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Symplectic Geometry

Description: This quiz covers the fundamental concepts and applications of Symplectic Geometry, a branch of differential geometry that studies symplectic manifolds and their properties.
Number of Questions: 14
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Tags: symplectic geometry symplectic manifolds hamiltonian mechanics poisson brackets
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What is a symplectic manifold?

  1. A manifold equipped with a symplectic form.

  2. A manifold with a Riemannian metric.

  3. A manifold with a complex structure.

  4. A manifold with a flat connection.

Show answer
Correct Option: A
Explanation:

A symplectic manifold is a manifold equipped with a closed, non-degenerate 2-form (\omega), called the symplectic form.

What is a symplectic form?

  1. A closed, non-degenerate 2-form.

  2. A closed, non-degenerate 1-form.

  3. An exact 2-form.

  4. A harmonic 2-form.

Show answer
Correct Option: A
Explanation:

A symplectic form is a closed, non-degenerate 2-form (\omega) on a manifold, meaning that (d\omega = 0) and (\omega(X, Y) = 0) for all tangent vectors (X, Y) implies (X = Y).

What is the symplectic gradient of a function?

  1. The vector field (X_f) defined by (\omega(X_f, Y) = df(Y)) for all tangent vectors (Y).

  2. The vector field (X_f) defined by (\omega(X_f, Y) = -df(Y)) for all tangent vectors (Y).

  3. The vector field (X_f) defined by (\omega(X_f, Y) = f(Y)) for all tangent vectors (Y).

  4. The vector field (X_f) defined by (\omega(X_f, Y) = -f(Y)) for all tangent vectors (Y).

Show answer
Correct Option: A
Explanation:

The symplectic gradient of a function (f) on a symplectic manifold is the vector field (X_f) defined by (\omega(X_f, Y) = df(Y)) for all tangent vectors (Y).

What is the Hamiltonian vector field of a function?

  1. The vector field (X_H) defined by (\omega(X_H, Y) = dH(Y)) for all tangent vectors (Y).

  2. The vector field (X_H) defined by (\omega(X_H, Y) = -dH(Y)) for all tangent vectors (Y).

  3. The vector field (X_H) defined by (\omega(X_H, Y) = H(Y)) for all tangent vectors (Y).

  4. The vector field (X_H) defined by (\omega(X_H, Y) = -H(Y)) for all tangent vectors (Y).

Show answer
Correct Option: A
Explanation:

The Hamiltonian vector field of a function (H) on a symplectic manifold is the vector field (X_H) defined by (\omega(X_H, Y) = dH(Y)) for all tangent vectors (Y).

What is the Poisson bracket of two functions?

  1. The function ({f, g}) defined by ({f, g}(x) = \omega(X_f(x), X_g(x))).

  2. The function ({f, g}) defined by ({f, g}(x) = -\omega(X_f(x), X_g(x))).

  3. The function ({f, g}) defined by ({f, g}(x) = f(x)g(x)).

  4. The function ({f, g}) defined by ({f, g}(x) = -f(x)g(x)).

Show answer
Correct Option: A
Explanation:

The Poisson bracket of two functions (f) and (g) on a symplectic manifold is the function ({f, g}) defined by ({f, g}(x) = \omega(X_f(x), X_g(x))), where (X_f) and (X_g) are the Hamiltonian vector fields of (f) and (g), respectively.

What is the symplectic form on the cotangent bundle of a manifold?

  1. The canonical symplectic form (\omega = d\theta) on the cotangent bundle (T^*M).

  2. The canonical symplectic form (\omega = -d\theta) on the cotangent bundle (T^*M).

  3. The canonical symplectic form (\omega = \theta) on the cotangent bundle (T^*M).

  4. The canonical symplectic form (\omega = -\theta) on the cotangent bundle (T^*M).

Show answer
Correct Option: A
Explanation:

The canonical symplectic form on the cotangent bundle (T^*M) of a manifold (M) is the 2-form (\omega = d\theta), where (\theta) is the canonical 1-form on (T^*M).

