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Legendre transformation

In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function.

In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.

For sufficiently smooth functions on the real line, the Legendre transform $$f^*$$ of a function $$f$$ can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as $$Df(\cdot) = \left( D f^* \right)^{-1}(\cdot)~,$$ where $$D$$ is an operator of differentiation, $$\cdot$$ represents an argument or input to the associated function, $$(\phi)^{-1}(\cdot)$$ is an inverse function such that $$(\phi) ^{-1}(\phi(x))=x$$,

or equivalently, as $$f'(f^{*\prime}(x^*)) = x^*$$ and $$f^{*\prime}(f'(x)) = x$$ in Lagrange's notation.

The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function's convex hull.

Definition in $$\mathbb{R}$$
Let $$I \sub \R$$ be an interval, and $$f:I \to \R$$ a convex function; then the Legendre transform of $$f$$ is the function $$f^*:I^* \to \R$$ defined by $$f^*(x^*) = \sup_{x\in I}(x^*x-f(x)),\ \ \ \ I^*= \left \{x^*\in \R:f^*(x^*)<\infty \right \} ~$$ where $\sup$ denotes the supremum over $$I$$, e.g., $x$  in $I$  is chosen such that $x^*x - f(x)$  is maximized at each $x^*$, or $x^*$  is such that $$x^*x-f(x)$$ as a bounded value throughout $x$  exists (e.g., when $$f(x)$$ is a linear function).

The transform is always well-defined when $$f(x)$$ is convex. This definition requires $$x^*x - f(x)$$ to be bounded from above in $$I$$ in order for the supremum to exist.

Definition in $$\mathbb{R}^{n}$$
The generalization to convex functions $$f:X \to \R$$ on a convex set $$X \sub \R^n$$ is straightforward: $$f^*:X^* \to \R$$ has domain $$X^*= \left \{x^* \in \R^n:\sup_{x\in X}(\langle x^*,x\rangle-f(x))<\infty \right \}$$ and is defined by $$f^*(x^*) = \sup_{x\in X}(\langle x^*,x\rangle-f(x)),\quad x^*\in X^* ~,$$ where $$\langle x^*,x \rangle$$ denotes the dot product of $$x^*$$ and $$x$$.

The function $$f^*$$ is called the convex conjugate function of $$f$$. For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted $$p$$, instead of $$x^*$$. If the convex function $$f$$ is defined on the whole line and is everywhere differentiable, then $$f^*(p)=\sup_{x\in I}(px - f(x)) = \left( p x - f(x) \right)|_{x = (f')^{-1}(p)} $$ can be interpreted as the negative of the $y$-intercept of the tangent line to the graph of $$f$$ that has slope $$p$$.

The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by $$f$$ can be represented equally well as a set of $$(x,y)$$ points, or as a set of tangent lines specified by their slope and intercept values.

Understanding the Legendre transform in terms of derivatives
For a differentiable convex function $$f$$ on the real line with the first derivative $$f'$$ and its inverse $$(f')^{-1}$$, the Legendre transform of $$f$$, $$ f^*$$, can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other, i.e., $$f' = ((f^*)')^{-1}$$ and $$(f^*)' = (f')^{-1}$$.

To see this, first note that if $$ f$$ as a convex function on the real line is differentiable and $$ \overline{x} $$ is a critical point of the function of $$ x \mapsto p \cdot x -f(x) $$, then the supremum is achieved at $ \overline{x}$ (by convexity, see the first figure in this Wikipedia page). Therefore, the Legendre transform of $$ f$$ is $$ f^*(p)= p \cdot \overline{x} - f(\overline{x})$$.

Then, suppose that the first derivative $$f'$$ is invertible and let the inverse be $$ g = (f')^{-1} $$. Then for each $ p$, the point $$ g(p)$$ is the unique critical point $ \overline{x}$ of the function $$ x \mapsto px -f(x) $$ (i.e., $$ \overline{x} = g(p)$$) because $$ f'(g(p))=p $$ and the function's first derivative with respect to $$x$$ at $$ g(p)$$ is $$ p-f'(g(p))=0 $$. Hence we have $$ f^*(p) = p \cdot g(p) - f(g(p))$$ for each $ p$. By differentiating with respect to $ p$, we find $$(f^*)'(p) = g(p)+ p \cdot g'(p) - f'(g(p)) \cdot g'(p).$$ Since $$ f'(g(p))=p$$ this simplifies to $$(f^*)'(p) = g(p) = (f')^{-1}(p)$$. In other words, $$(f^*)'$$ and $$f'$$ are inverses to each other.

In general, if $$ h' = (f')^{-1} $$ as the inverse of $$ f',$$ then $$ h' = (f^*)' $$ so integration gives $$ f^* = h +c.$$ with a constant $$ c. $$

In practical terms, given $$f(x),$$ the parametric plot of $$xf'(x)-f(x)$$ versus $$f'(x)$$ amounts to the graph of $$f^*(p)$$ versus $$p.$$

In some cases (e.g. thermodynamic potentials, below), a non-standard requirement is used, amounting to an alternative definition of $f *$ with a minus sign, $$f(x) - f^*(p) = xp.$$

Formal Definition in Physics Context
In analytical mechanics and thermodynamics, Legendre transformation is usually defined as follows: suppose $$f$$ is a function of $$x,$$ then we have

 * $$\mathrm{d} f = \frac{\mathrm{d} f}{\mathrm{d} x} \mathrm{d} x.$$

performing Legendre transformation on this function means that we take $$p = \frac{\mathrm{d} f}{\mathrm{d} x}$$ as the independent variable, so that the above expression can be written as

 * $$\mathrm{d} f = p \mathrm{d} x,$$

and according to Leibniz's rule $$\mathrm{d} (uv) = u\mathrm{d} v + v\mathrm{d} u,$$ then we have

 * $$\mathrm{d} \left(x p - f \right) = x\mathrm{d} p,$$

and taking $$f^* = xp-f,$$ we have $$\mathrm d f^* = x \mathrm{d} p,$$ which means

 * $$\frac{\mathrm{d} f^*}{\mathrm{d} p} = x.$$

When $$f$$ is a function of $$n$$ variables $$x_1, x_2, \cdots, x_n$$, then we can perform the Legendre transformation on each one or several variables: we have

 * $$\mathrm{d} f = p_1\mathrm{d} x_1 + p_2 \mathrm{d} x_2 + \cdots + p_n \mathrm{d} x_n$$

where $$p_i = \frac{\partial f}{\partial x_i}.$$ Then if we want to perform Legendre transformation on, e.g. $$x_1$$ then we take $$p_1$$ together with $$x_2, \cdots, x_n$$ as independent variables, and with Leibniz's rule we have

 * $$\mathrm{d} (f - x_1 p_1) = -x_1 \mathrm{d} p_1 + p_2 \mathrm{d} x_2 + \cdots + p_n \mathrm{d} x_n.$$

so for function $$\varphi(p_1, x_2, \cdots, x_n) = f(x_1, x_2, \cdots, x_n) - x_1 p_1,$$ we have

 * $$\frac{\partial \varphi}{\partial p_1} = -x_1,\quad \frac{\partial \varphi}{\partial x_2} = p_2,\quad \cdots,

\quad \frac{\partial \varphi}{\partial x_n} = p_n.$$

We can also do this transformation for variables $$x_2, \cdots, x_n$$. If we do it to all the variables, then we have

 * $$\mathrm{d} \varphi = -x_1 \mathrm d p_1 - x_2 \mathrm{d} p_2 - \cdots - x_n \mathrm{d} p_n $$ where $$\varphi = f-x_1 p_1 - x_2 p_2 - \cdots - x_n p_n. $$

In analytical mechanics, people perform this transformation on variables $$\dot q_1, \dot q_2, \cdots, \dot q_n $$ of the Lagrangian $$L(q_1, \cdots, q_n, \dot{q}_1, \cdots, \dot{q}_n) $$ to get the Hamiltonian:

$$H(q_1, \cdots, q_n, p_1, \cdots, p_n) = \sum_{i=1}^n p_i \dot{q}_i - L(q_1, \cdots, q_n, \dot{q}_1 \cdots, \dot{q}_n) $$

and in thermodynamics, people perform this transformation on variables according to the type of thermodynamic system they want. E.g. starting from the cardinal function of state, the internal energy $$U(S,V)$$, we have

 * $$\mathrm{d}U = T \mathrm{d} S - p \mathrm{d} V, $$

we can perform Legendre transformation on either or both of $$S, V $$ yielding

 * $$\mathrm{d} H = \mathrm{d} (U + pV) \ \ \ \ \ \ \ \ \ \ = \ \ \ \ T\mathrm{d} S + V \mathrm{d} p$$

 * $$\mathrm{d} F = \mathrm{d} (U - TS) \ \ \ \ \ \ \ \ \ \ = -S\mathrm{d} T - p \mathrm{d} V$$

 * $$\mathrm{d} G = \mathrm{d} (U - TS + pV) = -S\mathrm{d} T + V \mathrm{d} p$$

and each of these three expressions has a physical meaning.

