Classical Mechanics

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

In this post we examine the historical (physical) background that led to the introduction of symplectic manifolds in symplectic geometry. As mathematicians compiling bits and pieces picked up here and there, we may have introduced some physical inaccuracies.

Conservation of Energy

Physics, and especially mechanics, provides various formulas describing the motion of objects. Newton’s law of motion \(F=ma\), for example, is familiar even to those who have never studied physics.

One fact that became clear to physicists in the 19th century is that it is not these individual formulas but a single function that determines the motion of an object. In a sense, this fact is also somewhat related to the law of conservation of energy.

To see this, consider a phase space with one axis representing the position of an object and the other representing its momentum. For example, the phase space corresponding to a 1-dimensional space is a 2-dimensional space consisting of one position axis and one momentum axis; in general, the phase space of an \(n\)-dimensional space is a \(2n\)-dimensional space consisting of \(n\)-dimensional position coordinates and \(n\)-dimensional momentum coordinates.

In this setting, kinetic energy \(K=\frac{1}{2}mv^2\) or potential energy \(P=mgh\), once the constants \(m\) and \(g\) are fixed, are physical quantities described by position and velocity coordinates. That is, these energies are functions from phase space to \(\mathbb{R}\). The law of conservation of energy then means that when the motion of an object is described in phase space, its trajectory must lie entirely within a level set of the energy function.

For example, consider the motion of an object in one dimension subject to no force. Since the energy of this object consists only of kinetic energy, we may take the energy function \(E:\mathbb{R}^2\rightarrow\mathbb{R}\) to be \(E(x,v)=v^2\). Then the level sets of \(E\) are line segments parallel to the position axis.

By the law of conservation of energy, an object starting at a point in phase space cannot leave this level set, no matter how much time elapses. Physically, this means that when no external force acts on the object, its acceleration is zero.

The law of conservation of mechanical energy taught in high school states that the sum of kinetic and potential energy remains constant. Drawing a similar picture, the level sets of the energy function look roughly as follows.

Again, by the law of conservation of energy, the motion of the object must always remain on this level set of energy.

Finally, consider the motion of an object attached to a spring. By Hooke’s law, the potential energy imparted to the object by the spring is given by \(\frac{1}{2}kx^2\). Since the kinetic energy of the object is \(\frac{1}{2}mv^2\), we define the energy function \(E\) by \(E(x,v)=\frac{1}{2}kx^2+\frac{1}{2}mv^2\). Plotting this in phase space yields an ellipse, as shown below.

This suggests that the motion of the object will be periodic.

The pictures above are adequate when the object moves in one dimension, but as soon as the object moves in two dimensions the phase space becomes 4-dimensional, and the energy level set becomes 3-dimensional. Consequently, additional tools are needed to describe the actual curve the object follows within it.

Principle of Least Action

What plays an important role in this process is the following principle of least action.

When an object moves from \(x_0\) to \(x_1\), it follows the extremum \(z(t)=(x(t),y(t))\) (\(t_0\leq t\leq t_1\)) of the following action.

\[\mathcal{A}_H(z)=\int_{t_0}^{t_1}\langle y,\dot{x}\rangle-H(z)\mathop{dt}\]

In this formula, the newly introduced \(H\) denotes the Hamiltonian, and for our purposes we may simply regard it as the energy. This principle holds locally even when \(H\) depends on time; in that case one merely replaces \(H\) by \(H_t\). The mathematical treatment of such problems is very well known.

Proposition 1 A path \(z(t)=(x(t),y(t))\) (\(t_0\leq t\leq t_1\)) in phase space is an extremum of \(\mathcal{A}_H\) among paths satisfying the boundary conditions \(x(t_0)=x_0\), \(x(t_1)=x_1\) if and only if \(z\) satisfies the following Hamilton’s equations

\[\dot{x}=\frac{\partial H_t}{\partial y},\quad \dot{y}=-\frac{\partial H_t}{\partial x}\]

Proof

To prove this, suppose a 1-parameter family of paths \((z_s)=(x_s,y_s)\) satisfying the boundary conditions \(x_s(t_0)=x_0\), \(x_s(t_1)=x_1\) is given, and let \(z_0=z\). Then

\[\begin{aligned}\frac{\partial}{\partial s}\bigg\vert_{s=0}\mathcal{A}_H(z_s)&=\frac{\partial}{\partial s}\bigg\vert_{s=0}\int_{t_0}^{t_1}\langle y_s,\dot{x}_s\rangle-H_t(x_s,y_s)\mathop{dt}\\&=\int_{t_0}^{t_1}\frac{\partial}{\partial s}\bigg\vert_{s=0}\left(\langle y_s,\dot{x}_s\rangle-H_t(x_s,y_s)\right)\mathop{dt}\\&=\int_{t_0}^{t_1}\bigl\langle\partial_s y_s\vert_0,\dot{x}\bigr\rangle+\bigl\langle y,\partial_s\dot{x}\vert_0\bigr\rangle-\bigl\langle\partial_sx_s\vert_0,\partial_x H_t\bigr\rangle-\bigl\langle\partial_sy_s\vert_0,\partial_yH_t\bigr\rangle\mathop{dt}\end{aligned}\]

Now consider integration by parts:

\[\int_{t_0}^{t_1}\langle y,\partial_s\dot{x}_s\vert_0\rangle\mathop{dt}=\bigl[\langle y,\partial_sx_s\vert_0\rangle\bigr]_{t_0}^{t_1}-\int_{t_0}^{t_1}\langle\dot{y},\partial_sx_s\vert_0\rangle\mathop{dt}\]

