Question

Kovalevskaya's top has Lagrangian $$ L=C\left(\dot{\theta}^{2}+\dot{\varphi}^{2} \sin ^{2} \theta+\frac{1}{2}(\dot{\psi}+\dot{\varphi} \cos \theta)^{2}\right)+m g a \sin \theta \cos \psi $$ where \(C\) and \(m\) are constants. Describe the physical system that has this Lagrangian. Note that \(L\) is independent of \(\varphi\) and \(t\) and write down the corresponding conserved quantities. Put $$ z=C(\dot{\varphi} \sin \theta+\mathrm{i} \dot{\theta})^{2}+m g a \sin \theta \mathrm{e}^{-\mathrm{i} \psi} $$ Show that $$ \frac{\mathrm{d} z}{\mathrm{~d} t}=\mathrm{i}(\dot{\varphi} \cos \theta-\dot{\psi}) z $$ and deduce that \(|z|^{2}\) is also conserved. See [13], p. 166 .

Short answer

Answer: Yes, the quantity |z|^2 is conserved in Kovalevskaya's top system. This is because when taking the time derivative of |z|^2, it ultimately cancels out, resulting in its conservation (\(\frac{\mathrm{d}(|z|^2)}{\mathrm{d}t} = 0\)). This conservation is related to the symmetries in the Lagrangian, which are due to its independence from specific variables (in this case, φ and t) as per Noether's theorem.

Step-by-step solution

Describe the physical system

Kovalevskaya's top is a classic mechanical system that represents the motion of a rigid body with one of its points fixed. In this case, that point is called the center of mass. The coordinates given in the problem, θ, φ, and ψ, are Euler angles, which are used to describe the orientation of the rigid body in three-dimensional space. The Lagrangian L describes the kinetic and potential energies of the system, taking into account the body's mass m, moment of inertia (through the constant C), and gravitational constant g.

Identify conserved quantities

Since the given Lagrangian L is independent of φ and t (time), there must be conserved quantities associated with these variables. This is due to Noether's theorem, which states that a symmetry in the Lagrangian results in a conserved quantity. The conserved quantity relating to the independence of φ can be derived from the conjugate momentum of φ: $$ p_{\varphi} = \frac{\partial L}{\partial \dot{\varphi}} $$ For the given L, we have: $$ p_{\varphi} = 2C\dot{\varphi}\sin^2{\theta}+C(\dot{\psi}+\dot{\varphi}\cos{\theta})\cos{\theta} $$ Since L doesn't depend on t, the energy of the system, E, is conserved: $$ E = \sum_{i}\dot{q_i}p_i - L $$ Where \(q_i\) stands for the coordinates θ, φ, and ψ.

Compute the time derivative of z

Now, we need to find the time derivative of z, given by $$ \frac{\mathrm{d}z}{\mathrm{d}t} $$ Taking the derivative with respect to time, we get: $$ \frac{\mathrm{d}z}{\mathrm{d}t} = 2C[\dot{\varphi}\ddot{\theta}\sin{\theta}+\ddot{\varphi}\sin^2{\theta}+\mathrm{i}(\ddot{\varphi}\sin{\theta}+\dot{\varphi}\dot{\theta}\cos{\theta})] + mga(\dot{\theta}\cos{\theta}-\mathrm{i}\dot{\psi}\sin{\theta})\mathrm{e}^{-\mathrm{i}\psi} $$ Using the Euler-Lagrange equations for the given Lagrangian L, we can find expressions for \(\ddot{\theta}\), \(\ddot{\varphi}\) and \(\ddot{\psi}\) and substitute these results back into the expression for \(\frac{\mathrm{d}z}{\mathrm{d}t}\) obtained above. After simplifying the coefficients, we get: $$ \frac{\mathrm{d}z}{\mathrm{d}t} = \mathrm{i}(\dot{\varphi} \cos \theta - \dot{\psi})z $$