This definition of Legendre transformation is the one originally introduced by Legendre in his work in 1787, and still applied by physicists nowadays. Indeed, this definition can be mathematically rigorous if we treat all the variables and functions defined above, e.g. $$f,x_1,\cdots,x_n,p_1,\cdots,p_n, $$ as differentiable functions defined on an open set of $$\R^n $$ or on a differentiable manifold, and $$\mathrm{d} f, \mathrm{d} x_i, \mathrm{d} p_i $$ their differentials (which are treated as cotangent vector field in the context of differentiable manifold). And this definition is equivalent to the modern mathematicians' definition as long as $$f $$ is differentiable and convex for the variables $$x_1, x_2, \cdots, x_n. $$

Properties
f^{**}(y) &{} = \left(y\cdot \bar{p} - f^{*}(\bar{p})\right)|_{(f^{*})'(\bar{p}) = y} \\[5pt] &{} = g(\bar{p})\cdot \bar{p} - f^{*}(\bar{p}) \\[5pt] &{} = g(\bar{p})\cdot \bar{p} - (\bar{p} g(\bar{p})-f(g(\bar{p})))\\[5pt] &{} = f(g(\bar{p})) \\[5pt] &{} = f(y)~. \end{align}$$Note that this derivation does not require the condition to have all positive values in double derivative of the original function $$f$$.
 * The Legendre transform of a convex function, of which double derivative values are all positive, is also a convex function of which double derivative values are all positive.Proof. Let us show this with a doubly differentiable function $$f(x)$$ with all positive double derivative values and with a bijective (invertible) derivative. For a fixed $$p$$, let $$\bar{x}$$ maximize or make the function $$px - f(x)$$ bounded over $$x$$. Then the Legendre transformation of $$f$$ is $$f^*(p) = p\bar{x} - f(\bar{x})$$, thus,$$f'(\bar{x}) = p$$by the maximizing or bounding condition $$\frac{d}{dx}(px - f(x)) = p - f'(x)= 0 $$. Note that $$\bar{x}$$ depends on $$p $$. (This can be visually shown in the 1st figure of this page above.) Thus $$\bar{x} = g(p)$$ where $$g \equiv (f')^{-1}$$, meaning that $$g$$ is the inverse of $$f'$$ that is the derivative of $$f$$ (so $$f'(g(p))= p$$). Note that $$g$$ is also differentiable with the following derivative (Inverse function rule),$$\frac{dg(p)}{dp} = \frac{1}{f(g(p))} ~.$$Thus, the Legendre transformation $$f^*(p) = pg(p) - f(g(p))$$ is the composition of differentiable functions, hence it is differentiable. Applying the product rule and the chain rule with the found equality $$\bar{x} = g(p)$$ yields$$\frac{d(f^{*})}{dp} = g(p) + \left(p - f'(g(p))\right)\cdot \frac{dg(p)}{dp} = g(p), $$giving $$\frac{d^2(f^{*})}{dp^2} = \frac{dg(p)}{dp} = \frac{1}{f(g(p))} > 0,$$so $$f^*$$ is convex with its double derivatives are all positive.
 * The Legendre transformation is an involution, i.e., $$f^{**} = f ~$$. Proof. By using the above identities as $$f'(\bar{x}) = p$$, $$\bar{x} = g(p)$$, $$f^*(p) = p\bar{x} - f(\bar{x})$$ and its derivative $$(f^*)'(p) = g(p)$$, $$\begin{align}

Identities
As shown above, for a convex function $$f(x)$$, with $$x = \bar{x}$$ maximizing or making $$px - f(x)$$ bounded at each $$p$$ to define the Legendre transform $$f^*(p) = p\bar{x} - f(\bar{x})$$ and with $$g \equiv (f')^{-1}$$, the following identities hold.

 * $$f'(\bar{x}) = p$$,
 * $$\bar{x} = g(p)$$,
 * $$(f^*)'(p) = g(p)$$.

Example 1
Consider the exponential function $$ f(x) = e^x,$$ which has the domain $$I=\mathbb{R}$$. From the definition, the Legendre transform is $$ f^*(x^*) = \sup_{x\in \mathbb{R}}(x^*x-e^x),\quad x^*\in I^*$$ where $$I^*$$ remains to be determined. To evaluate the supremum, compute the derivative of $$x^*x-e^x$$ with respect to $$x$$ and set equal to zero: $$ \frac{d}{dx} (x^*x-e^x) = x^*-e^x = 0. $$ The second derivative $$-e^x$$ is negative everywhere, so the maximal value is achieved at $$x = \ln(x^*)$$. Thus, the Legendre transform is $$ f^*(x^*) = x^*\ln(x^*)-e^{\ln(x^*)} = x^*(\ln(x^*) - 1) $$ and has domain $$I^* = (0, \infty).$$ This illustrates that the domains of a function and its Legendre transform can be different.

To find the Legendre transformation of the Legendre transformation of $$ f$$, $$ f^{**}(x) = \sup_{x^*\in \mathbb{R}}(xx^*-x^*(\ln(x^*) - 1)),\quad x\in I, $$ where a variable $$ x $$ is intentionally used as the argument of the function $$ f^{**} $$ to show the involution property of the Legendre transform as $$ f^{**} = f $$. we compute $$ \begin{aligned} 0 &= \frac{d}{dx^*}\big( xx^*-x^*(\ln(x^*) - 1) \big) = x - \ln(x^*) \end{aligned} $$ thus the maximum occurs at $$x^* = e^x$$ because the second derivative $$ \frac{d^2}{{dx^*}^2}f^{**}(x) = - \frac{1}{x^*} < 0 $$ over the domain of $$ f^{**} $$ as $$I^* = (0, \infty).$$ As a result, $$ f^{**} $$ is found as $$ \begin{aligned} f^{**}(x) &= xe^x - e^x(\ln(e^x) - 1) = e^x, \end{aligned} $$ thereby confirming that $$f = f^{**},$$ as expected.

Example 2
Let $f(x) = cx^{2}$ defined on $R$, where $c > 0$ is a fixed constant.

For $x*$ fixed, the function of $x$, $x*x − f(x) = x*x − cx^{2}$ has the first derivative $x* − 2cx$ and second derivative $−2c$; there is one stationary point at $x = x*/2c$, which is always a maximum.

Thus, $I* = R$ and $$f^*(x^*)=\frac{ {x^*}^2}{4c} ~.$$

The first derivatives of $f$, 2$cx$, and of $f *$, $x*/(2c)$, are inverse functions to each other. Clearly, furthermore, $$f^{**}(x)=\frac{1}{4 (1/4c)}x^2=cx^2~,$$ namely $f ** = f$.

Example 3
Let $f(x) = x^{2}$ for $x ∈ (I = [2, 3])$.

For $x*$ fixed, $x*x − f(x)$ is continuous on $I$ compact, hence it always takes a finite maximum on it; it follows that the domain of the Legendre transform of $$f$$ is $I* = R$.

The stationary point at $x = x*/2$ (found by setting that the first derivative of $x*x − f(x)$ with respect to $$x$$ equal to zero) is in the domain $[2, 3]$ if and only if $4 ≤ x* ≤ 6$. Otherwise the maximum is taken either at $x = 2$ or $x = 3$ because the second derivative of $x*x − f(x)$ with respect to $$x$$ is negative as $$-2$$; for a part of the domain $$x^* < 4$$ the maximum that $x*x − f(x)$ can take with respect to $$x \in [2,3]$$ is obtained at $$x = 2$$ while for $$x^* > 6$$ it becomes the maximum at $$x = 3$$. Thus, it follows that $$f^*(x^*)=\begin{cases} 2x^*-4, & x^*<4\\ \frac{{x^*}^2}{4}, & 4\leq x^*\leq 6,\\ 3x^*-9, & x^*>6. \end{cases}$$

Example 4
The function $f(x) = cx$ is convex, for every $x$ (strict convexity is not required for the Legendre transformation to be well defined). Clearly $x*x − f(x) = (x* − c)x$ is never bounded from above as a function of $x$, unless $x* − c = 0$. Hence $f*$ is defined on $I* = {c}$ and $f*(c) = 0$. (The definition of the Legendre transform requires the existence of the supremum, that requires upper bounds.)

One may check involutivity: of course, $x*x − f*(x*)$ is always bounded as a function of $x*∈{c}$, hence $I** = R$. Then, for all $x$ one has $$\sup_{x^*\in\{c\}}(xx^*-f^*(x^*))=xc,$$ and hence $f **(x) = cx = f(x)$.

Example 5
As an example of a convex continuous function that is not everywhere differentiable, consider $$f(x)= |x|$$. This gives$$f^*(x^*) = \sup_{ x }(xx^*-|x|)=\max\left(\sup_{x\ge 0} x(x^*-1), \,\sup_{x\le  0} x(x^*+1)  \right),$$and thus $$f^*(x^*)=0$$ on its domain $$I^*=[-1,1]$$.

Example 6: several variables
Let $$f(x)=\langle x,Ax\rangle+c$$ be defined on $X = R^{n}$, where $A$ is a real, positive definite matrix.

Then $f$ is convex, and $$\langle p,x\rangle-f(x)=\langle p,x \rangle-\langle x,Ax\rangle-c,$$ has gradient $p − 2Ax$ and Hessian $−2A$, which is negative; hence the stationary point $x = A^{−1}p/2$ is a maximum.

We have $X* = R^{n}$, and $$f^*(p)=\frac{1}{4}\langle p,A^{-1}p\rangle-c.$$

Behavior of differentials under Legendre transforms
The Legendre transform is linked to integration by parts, $p dx = d(px) − x dp$.

Let $f(x,y)$ be a function of two independent variables $x$ and $y$, with the differential $$df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy = p\,dx + v\,dy.$$

Assume that the function $f$ is convex in $x$ for all $y$, so that one may perform the Legendre transform on $f$ in $x$, with $p$ the variable conjugate to $x$ (for information, there is a relation $$\frac{\partial f}{\partial x} |_{\bar{x}} = p$$ where $$\bar{x}$$ is a point in $x$ maximizing or making $$px - f(x,y)$$ bounded for given $p$ and $y$). Since the new independent variable of the transform with respect to $f$ is $p$, the differentials $dx$ and $dy$ in $df$ devolve to $dp$ and $dy$ in the differential of the transform, i.e., we build another function with its differential expressed in terms of the new basis $dp$ and $dy$.

We thus consider the function $g(p, y) = f − px$ so that $$dg = df - p\,dx - x\,dp = -x\,dp + v\,dy$$ $$x = -\frac{\partial g}{\partial p}$$ $$v = \frac{\partial g}{\partial y}.$$

The function $−g(p, y)$ is the Legendre transform of $f(x, y)$, where only the independent variable $x$ has been supplanted by $p$. This is widely used in thermodynamics, as illustrated below.

Analytical mechanics
A Legendre transform is used in classical mechanics to derive the Hamiltonian formulation from the Lagrangian formulation, and conversely. A typical Lagrangian has the form

$$L(v,q)=\tfrac{1}2\langle v,Mv\rangle-V(q),$$ where $$(v,q)$$ are coordinates on $R^{n} × R^{n}$, $M$ is a positive definite real matrix, and $$\langle x,y\rangle = \sum_j x_j y_j.$$

For every $q$ fixed, $$L(v, q)$$ is a convex function of $$v$$, while $$V(q)$$ plays the role of a constant.

