in generalized velocity.