What is the symplectic form on the phase space of a Hamiltonian system?

  1. The canonical symplectic form (\omega = d\theta) on the phase space (\mathbb{R}^{2n}).

  2. The canonical symplectic form (\omega = -d\theta) on the phase space (\mathbb{R}^{2n}).

  3. The canonical symplectic form (\omega = \theta) on the phase space (\mathbb{R}^{2n}).

  4. The canonical symplectic form (\omega = -\theta) on the phase space (\mathbb{R}^{2n}).

Show answer
Correct Option: A
Explanation:

The canonical symplectic form on the phase space (\mathbb{R}^{2n}) of a Hamiltonian system is the 2-form (\omega = d\theta), where (\theta) is the canonical 1-form on (\mathbb{R}^{2n}).

What is the symplectic form on the space of loops in a symplectic manifold?

  1. The Weil-Petersson symplectic form (\omega = \int_\gamma \omega) on the space of loops in a symplectic manifold (M).

  2. The Weil-Petersson symplectic form (\omega = -\int_\gamma \omega) on the space of loops in a symplectic manifold (M).

  3. The Weil-Petersson symplectic form (\omega = \int_\gamma \theta) on the space of loops in a symplectic manifold (M).

  4. The Weil-Petersson symplectic form (\omega = -\int_\gamma \theta) on the space of loops in a symplectic manifold (M).

Show answer
Correct Option: A
Explanation:

The Weil-Petersson symplectic form on the space of loops in a symplectic manifold (M) is the 2-form (\omega = \int_\gamma \omega), where (\omega) is the symplectic form on (M) and (\gamma) is a loop in (M).

What is the symplectic form on the space of paths in a symplectic manifold?

  1. The Hofer-Zehnder symplectic form (\omega = \int_\gamma \omega) on the space of paths in a symplectic manifold (M).

  2. The Hofer-Zehnder symplectic form (\omega = -\int_\gamma \omega) on the space of paths in a symplectic manifold (M).

  3. The Hofer-Zehnder symplectic form (\omega = \int_\gamma \theta) on the space of paths in a symplectic manifold (M).

  4. The Hofer-Zehnder symplectic form (\omega = -\int_\gamma \theta) on the space of paths in a symplectic manifold (M).

Show answer
Correct Option: A
Explanation:

The Hofer-Zehnder symplectic form on the space of paths in a symplectic manifold (M) is the 2-form (\omega = \int_\gamma \omega), where (\omega) is the symplectic form on (M) and (\gamma) is a path in (M).

What is the symplectic form on the space of Hamiltonian diffeomorphisms of a symplectic manifold?

  1. The Arnold-Liouville symplectic form (\omega = \int_M \omega) on the space of Hamiltonian diffeomorphisms of a symplectic manifold (M).

  2. The Arnold-Liouville symplectic form (\omega = -\int_M \omega) on the space of Hamiltonian diffeomorphisms of a symplectic manifold (M).

  3. The Arnold-Liouville symplectic form (\omega = \int_M \theta) on the space of Hamiltonian diffeomorphisms of a symplectic manifold (M).

  4. The Arnold-Liouville symplectic form (\omega = -\int_M \theta) on the space of Hamiltonian diffeomorphisms of a symplectic manifold (M).

Show answer
Correct Option: A
Explanation:

The Arnold-Liouville symplectic form on the space of Hamiltonian diffeomorphisms of a symplectic manifold (M) is the 2-form (\omega = \int_M \omega), where (\omega) is the symplectic form on (M).

What is the symplectic form on the space of Lagrangian submanifolds of a symplectic manifold?

  1. The Weinstein symplectic form (\omega = \int_L \omega) on the space of Lagrangian submanifolds of a symplectic manifold (M).