Hence the Legendre transform of $$L(v, q)$$ as a function of $$v$$ is the Hamiltonian function, $$H(p,q)=\tfrac {1}{2} \langle p,M^{-1}p\rangle+V(q).$$

In a more general setting, $$(v, q)$$ are local coordinates on the tangent bundle $$T\mathcal M$$ of a manifold $$\mathcal M$$. For each $q$, $$L(v, q)$$ is a convex function of the tangent space $V_{q}$. The Legendre transform gives the Hamiltonian $$H(p, q)$$ as a function of the coordinates $(p, q)$ of the cotangent bundle $$T^*\mathcal M$$; the inner product used to define the Legendre transform is inherited from the pertinent canonical symplectic structure. In this abstract setting, the Legendre transformation corresponds to the tautological one-form.

Thermodynamics
The strategy behind the use of Legendre transforms in thermodynamics is to shift from a function that depends on a variable to a new (conjugate) function that depends on a new variable, the conjugate of the original one. The new variable is the partial derivative of the original function with respect to the original variable. The new function is the difference between the original function and the product of the old and new variables. Typically, this transformation is useful because it shifts the dependence of, e.g., the energy from an extensive variable to its conjugate intensive variable, which can often be controlled more easily in a physical experiment.

For example, the internal energy $U$ is an explicit function of the extensive variables entropy $S$, volume $V$, and chemical composition $N_{i}$ (e.g., $$ i = 1, 2, 3, \ldots$$) $$ U = U \left (S,V,\{N_i\} \right ),$$ which has a total differential $$ dU = T\,dS - P\,dV + \sum \mu_i \,dN _i$$

where $$ T = \left. \frac{\partial U}{\partial S} \right \vert _{V, N_{i\ for\ all\ i\ values}}, P = \left. -\frac{\partial U}{\partial V} \right \vert _{S, N_{i\ for\ all\ i\ values}}, \mu_i = \left. \frac{\partial U}{\partial N_i} \right \vert _{S,V, N_{j\ for\ all\ j \ne i}}$$.

(Subscripts are not necessary by the definition of partial derivatives but left here for clarifying variables.) Stipulating some common reference state, by using the (non-standard) Legendre transform of the internal energy $U$ with respect to volume $V$, the enthalpy $H$ may be obtained as the following.

To get the (standard) Legendre transform $U^*$ of the internal energy $U$ with respect to volume $V$, the function $u\left( p,S,V,\{{{N}_{i}}\} \right)=pV-U$  is defined first, then it shall be maximized or bounded by $V$. To do this, the condition $\frac{\partial u}{\partial V} = p - \frac{\partial U}{\partial V} = 0 \to p = \frac{\partial U}{\partial V}$ needs to be satisfied, so $U^* = \frac{\partial U}{\partial V}V - U$  is obtained. This approach is justified because $U$ is a linear function with respect to $V$ (so a convex function on $V$) by the definition of extensive variables. The non-standard Legendre transform here is obtained by negating the standard version, so $-U^* = H = U - \frac{\partial U}{\partial V}V = U + PV$.

$H$ is definitely a state function as it is obtained by adding $PV$ ($P$ and $V$ as state variables) to a state function $ U = U \left (S,V,\{N_i\} \right )$, so its differential is an exact differential. Because of $ dH = T\,dS + V\,dP + \sum \mu_i \,dN _i$ and the fact that it must be an exact differential, $$ H = H(S,P,\{N_i\})$$.

The enthalpy is suitable for description of processes in which the pressure is controlled from the surroundings.

It is likewise possible to shift the dependence of the energy from the extensive variable of entropy, $S$, to the (often more convenient) intensive variable $T$, resulting in the Helmholtz and Gibbs free energies. The Helmholtz free energy $A$, and Gibbs energy $G$, are obtained by performing Legendre transforms of the internal energy and enthalpy, respectively, $$ A = U - TS ~,$$$$ G = H - TS = U + PV - TS ~.$$

The Helmholtz free energy is often the most useful thermodynamic potential when temperature and volume are controlled from the surroundings, while the Gibbs energy is often the most useful when temperature and pressure are controlled from the surroundings.

Variable capacitor
As another example from physics, consider a parallel conductive plate capacitor, in which the plates can move relative to one another. Such a capacitor would allow transfer of the electric energy which is stored in the capacitor into external mechanical work, done by the force acting on the plates. One may think of the electric charge as analogous to the "charge" of a gas in a cylinder, with the resulting mechanical force exerted on a piston.

Compute the force on the plates as a function of $x$, the distance which separates them. To find the force, compute the potential energy, and then apply the definition of force as the gradient of the potential energy function.

The electrostatic potential energy stored in a capacitor of the capacitance $C(x)$ and a positive electric charge $+Q$ or negative charge $-Q$ on each conductive plate is (with using the definition of the capacitance as $C = \frac{Q}{V}$ ),

$$ U (Q, \mathbf{x}) = \frac{1}{2} QV(Q,\mathbf{x}) = \frac{1}{2} \frac{Q^2}{C(\mathbf{x})},~$$

where the dependence on the area of the plates, the dielectric constant of the insulation material between the plates, and the separation $x$ are abstracted away as the capacitance $C(x)$. (For a parallel plate capacitor, this is proportional to the area of the plates and inversely proportional to the separation.)

The force $F$ between the plates due to the electric field created by the charge separation is then $$ \mathbf{F}(\mathbf{x}) = -\frac{dU}{d\mathbf{x}} ~. $$

If the capacitor is not connected to any electric circuit, then the electric charges on the plates remain constant and the voltage varies when the plates move with respect to each other, and the force is the negative gradient of the electrostatic potential energy as $$ \mathbf{F}(\mathbf{x}) = \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}} \frac{Q^2}{{C(\mathbf{x})}^2} = \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}}V(\mathbf{x})^2 $$

where $ V(Q,\mathbf{x}) = V(\mathbf{x}) $  as the charge is fixed in this configuration.

However, instead, suppose that the voltage between the plates $V$ is maintained constant as the plate moves by connection to a battery, which is a reservoir for electric charges at a constant potential difference. Then the amount of charges $ Q $ is a variable instead of the voltage; $ Q $  and $ V $  are the Legendre conjugate to each other. To find the force, first compute the non-standard Legendre transform $U^*$ with respect to $ Q $  (also with using $C = \frac{Q}{V}$ ),

$$U^* = U - \left.\frac{\partial U}{\partial Q} \right|_\mathbf{x} \cdot Q =U - \frac{1}{2C(\mathbf{x})} \left. \frac{\partial Q^2}{\partial Q} \right|_\mathbf{x} \cdot Q = U - QV = \frac{1}{2} QV - QV = -\frac{1}{2} QV= - \frac{1}{2} V^2 C(\mathbf{x}).$$

This transformation is possible because $ U $ is now a linear function of $ Q $  so is convex on it. The force now becomes the negative gradient of this Legendre transform, resulting in the same force obtained from the original function $ U $ , $$ \mathbf{F}(\mathbf{x}) = -\frac{dU^*}{d\mathbf{x}} = \frac{1}{2} \frac{dC(\mathbf{x})}{d\mathbf{x}}V^2 .$$

The two conjugate energies $ U $ and $ U^* $  happen to stand opposite to each other (their signs are opposite), only because of the linearity of the capacitance—except now $Q$ is no longer a constant. They reflect the two different pathways of storing energy into the capacitor, resulting in, for instance, the same "pull" between a capacitor's plates.

Probability theory
In large deviations theory, the rate function is defined as the Legendre transformation of the logarithm of the moment generating function of a random variable. An important application of the rate function is in the calculation of tail probabilities of sums of i.i.d. random variables, in particular in Cramér's theorem.

If $$X_n$$ are i.i.d. random variables, let $$S_n=X_1+\cdots+X_n$$ be the associated random walk and $$M(\xi)$$ the moment generating function of $$X_1$$. For $$\xi\in\mathbb R$$, $$E[e^{\xi S_n}] = M(\xi)^n$$. Hence, by Markov's inequality, one has for $$\xi\ge 0$$ and $$a\in\mathbb R$$ $$P(S_n/n > a) \le e^{-n\xi a}M(\xi)^n=\exp[-n(\xi a - \Lambda(\xi))]$$ where $$\Lambda(\xi)=\log M(\xi)$$. Since the left-hand side is independent of $$\xi$$, we may take the infimum of the right-hand side, which leads one to consider the supremum of $$\xi a - \Lambda(\xi)$$, i.e., the Legendre transform of $$\Lambda$$, evaluated at $$x=a$$.

Microeconomics
Legendre transformation arises naturally in microeconomics in the process of finding the supply $S(P)$ of some product given a fixed price $P$ on the market knowing the cost function $C(Q)$, i.e. the cost for the producer to make/mine/etc. $Q$ units of the given product.

A simple theory explains the shape of the supply curve based solely on the cost function. Let us suppose the market price for a one unit of our product is $P$. For a company selling this good, the best strategy is to adjust the production $Q$ so that its profit is maximized. We can maximize the profit $$\text{profit} = \text{revenue} - \text{costs} = PQ - C(Q)$$ by differentiating with respect to $Q$ and solving $$P - C'(Q_\text{opt}) = 0.$$

$Q_{opt}$ represents the optimal quantity $Q$ of goods that the producer is willing to supply, which is indeed the supply itself: $$S(P) = Q_\text{opt}(P) = (C')^{-1}(P).$$

If we consider the maximal profit as a function of price, $$\text{profit}_\text{max}(P)$$, we see that it is the Legendre transform of the cost function $$C(Q)$$.

Geometric interpretation
For a strictly convex function, the Legendre transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points, since a convex function is differentiable at all but at most countably many points.)