The first term on the right-hand side vanishes by the boundary conditions \(x_s(t_0)=x_0\), \(x_s(t_1)=x_1\). Substituting this into the preceding equation and rearranging, we obtain

\[\frac{\partial}{\partial s}\bigg\vert_{s=0}\mathcal{A}_H(z)=\int_{t_0}^{t_1}\langle\partial_sy_s\vert_0,\dot{x}-\partial_yH_t\rangle\mathop{dt}-\int_{t_0}^{t_1}\langle\partial_sx_s\vert_0,\dot{y}+\partial_xH_t\rangle\mathop{dt}\]

Since \(\partial_sx_s\vert_0\) and \(\partial_sy_s\vert_0\) can vary arbitrarily, \(z\) being an extremum of \(\mathcal{A}_H\) is equivalent to the two equations

\[\dot{x}-\partial_yH_t=0,\qquad\dot{y}+\partial_xH_t=0\]

holding.

Consider the (linear) complex structure¹ \(J_0\in\End(\mathbb{R}^{2n})\) defined on the vector space \(\mathbb{R}^{2n}\). With respect to the basis \(\{x_1,\ldots, x_n,y_1,\ldots, y_n\}\), this linear map is given by the matrix

\[J_0=\begin{pmatrix}0&-I\\I&0\end{pmatrix}\]

This represents multiplication by the imaginary unit \(i\) when we identify \(\mathbb{R}^{2n}\) with \(\mathbb{C}^n\) via \(z_j:=x_j+iy_j\). Regarding \(\mathbb{R}^{2n}\) as a manifold and \(T_p\mathbb{R}^{2n}\cong\mathbb{R}^{2n}\) as the tangent space at each point \(p\in\mathbb{R}^{2n}\), we may view \(J_0\) as an element of \(\End(T\mathbb{R}^{2n})\) defined by the formulas

\[J_0\left(\frac{\partial}{\partial x^j}\bigg\vert_p\right)=\frac{\partial}{\partial y^j}\bigg\vert_p,\qquad J_0\left(\frac{\partial}{\partial y^j}\bigg\vert_p\right)=-\frac{\partial}{\partial x^j}\bigg\vert_p\tag{1}\]

Now

\[\nabla H=\begin{pmatrix}\partial H/\partial x\\ \partial H/\partial y\end{pmatrix}\]

so Hamilton’s equations can be written compactly as

\[\dot{z}=-J_0\nabla H(z)\]

Intuitively, the gradient \(\nabla H\) points in the direction of greatest increase of \(H\), that is, in the direction normal to the level sets of \(H\); applying \(J_0\) rotates this direction by 90 degrees, making it tangent to the level sets of \(H\), and this is precisely \(\dot{z}\).

Therefore, finding the actual trajectory of the object in phase space—that is, finding \(z\)—is exactly the problem of finding the integral flow of the following Hamiltonian vector field

\[X_H=-J_0\nabla H(z)\]

and we know that this is always possible. (§Vector Fields, ⁋Theorem 6)

Symplectic Form

To summarize the discussion above, the Hamiltonian \(H\) describes the motion of the object through the formula

\[dH=\langle-J_0\nabla H(z), -\rangle\]

Now define a \(2\)-form on \(\mathbb{R}^{2n}\) by

\[\omega_0(-,-):=\langle J_0-, -\rangle\]

Then the above formula can be written, by analogy with the definition of the gradient of a function \(f\) via \(df=\langle\nabla f,-\rangle\), as

\[dH=\omega_0(X_H, -)\]

We call \(\omega_0\) the canonical symplectic form on \(\mathbb{R}^{2n}\), and from this point of view \(X_H\) is sometimes called the symplectic gradient.

In the standard coordinate system on \(\mathbb{R}^{2n}\),

\[\langle-,-\rangle=\sum_{j=1}^n dx^j\otimes dx^j+\sum_{j=1}^n dy^j\otimes dy^j\]

so using equation (1) we can compute \(\omega_0\) on the basis vectors \(\partial/\partial x^j,\partial/\partial y^j\). For instance,

\[(\omega_0)_p\left(\frac{\partial}{\partial x^j}\bigg\vert_p,\frac{\partial}{\partial y^k}\bigg\vert_p\right)=\left\langle\frac{\partial}{\partial y^j}\bigg\vert_p,\frac{\partial}{\partial y^k}\bigg\vert_p\right\rangle_p=\delta_{jk}\]

and carrying out the remaining computations, we find that \(\omega_0\) is expressed in standard coordinates as

\[\omega_0=\sum_{j=1}^n dx^j\wedge dy^j\]

Before long we shall see that the entire discussion above remains valid when \(\mathbb{R}^{2n}\) is replaced by an arbitrary manifold \(M\), its cotangent bundle \(T^\ast M\), a Riemannian metric \(g\), and an almost complex structure \(J\).

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References

[MS] D. Mcduff and D. Salamon. Introduction to symplectic topology. Oxford graduate texts in mathematics. Oxford University Press, 2017.

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1. A linear complex structure on a vector space \(V\) is an endomorphism \(J\in\End(V)\) satisfying \(J^2=-\id\). When such a \(J\) is given, one can verify that \(V\) acquires the structure of a \(\mathbb{C}\)-vector space via the formula \((a+bi)\cdot v:=av+bJv\). ↩