  2. The Weinstein symplectic form (\omega = -\int_L \omega) on the space of Lagrangian submanifolds of a symplectic manifold (M).

  3. The Weinstein symplectic form (\omega = \int_L \theta) on the space of Lagrangian submanifolds of a symplectic manifold (M).

  4. The Weinstein symplectic form (\omega = -\int_L \theta) on the space of Lagrangian submanifolds of a symplectic manifold (M).

Show answer
Correct Option: A
Explanation:

The Weinstein symplectic form on the space of Lagrangian submanifolds of a symplectic manifold (M) is the 2-form (\omega = \int_L \omega), where (\omega) is the symplectic form on (M) and (L) is a Lagrangian submanifold of (M).

What is the symplectic form on the space of symplectic embeddings of a symplectic manifold into another symplectic manifold?

  1. The Gromov symplectic form (\omega = \int_M \omega_1 - \int_N \omega_2) on the space of symplectic embeddings of a symplectic manifold (M) into another symplectic manifold (N).

  2. The Gromov symplectic form (\omega = -\int_M \omega_1 + \int_N \omega_2) on the space of symplectic embeddings of a symplectic manifold (M) into another symplectic manifold (N).

  3. The Gromov symplectic form (\omega = \int_M \theta_1 - \int_N \theta_2) on the space of symplectic embeddings of a symplectic manifold (M) into another symplectic manifold (N).

  4. The Gromov symplectic form (\omega = -\int_M \theta_1 + \int_N \theta_2) on the space of symplectic embeddings of a symplectic manifold (M) into another symplectic manifold (N).

Show answer
Correct Option: A
Explanation:

The Gromov symplectic form on the space of symplectic embeddings of a symplectic manifold (M) into another symplectic manifold (N) is the 2-form (\omega = \int_M \omega_1 - \int_N \omega_2), where (\omega_1) and (\omega_2) are the symplectic forms on (M) and (N), respectively.

What is the symplectic form on the space of Hamiltonian actions of a Lie group on a symplectic manifold?

  1. The Marsden-Weinstein symplectic form (\omega = \int_M \omega - \int_G \theta) on the space of Hamiltonian actions of a Lie group (G) on a symplectic manifold (M).

  2. The Marsden-Weinstein symplectic form (\omega = -\int_M \omega + \int_G \theta) on the space of Hamiltonian actions of a Lie group (G) on a symplectic manifold (M).

  3. The Marsden-Weinstein symplectic form (\omega = \int_M \theta - \int_G \omega) on the space of Hamiltonian actions of a Lie group (G) on a symplectic manifold (M).

  4. The Marsden-Weinstein symplectic form (\omega = -\int_M \theta + \int_G \omega) on the space of Hamiltonian actions of a Lie group (G) on a symplectic manifold (M).

Show answer
Correct Option: A
Explanation:

The Marsden-Weinstein symplectic form on the space of Hamiltonian actions of a Lie group (G) on a symplectic manifold (M) is the 2-form (\omega = \int_M \omega - \int_G \theta), where (\omega) is the symplectic form on (M) and (\theta) is the canonical 1-form on (G).

What is the symplectic form on the space of symplectic vector spaces?

  1. The Kostant-Kirillov symplectic form (\omega = \int_V \omega) on the space of symplectic vector spaces (V).

  2. The Kostant-Kirillov symplectic form (\omega = -\int_V \omega) on the space of symplectic vector spaces (V).

  3. The Kostant-Kirillov symplectic form (\omega = \int_V \theta) on the space of symplectic vector spaces (V).

  4. The Kostant-Kirillov symplectic form (\omega = -\int_V \theta) on the space of symplectic vector spaces (V).

Show answer
Correct Option: A
Explanation:

The Kostant-Kirillov symplectic form on the space of symplectic vector spaces (V) is the 2-form (\omega = \int_V \omega), where (\omega) is the symplectic form on (V).

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