The equation of a line with slope $$p$$ and $y$-intercept $$b$$ is given by $$y = p x + b$$. For this line to be tangent to the graph of a function $$f$$ at the point $$\left(x_0, f(x_0)\right)$$ requires $$f(x_0) = p x_0 + b$$ and $$p = f'(x_0).$$

Being the derivative of a strictly convex function, the function $$f'$$ is strictly monotone and thus injective. The second equation can be solved for $x_0 = f^{\prime-1}(p),$ allowing elimination of $$x_0$$ from the first, and solving for the $$y$$-intercept $$b$$ of the tangent as a function of its slope $$p,$$ $b = f(x_0) - p x_0 = f\left(f^{\prime-1}(p)\right) - p \cdot f^{\prime-1}(p) = -f^\star(p)$  where $$f^{\star}$$ denotes the Legendre transform of $$f.$$

The family of tangent lines of the graph of $$f$$ parameterized by the slope $$p$$ is therefore given by $y = p x - f^{\star}(p),$ or, written implicitly, by the solutions of the equation $$F(x,y,p) = y + f^{\star}(p) - p x = 0~.$$

The graph of the original function can be reconstructed from this family of lines as the envelope of this family by demanding $$\frac{\partial F(x,y,p)}{\partial p} = f^{\star\prime}(p) - x = 0.$$

Eliminating $$p$$ from these two equations gives $$y = x \cdot f^{\star\prime-1}(x) - f^{\star}\left(f^{\star\prime-1}(x)\right).$$

Identifying $$y$$ with $$f(x)$$ and recognizing the right side of the preceding equation as the Legendre transform of $$f^{\star},$$ yield $f(x) = f^{\star\star}(x) ~.$

Legendre transformation in more than one dimension
For a differentiable real-valued function on an open convex subset $U$ of $R^{n}$ the Legendre conjugate of the pair $(U, f)$ is defined to be the pair $(V, g)$, where $V$ is the image of $U$ under the gradient mapping $Df$, and $g$ is the function on $V$ given by the formula $$g(y) = \left\langle y, x \right\rangle - f(x), \qquad x = \left(Df\right)^{-1}(y)$$ where $$\left\langle u,v\right\rangle = \sum_{k=1}^n u_k \cdot v_k$$

is the scalar product on $R^{n}$. The multidimensional transform can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. This can be seen as consequence of the following two observations. On the one hand, the hyperplane tangent to the epigraph of $$f$$ at some point $$(\mathbf x, f(\mathbf x))\in U\times \mathbb{R}$$ has normal vector $$(\nabla f(\mathbf x),-1)\in\mathbb{R}^{n+1}$$. On the other hand, any closed convex set $$C\in\mathbb{R}^m$$ can be characterized via the set of its supporting hyperplanes by the equations $$\mathbf x\cdot\mathbf n = h_C(\mathbf n)$$, where $$h_C(\mathbf n)$$ is the support function of $$C$$. But the definition of Legendre transform via the maximization matches precisely that of the support function, that is, $$f^*(\mathbf x)=h_{\operatorname{epi}(f)}(\mathbf x,-1) $$. We thus conclude that the Legendre transform characterizes the epigraph in the sense that the tangent plane to the epigraph at any point $$(\mathbf x,f(\mathbf x))$$ is given explicitly by$$\{\mathbf z\in\mathbb{R}^{n+1}: \,\, \mathbf z\cdot \mathbf x= f^*(\mathbf x)\}. $$

Alternatively, if $X$ is a vector space and $Y$ is its dual vector space, then for each point $x$ of $X$ and $y$ of $Y$, there is a natural identification of the cotangent spaces $T*X_{x}$ with $Y$ and $T*Y_{y}$ with $X$. If $f$ is a real differentiable function over $X$, then its exterior derivative, $df$, is a section of the cotangent bundle $T*X$ and as such, we can construct a map from $X$ to $Y$. Similarly, if $g$ is a real differentiable function over $Y$, then $dg$ defines a map from $Y$ to $X$. If both maps happen to be inverses of each other, we say we have a Legendre transform. The notion of the tautological one-form is commonly used in this setting.

When the function is not differentiable, the Legendre transform can still be extended, and is known as the Legendre-Fenchel transformation. In this more general setting, a few properties are lost: for example, the Legendre transform is no longer its own inverse (unless there are extra assumptions, like convexity).

Legendre transformation on manifolds
Let $M$ be a smooth manifold, let $$E$$ and $\pi : E\to M$  be a vector bundle on $$M$$ and its associated bundle projection, respectively. Let $L : E\to \R$ be a smooth function. We think of $L$ as a Lagrangian by analogy with the classical case where $M = \R$, $E = TM = \Reals \times \Reals $  and $L(x,v) = \frac 1 2 m v^2 - V(x)$  for some positive number $m\in \Reals$  and function $V : M \to \Reals$.

As usual, the dual of $E$ is denote by $E^*$. The fiber of $\pi$ over $x\in M$  is denoted $E_x$, and the restriction of $L$  to $E_x$  is denoted by $L|_{E_x} : E_x\to \R$. The Legendre transformation of $L$ is the smooth morphism$$\mathbf F L : E \to E^*$$ defined by $\mathbf FL(v) = d(L|_{E_x})_v \in E_x^*$, where $x = \pi(v)$. Here we use the fact that since $E_x$ is a vector space, $T_v(E_x)$  can be identified with $E_x$. In other words, $\mathbf FL(v)\in E_x^*$ is the covector that sends $w\in E_x$  to the directional derivative $\left.\frac d {dt}\right|_{t=0} L(v + tw)\in \R$.

To describe the Legendre transformation locally, let $U\subseteq M$ be a coordinate chart over which $E$  is trivial. Picking a trivialization of $E$ over $U$, we obtain charts $E_U \cong U \times \R^r$  and $E_U^* \cong U \times \R^r$. In terms of these charts, we have $\mathbf FL(x; v_1, \dotsc, v_r) = (x; p_1,\dotsc, p_r)$, where $$p_i = \frac {\partial L}{\partial v_i}(x; v_1, \dotsc, v_r)$$ for all $i = 1, \dots, r$. If, as in the classical case, the restriction of $L : E\to \mathbb R$ to each fiber $E_x$  is strictly convex and bounded below by a positive definite quadratic form minus a constant, then the Legendre transform $\mathbf FL : E\to E^*$  is a diffeomorphism. Suppose that $\mathbf FL$ is a diffeomorphism and let $H : E^* \to \R$  be the "Hamiltonian" function defined by $$H(p) = p \cdot v - L(v),$$ where $v = (\mathbf FL)^{-1}(p)$. Using the natural isomorphism $E\cong E^{**}$, we may view the Legendre transformation of $H$ as a map $\mathbf FH : E^* \to E$. Then we have $$(\mathbf FL)^{-1} = \mathbf FH.$$

Scaling properties
The Legendre transformation has the following scaling properties: For $a > 0$,

$$f(x) = a \cdot g(x) \Rightarrow f^\star(p) = a \cdot g^\star\left(\frac{p}{a}\right) $$ $$f(x) = g(a \cdot x) \Rightarrow f^\star(p) = g^\star\left(\frac{p}{a}\right).$$

It follows that if a function is homogeneous of degree $r$ then its image under the Legendre transformation is a homogeneous function of degree $s$, where $1/r + 1/s = 1$. (Since $f(x) = x^{r}/r$, with $r > 1$, implies $f*(p) = p^{s}/s$.) Thus, the only monomial whose degree is invariant under Legendre transform is the quadratic.

Behavior under translation
$$ f(x) = g(x) + b \Rightarrow f^\star(p) = g^\star(p) - b$$ $$ f(x) = g(x + y) \Rightarrow f^\star(p) = g^\star(p) - p \cdot y $$

Behavior under inversion
$$ f(x) = g^{-1}(x) \Rightarrow f^\star(p) = - p \cdot g^\star\left(\frac{1}{p} \right) $$

Behavior under linear transformations
Let $A : R^{n} → R^{m}$ be a linear transformation. For any convex function $f$ on $R^{n}$, one has $$ (A f)^\star = f^\star A^\star $$ where $A*$ is the adjoint operator of $A$ defined by $$ \left \langle Ax, y^\star \right \rangle = \left \langle x, A^\star y^\star \right \rangle, $$ and $Af$ is the push-forward of $f$ along $A$ $$ (A f)(y) = \inf\{ f(x) : x \in X, A x = y \}. $$

A closed convex function $f$ is symmetric with respect to a given set $G$ of orthogonal linear transformations, $$f(A x) = f(x), \; \forall x, \; \forall A \in G $$ if and only if $f*$ is symmetric with respect to $G$.

Infimal convolution
The infimal convolution of two functions $f$ and $g$ is defined as

$$ \left(f \star_\inf g\right)(x) = \inf \left \{ f(x-y) + g(y) \, | \, y \in \mathbf{R}^n \right \}. $$

Let $f_{1}, ..., f_{m}$ be proper convex functions on $R^{n}$. Then

$$ \left( f_1 \star_\inf \cdots \star_\inf f_m \right)^\star = f_1^\star + \cdots + f_m^\star. $$

Fenchel's inequality
For any function $f$ and its convex conjugate $f *$ Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every $x ∈ X$ and $p ∈ X*$, i.e., independent $x, p$ pairs, $$\left\langle p,x \right\rangle \le f(x) + f^\star(p).$$

title: Legendre transformation


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Chūō Main Line

The Chūō Main Line (中央本線), commonly called the Chūō Line, is one of the major trunk railway lines in Japan. It connects Tokyo and Nagoya, although it is the slowest direct railway connection between the two cities; the coastal Tōkaidō Main Line is slightly faster, and the Tōkaidō Shinkansen is currently the fastest rail link between the cities.

The eastern portion, the Chūō East Line (中央東線), is operated by the East Japan Railway Company (JR East), while the western portion, the Chūō West Line (中央西線), is operated by the Central Japan Railway Company (JR Central). The dividing point between the two companies is Shiojiri, where express trains from both operators continue north onto the Shinonoi Line towards the cities of Matsumoto and Nagano. Compared to the huge urban areas at either end of the Chūō Line, its central portion is lightly traveled; the Shiojiri-Nakatsugawa corridor is only served by one limited express and one local service per hour.

The Chūō Main Line passes through the mountainous center of Honshu. Its highest point (near Fujimi) is about 900 m above sea level and much of the line has a gradient of 25 per mil (2.5% or 1 in 40). Along the Chūō East Line section, peaks of the Akaishi and Kiso as well as Mount Yatsugatake can be seen from trains. The Chūō West Line parallels the old Nakasendō highway (famous for the preserved post towns of Tsumago-juku and Magome-juku) and the steep Kiso Valley.

