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1. Assume

• (i) that the kinetic term $T$ is quadratic in generalized velocities $\dot{\bf u}$;

• (ii) that the potential term $U$ is independent of the generalized velocities $\dot{\bf u}$; and

• (iii) that the Lagrangian $L=T-U$ does not depend explicitly on time.

2. Case without holonomic constraints. The energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. e.g this Phys.SE post.

3. Case with holonomic constraints without explicit time dependence.

• (i) Either we can formally eliminate variables such that there are no holonomic constraints left; or

• (ii) Alternatively, we can impose the constraints by introducing Lagrange multipliers, which we add to the list of variables ${\bf u}$.

In both cases, the form 1 is maintained, and we can apply the conclusion from section 2: Energy is still conserved.

1. Assume

• (i) that the kinetic term $T$ is quadratic in generalized velocities $\dot{\bf u}$;

• (ii) that the potential term $U$ is independent of the generalized velocities $\dot{\bf u}$; and

• (iii) that the Lagrangian $L=T-U$ does not depend explicitly on time.

2. Case without holonomic constraints. The energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. e.g this Phys.SE post.

3. Case with holonomic constraints $G({\bf u})\approx 0$ without explicit time dependence.

• (i) Either we can formally eliminate variables such that there are no holonomic constraints left; or

• (ii) alternatively, we can introduce Lagrange multipliers, which we add to the list of variables ${\bf u}$, and add terms of the form 'Lagrange multiplier times $G$' to the potential term $U$. (The notion of potential energy will be unaltered on-shell.)

In both cases, the form 1 is maintained, and we can apply the conclusion from section 2: Energy is still conserved.

1. Assume (i) that the kinetic term $T$ is quadratic in generalized velocities $\dot{\bf u}$; (ii) that the potential term $U$ is independent of the generalized velocities $\dot{\bf u}$; and (iii) that the Lagrangian $L=T-U$ does not depend explicitly on time.

2. Case without holonomic constraints. The energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. Noether's theorem.

3. Case with holonomic constraints without explicit time dependence. We can formally eliminate variables such that there are no holonomic constraints left, and such that the form 1 is maintained. Then we can apply the conclusion from section 2: Energy is still conserved.

1. Assume

• (i) that the kinetic term $T$ is quadratic in generalized velocities $\dot{\bf u}$;

• (ii) that the potential term $U$ is independent of the generalized velocities $\dot{\bf u}$; and

• (iii) that the Lagrangian $L=T-U$ does not depend explicitly on time.

2. Case without holonomic constraints. The energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. e.g this Phys.SE post.

3. Case with holonomic constraints without explicit time dependence.

• (i) Either we can formally eliminate variables such that there are no holonomic constraints left; or

• (ii) Alternatively, we can impose the constraints by introducing Lagrange multipliers, which we add to the list of variables ${\bf u}$.

In both cases, the form 1 is maintained, and we can apply the conclusion from section 2: Energy is still conserved.

Yes, energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. Noether's theorem.

1. Assume (i) that the kinetic term $T$ is quadratic in generalized velocities $\dot{\bf u}$; (ii) that the potential term $U$ is independent of the generalized velocities $\dot{\bf u}$; and (iii) that the Lagrangian $L=T-U$ does not depend explicitly on time.

2. Case without holonomic constraints. The energy $h=\dot{\bf u}\cdot \frac{\partial L}{\partial \dot{\bf u}}-L=T+U$ is conserved because the Lagrangian $L=T-U$ does not depend explicitly on time, cf. Noether's theorem.

3. Case with holonomic constraints without explicit time dependence. We can formally eliminate variables such that there are no holonomic constraints left, and such that the form 1 is maintained. Then we can apply the conclusion from section 2: Energy is still conserved.