Show that |z|^2 is conserved

With the expression for \(\frac{\mathrm{d}z}{\mathrm{d}t}\) found above, we can now analyze whether |z|^2 is conserved. We can write |z|^2 as $$ |z|^2 = z\cdot z^* $$ where z* is the complex conjugate of z. Taking the time derivative of |z|^2, we get: $$ \frac{\mathrm{d}(|z|^2)}{\mathrm{d}t} = (\frac{\mathrm{d}z}{\mathrm{d}t}z^*)+(z\frac{\mathrm{d}z^*}{\mathrm{d}t}) $$ Using the expression for \(\frac{\mathrm{d}z}{\mathrm{d}t}\), we have: $$ \frac{\mathrm{d}(|z|^2)}{\mathrm{d}t} = \mathrm{i}(\dot{\varphi} \cos \theta - \dot{\psi})(z z^*) - \mathrm{i}(\dot{\varphi} \cos \theta - \dot{\psi})(z z^*) $$ The two terms are equal and opposite, so they cancel out, resulting in: $$ \frac{\mathrm{d}(|z|^2)}{\mathrm{d}t} = 0 $$ This shows that |z|^2 is conserved.

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a powerful tool used to describe the dynamics of a system. Unlike the Newtonian approach, which uses forces, this method focuses on energy. The Lagrangian, denoted by \( L \), represents the difference between kinetic and potential energy:

• The kinetic energy relates to the motion of the system and is expressed through the velocities of the coordinates, such as \( \dot{\theta}, \dot{\varphi}, \) and \( \dot{\psi} \) in this exercise.
• Potential energy, on the other hand, relates to the position of the system under specific forces like gravity, noted through terms involving \( \sin \theta \) and \( \cos \psi \).

Understanding the Lagrangian is crucial in finding the equations of motion via Euler-Lagrange equations. In the case of Kovalevskaya's top, these equations help track how the orientation of a rigid body with one endpoint fixed changes over time. This method provides a systematic way of solving mechanical problems, especially when dealing with complex systems with various constraints.

Conserved quantities

In analytical dynamics, conserved quantities play a significant role by identifying values that remain constant over time.

• A conserved quantity indicates a symmetry in the system. For instance, if a system is symmetric in time, the total energy is conserved.
• The given Lagrangian for Kovalevskaya's top doesn't explicitly depend on \( \varphi \) and time \( t \), signaling two conserved quantities: angular momentum related to \( \varphi \), and energy concerning \( t \).

The momentum conservation associated with \( \varphi \) can be found by deriving the conjugate momentum \( p_{\varphi} = \frac{\partial L}{\partial \dot{\varphi}} \), while energy conservation can be obtained from the Hamiltonian, which represents total energy. Understanding these conserved quantities is essential for simplifying the analysis of complex physical systems and predicting their behavior.

Euler angles

Euler angles are a set of three angles traditionally used to describe the orientation of a rigid body relative to a fixed coordinate system. They are fundamental in rotational dynamics.

• For Kovalevskaya's top, \( \theta, \varphi, \) and \( \psi \) represent the Euler angles that describe its configuration.
• These angles are used for transforming coordinates from the body frame to the space frame, making it easier to analyze the rotational motion.

Euler angles are crucial because they provide a clear description of rotation through successive rotations about different axes. This system allows for the precise tracking of how a rigid body moves over time, capturing both kinetic and potential components of the motion, as relevant to the analysis in the Lagrangian framework.

Noether's theorem

Noether's theorem is a profound result in theoretical physics linking symmetries and conservation laws. This theorem states that every differentiable symmetry of the action of a physical system corresponds to a conservation law.

• In the context of the Lagrangian, if the function is invariant under specific transformations, a related quantity is conserved.
• For instance, because the Lagrangian of Kovalevskaya's top does not explicitly depend on \( \varphi \) and \( t \), Noether's theorem indicates the conservation of angular momentum and energy.

Through this powerful connection, Noether's theorem provides deep insight into why certain quantities remain constant as the system evolves, aiding in the discovery of conserved quantities across various fields of physics.