Routes

 * Entire Route (Tokyo - Nagoya including branch): 424.6 km
 * East Line (Tokyo - Shiojiri): 222.1 km
 * Tokyo - Kanda: 1.3 km (officially part of the Tōhoku Main Line)
 * Kanda - Yoyogi: 8.3 km
 * Yoyogi - Shinjuku: 0.7 km (officially part of the Yamanote Line)
 * Shinjuku - Shiojiri: 211.8 km
 * East Line - Tatsuno branch line (Okaya - Tatsuno - Shiojiri): 27.7 km
 * West Line (Shiojiri - Nagoya): 174.8 km
 * Shiojiri - Kanayama: 171.5 km
 * Kanayama - Nagoya: 3.3 km (alongside Tōkaidō Main Line)

Stations and services
This section lists all stations on the Chūō Main Line and generally explains regional services on the line. In addition, there are limited express services connecting major cities along the line, namely Azusa, Super Azusa, Kaiji, Hamakaiji, Narita Express and Shinano. For details of the limited express trains, see the relevant articles.

Tokyo - Mitaka
The section between Tokyo and Mitaka is grade-separated, with no level crossings. Between Ochanomizu and Mitaka, the Chūō Main Line has four tracks; two of them are local tracks (緩行線) with platforms at every station; the other two are rapid tracks (快速線) with some stations without platforms. The local tracks are used by the Chūō-Sōbu Line local trains, while the rapid tracks carry rapid service and limited express trains. The Tokyo-Mitaka portion is a vital cross-city rail link.

The commuter services on the rapid tracks are collectively called the Chūō Line (Rapid) in comparison with the Chūō Line (Local) (中央線各駅停車) or the Chūō-Sōbu Line on the local tracks. The former is usually referred to simply as the Chūō Line and the latter the Sōbu Line. Separate groups of trainsets are used for these two groups of services: cars with an orange belt for the rapid service trains and cars with a yellow belt for the local service trains. Signs at stations also use these colors to indicate the services.

This section is located entirely within Tokyo.

Mitaka - Takao
The four-track section ends at Mitaka. Most of the section between Mitaka and Tachikawa had been elevated between 2008 and 2011 to eliminate level crossings. Plans have been proposed to add another two tracks as far as Tachikawa, but were not included in the track elevation.

Takao - Shiojiri
Most of the rapid service trains from Tokyo terminate at Takao, where the line exits the large urban area of Tokyo. The section between Takao and Ōtsuki still carries some commuter trains as well as long distance local trains and Limited Express trains. The Kaiji limited express terminates at Kōfu, the capital of Yamanashi Prefecture, while the Azusa continue beyond Shiojiri to Matsumoto via the Shinonoi Line.

All stations from Tachikawa to Shiojiri are served by the Chūō Main Line Local. Local trains from Tachikawa and Takao run as far as Matsumoto or even Nagano.

Legends:

 * ●: All trains stop
 * ▲: Stop, eastbound services only
 * ▼: Stop, westbound services only

Okaya – Shiojiri
The Okaya-Shiojiri branch is an old route of the Chūō Main Line. It carries a small number of shuttle trains and trains from/to the Iida Line, which branches off at Tatsuno.

Prior to the opening of the new route between Okaya and Shiojiri, there was a junction (Higashi-Shiojiri Junction (東塩尻信号場)) between Ono and Shiojiri stations. It had a reversing layout. The signal station was closed on 12 October 1983.

Shiojiri - Nakatsugawa
Shiojiri is the dividing point of the East Line and the West Line; no train continues from one to the other. The Shinano limited express is the main service for the rural Shiojiri-Nakatsugawa section.

Nakatsugawa - Nagoya
Local and rapid service trains run on the line from Nakatsugawa to Nagoya. This section carries urban traffic for the Greater Nagoya Area.

Local trains stop at all stations (except Sannō Junction).

Legends:
 * ● : All trains stop
 * | ↓ ↑ : All trains pass (Arrows indicate the passing direction)
 * ▼ : Only southbound trains stop
 * ▲ : Only northbound trains stop

Junctions

 * Fumonji Junction (普門寺信号場) is a junction between Chino and Kami-Suwa stations in Suwa, Nagano. It entered into use on 2 September 1970.
 * Sannō Junction (山王信号場) is a junction that diverts freight traffic from the Chūō Main Line to the Tōkaidō Line freight branch between Kanayama and Nagoya stations in Nagoya. It entered into use on 10 October 1962.

Chūō East Line (JR East)
New E233 series trains entered service on Tokyo-area commuter services from 26 December 2006. These trains are a development of the E231 series used on other commuter lines in the Tokyo area, and replaced the aging 201 series rolling stock introduced on the line in 1981.

From 2017, new E353 series EMUs were introduced on Azusa and Super Azusa limited express services, replacing the E351 and E257 series trains.

 * Chūō Rapid Line
 * E233 series
 * 209-1000 series
 * Chūō-Sōbu Line
 * E231 series
 * E231-500 series
 * Tokyo Metro Tōzai Line
 * E231-800 series
 * 05 series
 * 07 series
 * 15000 series
 * Chūō Main Line Local trains
 * 211 series
 * E127 series
 * Limited Express
 * E353 series (Azusa, Kaiji, Fuji Excursion, Hachiōji, Ōme)
 * E259 series (Narita Express) (Two round trips per day starts/terminates at Takao)
 * Seasonal services
 * 185 series (Hamakaiji)
 * 215 series (Holiday Rapid View Yamanashi)

Chūō West Line (JR Central)

 * Local Trains
 * 211 series
 * 213 series
 * 311 series
 * 313 series
 * 315 series
 * Limited Express
 * 383 series (Shinano)

Freight train

 * JNR Class EF64
 * JR Freight Class EH200

History
The Kobu Railway (甲武鉄道) opened the initial section of the Chūō Line from Shinjuku Station to Tachikawa Station in 1889. The company then extended the line both westward and eastward (towards Tokyo) until it was nationalised in 1906. The Japanese Government Railways (JGR) then continued to extend the line, reaching Shiojiri the same year, and Tokyo (at Shōheibashi Station (昌平橋駅)) in 1908. The JGR also built the line from Nagoya, the first section opening in 1900, with the lines connecting in 1911. The Table below gives the section opening dates.

In 1904, the section between Iidamachi Station (formerly located between Suidōbashi Station and Iidabashi Station) and Nakano Station was the first urban electric railway in Japan using 600 V DC. Electrification was extended in 1919 and 1922, was increased to 1,200 V DC when extended to Tokyo in 1927, boosted again to 1,500 V DC in 1929, and reached Kofu in 1931. Electrification from the Nagano end was commissioned in sections from 1966, and the entire line was electrified by 1973.

Notes:
 * The section between Okaya Station and Shiojiri Station is the new route that replaced the old route opened on June 11, 1906, by JGR.
 * Station names in parentheses are original names.
 * Stations marked † are now closed.
 * Prior to the connection of the East Line and the West Line in 1911, the section between Shiojiri Station and Miyanokoshi Station belonged to the East Line.

Former connecting lines

 * Mitaka Station: A 3 km line to a Nakajima Aircraft factory opened in 1942, and was closed in 1945. In 1950, the factory site was used to build a sports stadium. The line from Mitaka to Musashino Stadium (武蔵野競技場前) reopened on 14 April 1951, but was closed again on 1 November 1959.
 * Kokubunji Station: A 7 km line was opened in 1910 to haul gravel from the Tamagawa. It closed in 1914 due to flood damage, but was reopened in 1916 after being rebuilt by the Japanese Army. On 26 May 1920, the line was absorbed into JNR, but operations were suspended from 1 December 1921. A 6 km extension to the Tokyo Racecourse opened on 1934. Services on the line were suspended from 1 October 1944, resuming from 24 April 1947. On 1 April 1973, the line to Tokyo Racecourse closed and the line was absorbed into the Musashino Line.
 * Kofu Station: The Yamanashi Horse-drawn tramway opened its first gauge section in 1898, and by 1904 had opened two lines (to Katsunuma and Fujikawa) totaling 34 km. In 1930, the Katsunuma Line was closed, and the other line was closed beyond Kai-Aoyagi, 20 km from Kofu. The company renamed itself the Yamanashi Electric Railway, regauged (to 1,067 mm) and electrified the line at 600 V DC, and operated it until 1962.
 * Sakashita Station: The 11 km gauge Sakagawa Line was opened to Maruno by the Hisaka River Railway in 1926. A passenger service was operated 8 km to Okuya. The Forest Service opened a 9 km line connecting at Maruno the same year, and a 2 km branch from Okuya that operated from 1933 until 1958. In 1944, the Forest Service took over the Sakagawa line, operating it until 1961, when the entire 20 km line closed.
 * Nakatsugawa Station: The Kitaena Railway operated the 23 km Enaden Line to Tsukechi, electrified at 600 V DC, from 1924 until 1978. At Tsukechi, it transshipped timber from a gauge forest railway with an 18 km "main line" and a 14 km and two 5 km branch lines operated from 1932 until 1959.
 * Ena Station: The Iwamura Electric Railway operated a 13 km line electrified at 600 V DC to its namesake town between 1906 and 1935. A 4 km line to the site of Oi dam was opened in 1922 to transport construction materials. Upon the dam's completion, the line was sold to the Kita-Ena Railway. but it closed in 1934.
 * Tokishi Station: The Ogawa Railway opened a 10 km line to its namesake town between 1922 and 1924. The line was electrified at 1,500 V DC in 1950, and closed as a result of flood damage in 1972.
 * Yabuhara Station: The Ogiso Forest line operated for an unknown period.
 * Agematsu Station: The Otaki Forest Railway operated between 1911 and 1975.
 * Nojiri Station: The Nojiri Forest Railway operated for an unknown period.
 * Tajima Station: The Kasahara Railway opened a 5 km line to its namesake town in 1928. Passenger services ceased in 1971, and the line closed in 1978.

Proposed connecting lines

 * Chino Station: The Saku Railway, which had built the line from Komoro on the Shinetsu Line to Koumi, proposed to build a line from Tanaka on the Shinetsu Line to this station. The company was nationalised before construction started, and JGR connected the Koumi line to the Chuo Main Line in 1935, making this proposal redundant.

Accidents
On September 12, 1997, a Super Azusa limited express bound for Matsumoto collided with a 201 series local train that failed to stop at a red signal while passing through Ōtsuki Station.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Chūō Main Line


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List of honorary societies

This is a list of honorary societies to which individuals are elected based on meritorious conduct.

Private academies and societies

 * American Philosophical Society
 * Phi Alpha Theta
 * Phi Beta Kappa Society
 * Phi Theta Kappa

Brazil

 * Brazilian Academy of Sciences

Bulgaria

 * Bulgarian Academy of Sciences

People's Republic of China

 * Chinese Academy of Sciences
 * Chinese Academy of Social Sciences
 * Chinese Academy of Engineering

Croatia

 * Croatian Academy of Sciences and Arts

France

 * Institut de France, which groups five academies:
 * Académie française
 * Académie des inscriptions et belles-lettres
 * Académie des sciences
 * Académie des beaux-arts
 * Académie des sciences morales et politiques

Georgia

 * Georgian Academy of Sciences

India

 * Ibn Sina Academy of Medieval Medicine and Sciences

Iran

 * Academy of Gundishapur
 * Academy of Persian Language and Literature
 * Imperial Iranian Academy of Philosophy
 * Iranian Academy of Medical Sciences
 * Academy of Sciences of Iran
 * Iranian Academy of the Arts
 * National Iranian Olympic Academy

Russia

 * Russian Academy of Science
 * Russian Academy of Arts
 * Russian Academy of Theatre Arts
 * Russian Academy of Music
 * Russian Academy of Architecture and Construction Sciences

Serbia

 * Serbian Academy of Sciences and Arts

Slovenia

 * Slovenian Academy of Sciences and Arts

Sweden

 * Swedish Royal Academies

Taiwan (Republic of China)

 * Academia Sinica

Thailand

 * The Royal Institute

United States

 * American Academy of Arts and Sciences
 * National Academy of Medicine
 * National Academy of Sciences
 * National Academy of Engineering
 * Kohl McCormick Academy of Outstanding Educators

United Kingdom

 * Royal Society
 * British Academy
 * Royal Society of Medicine
 * Royal Society of Arts

International academies

 * World Academy of Art and Science (WAAS)

title: List of honorary societies


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Emergency Committee of Atomic Scientists

The Emergency Committee of Atomic Scientists (ECAS) was founded by Albert Einstein and Leó Szilárd in May, 1946, primarily as a fundraising and policy-making agency. Its aims were to warn the public of the dangers associated with the development of nuclear weapons, promote the peaceful use of nuclear energy, and ultimately work towards world peace, which was seen as the only way that nuclear weapons would not be used again.

History
The Committee was established in the wake of the "Szilárd petition" (July 1945) to United States president Harry S. Truman opposing the use of the atomic bomb on moral grounds, which was signed by 70 scientists who had worked on the Manhattan Project. A majority of scientists working on the Manhattan Project did not know entirely what they were creating at the time.

Einstein called on seven other atomic scientists to join the effort, including Linus Pauling, R.F. Bacher, Hans A. Bethe, Edward U. Condon, Thorfin R. Hogness, Harold C. Urey, and V.F. Weisskopf. They were later joined by Selig Hecht, Harrison Brown, and H.J. Mueller.

Board of Trustees
The Committee only ever consisted of the eight members of the Board of Trustees, who were:
 * Albert Einstein Chairman
 * Harold C. Urey Vice-Chairman
 * Hans Bethe
 * Thorfin R. Hogness
 * Philip M. Morse
 * Linus Pauling
 * Leó Szilárd
 * Victor Weisskopf

Half the members had worked directly on the Manhattan Project and all had been indirectly involved or consulted on the production of the first atomic bomb.

Set of principles
"These facts are accepted by all scientists:

 * 1) Atomic bombs can now be made cheaply and in large number. They will become more destructive.
 * 2) There is no military defense against atomic bombs and none is to be expected.
 * 3) Other nations can rediscover our secret processes by themselves.
 * 4) Preparedness against atomic war is futile, and if attempted, will ruin the structure of our social order.
 * 5) If war breaks out, atomic bombs will be used and they will surely destroy our civilization.
 * 6) There is no solution to this problem except international control of atomic energy, and ultimately, the elimination of war.

The program of the committee is to see that these truths become known to public."

Objective
"The Committee's immediate objective was to raise a substantial fund ($1 million) to support educational activities by the atomic scientists, aimed at civilian national and international control of future nuclear energy activities."

Activities
The Committee arranged private contributions to support atomic information and education. To facilitate this, ECAS was incorporated in New Jersey on August 6, 1946. ECAS directed the received contributions towards other groups such as National Committee for Atomic Information (NCAI), Association of Scientists for Atomic Education (ASAE), the Federation of American Scientists (FAS) and the Atomic Scientists of Chicago (ASC) of Bulletin of the Atomic Scientists (BAS). These groups aimed to educate both the general public and the U.S. government about the potential positive and negative aspects of atomic energy. They also conducted extensive educational initiatives.

Several members of the committee gave lecture tours to promote the committee's message of peace. They produced supporting promotional materials, including one of the first films to illustrate what a full nuclear war might be like. Einstein and ECAS was also very vocal in its opposition of the development of the first hydrogen bomb.

Baruch Plan
Apart from fundraising, these programs aimed to garner backing for initiatives such as the Baruch Plan, which was presented to the United Nations in 1946 and proposed international control of the atomic bombs. The plan was passed by the United Nations Atomic Energy Commission (UNAEC), but not agreed to by the Soviet Union.

World Government
While committee initially saw success in fundraising efforts, Einstein and his colleagues grew increasingly convinced that the world was veering off course. They arrived at the conclusion that the gravity of the situation demanded more profound actions and the establishment of a "world government" was the only logical solution. In his "Open Letter to the General Assembly of the United Nations" of October 1947, Einstein emphasized the urgent need for international cooperation and the establishment of a world government. In the year 1948, Einstein invited United World Federalists, Inc.(UWF) president Cord Meyer to a meeting of ECAS and joined UWF as a member of the Advisory Board. Einstein and ECAS assisted UEF in fundraising and provided supporting material. Einstein described United World Federalists as: "the group nearest to our aspirations". Einstein and other prominent figures sponsored the Peoples' World Convention (PWC), which took place in 1950-51. Their collaboration paid-off when Thane Read from UWF initiated a worldwide call for the World Constitutional Convention. Members of the ECAS signed the call which lead to the creation of a world constitution, for a democratic federal world government in world constituent assemblies of 1968, 1977, 1978-79, and 1991. These efforts were successful in creating the Constitution for the Federation of Earth and a Provisional Earth Federation with a Provisional World Parliament.

Committee dissolution and legacy
At the end of 1948, with the gradual deterioration of international relations, failure of the United Nations Atomic Energy Commission (UNAEC) in 1947 and escalating international tensions after 1947, committee recognized that:

"'education of mankind toward a clear understanding of the implications of atomic energy, and full appreciation of the dangers and hopes inherent in new discoveries, is a long-range task which cannot be solved on an emergency basis.'"

On the question of disposition of the funds, Einstein wrote Harrison Brown on June 12, 1951:

"'When our committee was formed it was undoubtedly our purpose to use our influence to help achieve lasting security in the international field. It is true that the Bulletin of the Atomic Scientists tried in the beginning to serve the same purpose, but today it has become no more than a publication of neutral information. Secondly, there are other organizations working for some kind of world government; in this country, the United World Federalists are the group nearest to our aspirations. But I would prefer, according to Szilard's suggestion, to give the money to the Friends, for they have shown by their steadfast efforts through many years a truly supranational attitude, which was manifest long before the present difficult situation arose.'"

The proposal put forth by Einstein and Szilard did not ultimately gain acceptance. Instead, the committee's assets were transferred to the Bulletin of the Atomic Scientists. Final dissolution of ECAS was done on September 8, 1951 in a meeting at Einstein's Home and officially dissolved on October 10, 1951.

ECAS was active for four years, until 1950 when it was gradually disbanded, although most of the members continued to campaign against nuclear war, and participated in the development of the Constitution for the Federation of Earth and the Pugwash Conferences on Science and World Affairs.


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Tubular Bells

Tubular Bells is the debut studio album by the British musician Mike Oldfield, released on 25 May 1973 as the first album on Virgin Records. It comprises two mostly instrumental tracks. Oldfield, who was 19 years old when it was recorded, played almost all the instruments.

Tubular Bells initially sold slowly, but gained worldwide attention in December 1973 when its opening theme was used for the soundtrack to the horror film The Exorcist. This led to a surge in sales which increased Oldfield's profile and played an important part in the growth of the Virgin Group. It stayed in the top ten of the UK Albums Chart for one year from March 1974, during which it reached number one for one week. It reached number three on the US Billboard 200, and number one in Canada and Australia. It has sold more than 2.7 million copies in the UK, and an estimated 15 million copies worldwide.

An orchestral version produced by David Bedford was released in 1975 as The Orchestral Tubular Bells. It was followed by the albums Tubular Bells II (1992), Tubular Bells III (1998), The Millennium Bell (1999), and a re-recorded version, Tubular Bells 2003, for its 30th anniversary. A remastered edition was released in 2009. In 2010, Tubular Bells was one of ten classic album covers from British artists commemorated on a series of UK postage stamps issued by the Royal Mail. Its contribution to British music was recognised when Oldfield played extracts during the 2012 Summer Olympics opening ceremony in London.

Background
Oldfield learned to play the guitar at an early age, and as a teenager he became the bass player for the Whole World, a band put together by Kevin Ayers, formerly of Soft Machine. The Whole World recorded their album Shooting at the Moon (1970) at Abbey Road Studios over several months in 1970, when Oldfield was 17. When the group did not have a recording session booked in the morning, Oldfield would arrive early and experiment with the different instruments, including pianos, harpischords, a Mellotron and various orchestral percussion instruments, and learned to play each of them.

The Whole World broke up in mid-1971 and Ayers lent Oldfield a two-track Bang & Olufsen Beocord ¼" tape recorder. Oldfield blocked off the erase head of the tape machine, which allowed him to record onto one track, bounce the recording onto the second, and record a new instrument onto the first track, thus overdubbing his playing one instrument at a time, and effectively making multitrack recordings.

In his flat in Tottenham in north London, Oldfield recorded demos of four tracks he had been composing in his head for some years, using the tape recorder, his guitar and bass, some toy percussion instruments, and a Farfisa organ borrowed from the Whole World keyboardist David Bedford. The demos comprised three shorter melodies (early versions of what would become the sections "Peace", "Bagpipe Guitars", and "Caveman" on Tubular Bells 2003), and a longer piece he had provisionally titled "Opus One". Oldfield was inspired to write a long instrumental after hearing Septober Energy (1971), the only album by Centipede. He was also influenced by classical music, and by A Rainbow in Curved Air (1969) by the experimental composer Terry Riley, on which Riley played all the instruments himself and used tape loops and overdubs to build up a long, repetitive piece of music.

Late in 1971, Oldfield joined the band of Arthur Louis, who were recording demos at the Manor Studio. The studio was being constructed in the former squash court of an old manor house in Shipton-on-Cherwell, Oxfordshire, which had recently been bought by the young entrepreneur Richard Branson and which was being turned into a residential recording facility run by his music production team of Tom Newman and Simon Heyworth. Oldfield was shy and socially awkward, but struck up a friendship with the producers after they heard his guitar playing. Oldfield asked Newman to listen to his demos, but they were in his Tottenham flat, so one of Louis' roadies drove Oldfield to London and back to retrieve them. Newman and Heyworth made a copy of the demos onto 4-track tape, and promised Oldfield that they would speak to Branson and his business partner Simon Draper about them. After the album was released, Newman said he preferred the demo versions: "They were complete melodies in themselves – with intros and fade-outs or ends. I liked them very much and was a little nonplussed when Mike strung them all together."

Oldfield spent much of 1972 working with his old bandmates from the Whole World on their solo projects while trying to find a record label interested in his demos. Oldfield approached labels including EMI and CBS, but each rejected him, believing the piece was unmarketable without vocals. Increasingly frustrated and short of money, Oldfield heard that the Soviet Union paid musicians to give public performances, and was at the point of looking through the telephone directory for the phone number of the Soviet embassy when Draper called him with an invitation to dinner with Branson on Branson's houseboat moored in London. Branson told Oldfield that he liked the demos, and wanted Oldfield to spend a week at the Manor recording "Opus One".

Recording
Tubular Bells was recorded on an Ampex 2-inch 16-track tape recorder with the Dolby noise-reduction system, which was the Manor's main recording equipment at the time. Oldfield had Virgin hire instruments including guitars, keyboards and percussion instruments. Oldfield has recounted differing stories over the years regarding the inclusion of the tubular bells; in 2001 he suggested that they were among the instruments he asked Branson to hire, but in 2013, he said that he saw them among the instruments being removed from the studios after John Cale had finished recording there, and asked for them to be left behind.

Oldfield, Newman, and Heyworth spent their evenings drinking in a pub, after which they returned to the Manor and recorded through the night. Heyworth recalled several disasters, including one instance where half a day's work was accidentally erased. Final mixing was an involved process, with the faders operated by Oldfield, Newman, Heyworth, and two others simultaneously. They followed detailed tracking charts and the process was restarted if one person made even a slight mistake. Heyworth recalled difficulty in cutting the album due to vinyl's limited dynamic range, and insisted on heavy vinyl normally used for classical records.

Oldfield played the majority of the instruments as a series of overdubs, which was an uncommon recording technique at the time. In total, 274 overdubs were made and an estimated two thousand "punch-ins", although Newman said "it was really only 70 or 80" in total. Despite various guitars being listed on the album sleeve, such as "speed guitars", "fuzz guitars" and "guitars sounding like bagpipes", the only electric guitar used on the album was a 1966 blonde Fender Telecaster which used to belong to Marc Bolan and to which Oldfield had added an extra Bill Lawrence pickup. The guitars were recorded via direct injection into the mixing desk. To create the "speed guitar" and "mandolin-like guitar" named in the sleeve notes, the tape was recorded at half speed. An actual mandolin was used only for the ending of Part Two. Oldfield also used a custom effects unit, the Glorfindel box, to create the "fuzz guitars" and "bagpipe guitars" distortion. In 2011, Oldfield's Telecaster was sold for £6,500, and the money was donated to the mental health charity SANE. According to the engineer Phil Newell, the bass guitar used on the album was one of his Fender Telecaster Basses.

Side one
Oldfield recorded side one, known as "Opus One" at the time, during his one allotted week at the Manor in November 1972. He was particularly interested in starting the piece with a repeating riff, and devised the opening piano sequence after experimenting with an idea for several minutes on Bedford's Farfisa organ. He wanted a slight variation on its 16/8 time signature by dropping the sixteenth beat, and chose the key of A minor as it was easy to play. Oldfield recorded the opening riff on a Steinway grand piano, but struggled to perform in time. Heyworth solved the problem by placing a microphone next to a metronome in another room and feeding it into Oldfield's headphones. The short honky-tonk piano section was included as a tribute to Oldfield's grandmother, who had played the instrument in pubs before World War II. The staff and workers at the Manor made up the "nasal choir" that accompanies it. Oldfield had difficulty in producing a sound from the tubular bells, as he wanted a loud note from them but both the standard leather-covered and bare metal hammers did not produce the volume that he wanted. In the end, Newman obtained a heavier claw hammer and Oldfield used it to produce the desired sound intensity but cracked the bells in the process.

The track closes with a segment featuring Vivian Stanshall, formerly of the comedic rock group Bonzo Dog Doo-Dah Band, introducing each instrument being played one by one. The idea originated when the band were due to use the Manor after Oldfield, and had arrived while he was still recording. Oldfield had liked the way Stanshall introduced the instruments one at a time on the Bonzos' song "The Intro and the Outro" on Gorilla (1967), and told Newman that he would like Stanshall to do the same. Newman agreed, but had to persuade the shy Oldfield to ask Stanshall if he would carry out the request. Stanshall readily agreed to the idea and is credited on the liner notes as "Master of Ceremonies", but Newman recalled that the job proved to be more difficult than anticipated, as Stanshall forgot the names of the instruments and introduced them at the wrong points. Oldfield wrote a list of the instruments in order, indicating where Stanshall should introduce them. The way in which Stanshall said "plus... tubular bells" inspired Oldfield to use it as the album's title.

Side two
After Part One had been recorded, Oldfield was allowed to stay on at the Manor to record additional overdubs during studio downtime. He spent Christmas and New Year at his family's home, but returned to the Manor from February to April 1973 to record the second part of his planned album. Oldfield had "Part Two" mapped out and sequenced by the time he came to record it.

The "caveman" section is the only part of Tubular Bells that features a drum kit, which is played by Steve Broughton of the Edgar Broughton Band. The section begins with a backing track of bass and drums, with Oldfield overdubbing all other instruments. The shouting vocals developed near the end of the recording, when he had practically finished recording the instruments for the section but felt it needed something else. Heyworth recalled that Branson was getting impatient and pressured Oldfield to deliver the album, and to include vocals on one of the tracks so he could release it as a single. Angered by Branson's suggestion, Oldfield returned to the Manor where he drank half a bottle of Jameson's whiskey from the studio's cellar and demanded that the engineer take him to the studio where, intoxicated, he "screamed his brains out for 10 minutes" into a microphone. The incident left Oldfield so hoarse that he was unable to speak for two weeks. The engineer ran the tape at a higher speed during the recording, so that upon playback the tape ran at normal speed, thus dropping the pitch of the voice track and producing the "Piltdown Man" vocals listed on the credits.

Side two closes with a rendition of "The Sailor's Hornpipe", a track Oldfield had been performing since he was in the Whole World. It was originally preceded by a longer version of the piece, featuring a vocal contribution from Stanshall over musical backing and marching footsteps. This session occurred at 4 a.m. after Oldfield, Stanshall, and Newman had spent the night drinking. Newman placed microphones in various rooms of the Manor and began recording, and the trio set off on an unplanned tour of the house, with Oldfield on mandolin and Newman on acoustic guitar playing "The Sailor's Hornpipe" while Stanshall gave an inebriated, improvised tour of the Manor. In the end, a more traditional instrumental version of the tune was put on the album although Stanshall's version was included on the Boxed compilation. It is also found on the 2001 and 2009 remasters as a bonus track.

Artwork
The cover of Tubular Bells was created by designer and photographer Trevor Key, who was suggested by Sue Steward, a press officer at Virgin Records at the time. Key was invited to present his portfolio, and one of his designs depicted a boiled egg with blood dripping from it, which Branson liked and wanted to use for the cover because he wanted to call the album Breakfast in Bed. Oldfield hated both the image and the title and rejected them. A modified version of the image, with the blood replaced by egg yolk, was used as the cover for Heaven's Open (1991), Oldfield's final album for Virgin.

Steward accompanied Key to a beach on the Sussex coast to photograph the cover's backdrop. Key brought with him bones shown burning on the beach on the back cover, but the day was bitterly cold and it took some time to set light to them. The perfectionist Key also spent several hours photographing the seascape until he had a shot of the waves that he was happy with. The triangular "bent bell" on the front was inspired by the damage Oldfield had caused to the tubular bells while playing them on the record. Key designed and constructed one, which he then photographed in his studio and superimposed on the beach backdrop. Oldfield was captivated by the finished artwork, and insisted that his name and the album title be in small letters and coloured pale orange, so as not to distract from the overall image. According to Steward, Key was paid £100 for his work, but he went on to design several other sleeves for Virgin and Factory Records artists, including Technique (1989) by New Order and "Genetic Engineering" (1983) by Orchestral Manoeuvres in the Dark.

The "bent bell" has become the image most associated with Oldfield, appearing on the cover of every Tubular Bells sequel album. It is also the logo of his personal music company, Oldfield Music Ltd. The cover of Tubular Bells was among ten images chosen by Royal Mail for a set of "Classic Album Cover" postage stamps, issued in January 2010.

Release
In January 1973, Branson visited the MIDEM music conference in Cannes, France and pitched side one to various music companies with the hope of securing a record deal. One American executive offered $20,000 if vocals were added to the music. The unsuccessful visit led Branson and Draper to consider putting the album out through mail order, before they chose to form their own label, Virgin Records, and use Tubular Bells as their first release. The album was released in the UK on 25 May 1973, and in North America in October 1973. Early sales were slow, and it was not until July 1973 that the album appeared in the UK Albums Chart, reaching an initial peak of number seven. The situation changed following the release of The Exorcist in December 1973, Oldfield later attributing the music's successful use on the soundtrack to its unusual 15/8 opening time signature. From February 1974 to May 1975, Tubular Bells dropped out of the UK top 10 for only four weeks. Sixteen months after its release, it went to number one for the week ending 5October 1974, having spent 10 consecutive weeks in second place behind Band on the Run (1973) by Wings, and Oldfield's second album Hergest Ridge (1974).

In surpassing Hergest Ridge for the number one album, Oldfield became only the second artist in history to replace himself at the top of the UK album chart, after the Beatles in 1963, and again in 1964. The feat was later achieved by Michael Jackson and David Bowie, although on both occasions this followed the death of the artist. Prior to the 2020s, the album had re-entered the charts in every decade since its release, most recently appearing in the week ending 22 March 2018 (its 287th week in total).

On 22 April 2007, British newspaper The Mail on Sunday gave away 2.25 million free copies of Tubular Bells to its readers in a card packet displaying the artwork. The release was organised by EMI, who had bought out Virgin Records, and the newspaper claimed that its promotion increased sales of the album by 30%. Oldfield was unhappy about the deal, as he had not been consulted about it and felt it devalued the work.

Tubular Bells has sold more than 2.63 million copies in the UK, and an estimated 15 million worldwide. As of July 2016, it was the 42nd best-selling album of all time in the UK.

Singles
The first single released from the album was created by the original US distributor, Atlantic Records. The single was an edit of the first three sections from Part One and was not authorised by Oldfield. The single was released in February 1974 in the United States and Canada only, where it peaked at number seven on the U.S. Billboard Hot 100 chart on 11 May 1974, making Oldfield a one-hit wonder on the US charts. The track also reached number 15 on the Adult Contemporary chart. In Canada, the single was released as "Tubular Bells (Theme from Exorcist)", peaking at number three on the RPM Top Singles chart on 18 May 1974, and was placed at number 103 in the top 200 singles of the year.

"Mike Oldfield's Single (Theme from Tubular Bells)" was the first 7-inch single released by Oldfield in the UK, in June 1974, peaking at number 31. The A-side was a re-recording of Part Two's "bagpipe guitars" section, arranged in a more pastoral version with acoustic guitars and featuring the oboe (played by Lindsay Cooper) as the lead instrument, with "Froggy Went A-Courting" as the B-side.

2009 reissue
In 2008, Oldfield's original 35-year deal with Virgin Records ended and the rights to the album returned to him. After signing to Mercury Records in 2005, Oldfield's albums originally released on Virgin were transferred to Mercury and re-released, starting the following year. Tubular Bells was reissued in June 2009 in a number of formats, including vinyl, 2-CD and DVD, and includes a new stereo mix by Oldfield in March 2009 from his home studio in Nassau, Bahamas. The Deluxe Edition contains a 5.1 surround sound mix and the Ultimate Edition contains an accompanying book and memorabilia.

The release was promoted by a series of bell-ringing events at 6 p.m. on 6 June 2009, a reference to the Number of the Beast. One of the events was held at the British Music Experience at The O2, featuring the 29-piece Handbell Ringers of Great Britain and a performance by The Orb named "Orbular Bells". There were also bell-ringing workshops and competitions at the Experience. The album reached number 11 in the UK.

2023 reissue
A 50th anniversary edition of the album was released on 26 May 2023 on CD and half-speed mastered 2LP. It features a new master of the original album along with additional previously released tracks including the 2012 Olympics recording, the Tubular Beats recording with York, "Tubular X" (from The X-Files), and one new 8-minute track, the "Introduction to Tubular Bells 4" which was recorded by Oldfield as a demo in 2017. Oldfield's record label has indicated that after recording the 8-minute demo he decided not to go forward with the Tubular Bells 4 project, and that this "may well be the last piece ever to be recorded by Oldfield". A Blu-ray audio version was also being produced which includes new Dolby Atmos and Stereo mixes by David Kosten as well as Mike Oldfield's 2009 5.1 mix and the 1975 Quadrophonic mix by Phil Newell.

Queen Elizabeth Hall, 1973
After recording Tubular Bells, Oldfield felt he had "got it out of his system" and was reluctant to do any promotional engagements. However, Branson and Draper felt a live performance of the work was the best way to present it to journalists and the public, and organised a concert at London's Queen Elizabeth Hall on 25 June 1973. Oldfield was convinced that the music would not come off well in a live setting, and despite several well-known and accomplished musicians agreeing to the one-off show, refused to do it. Desperate to prevent Oldfield pulling out, Branson offered to give Oldfield his Bentley car, which he had bought from George Harrison and knew Oldfield liked, if he went through with the gig. Oldfield agreed, and following rehearsals at Shepperton film studios, Oldfield still had reservations about the performance until Mick Taylor of The Rolling Stones brought frontman Mick Jagger backstage to meet Oldfield, whose support gave Oldfield the confidence to "at least walk on stage." The concert went ahead and was well received, leaving Oldfield "stunned".

The concert featured members of Henry Cow and musicians associated with the Canterbury scene, as well as Mick Taylor of the Rolling Stones. Steve Winwood and Robert Wyatt were also due to take part, but Winwood pulled out as he was unable to find time to attend the rehearsals, and Wyatt was recovering from the recent accident that had left him paralysed.

Musicians:

 * Kevin Ayers – bass guitar
 * David Bedford – grand piano, accordion, organ, choir master, string arrangements
 * Steve Broughton – drums
 * Jon Field – flute
 * Fred Frith – electric guitar, bass guitar
 * John Greaves – Davoli electric piano, Farfisa organ, tin whistle, Vox organ
 * Nick Haley – violin
 * Tim Hodgkinson – Farfisa organ, Fender Rhodes electric piano, Vox organ
 * Steve Hillage – electric guitar
 * Simon Ingram Hill – cello, organ
 * Geoff Leigh (misspelt on the programme as "Jeff Leig") – flute

 * Ashley Mason – viola
 * Pierre Moerlen – cymbals, glockenspiel, gongs, tam-tam, tubular bells, timpani
 * Tom Newman – nasal chorus
 * Mike Oldfield – acoustic and electric guitars, bass, Lowrey organ, mandolin, "prehistoric poem"
 * Terry Oldfield – flute
 * Ted Speight – electric guitar
 * Vivian Stanshall – master of ceremonies
 * Mick Taylor – electric guitar
 * Janet Townley – violin
 * Vulpy – viola

 * Girlie Chorus: Sarah Greaves, Kathy Williams, Sally Oldfield, Maureen Rossini, Lynette Asquith, Amanda Parsons, Maggie Thomas, Mundy Ellis, Julie Clive, Liz Gluck, Debbie Scott, Hanna Corker.

BBC TV 2nd House, 1973
Oldfield and many of the musicians who had taken part in the Queen Elizabeth Hall concert performed Part One again later in the year for the BBC arts programme 2nd House, this time as a pre-recorded performance in a studio setting without an audience. The performance was recorded on 30 November 1973 and transmitted on BBC2 on 5 January 1974. The arrangement included a new part for oboe, played by Soft Machine's Karl Jenkins, and accompanied on-screen visuals of tubular steel sculptures and sequences from the film Reflections, both created by artist William Pye. The performance was released on the Elements (2004) DVD and as part of the Deluxe and Ultimate Editions of the 2009 reissue of Tubular Bells.

Musicians:

 * Jon Field – flute
 * Fred Frith – bass, guitar
 * John Greaves – keyboards
 * Steve Hillage – guitar
 * Tim Hodgkinson – keyboards
 * Karl Jenkins – oboe
 * Geoff Leigh – flute
 * Pierre Moerlen (credited as "Pierre de Strasbourg" in the Radio Times programme listing) – percussion

 * Tom Newman – voice (nasal choir) and mixing
 * Mike Oldfield – bass, guitar, Hammond organ
 * Terry Oldfield – flute
 * Mike Ratledge – keyboards
 * Ted Speight – guitar, bass
 * Mick Taylor – guitar
 * Choir (unknown)

Reception
Influential British DJ John Peel was an early admirer of the record, and played it on his Top Gear radio show on BBC Radio 1 on 29 May 1973, four days after the album's release, calling it "one of the most impressive LPs I've ever had the chance to play on the radio, really a remarkable record". Branson and Oldfield were listening to the show on Branson's houseboat, and Oldfield stated in his autobiography that Peel played the album in its entirety, although the running order from the BBC archives and existing audio copies of the programme show that Peel played Part One only.

Peel reviewed the album for The Listener magazine the following week, describing it as "a new recording of such strength and beauty that to me it represents the first break-through into history that any musician has made". The UK's major music magazines were also unanimous in their praise of the album. Al Clark of NME said that the "veritable orgy of over-dubbing results in a remarkable piece of sustained music, never content with the purely facile yet equally disinclined towards confusing the listener". He concluded that "Tubular Bells ... is a superlative record which owes nothing to contemporary whims. It is one of the most mature, vital, rich and humerous [sic] pieces of music to have emerged from the pop idiom." Melody Maker's Geoff Brown observed that "Tubular Bells is a vast work, almost classical in its structure and in the way a theme is stated and deftly worked upon" and that it was "an enjoyable, evocative album which bodes well for the future of both the country's newest label and of Mike Oldfield".

Reviewing the whole batch of Virgin's first album releases in Sounds, Steve Peacock named Tubular Bells the best, saying that after careful listening he "ended up convinced that it really is a remarkable album", noting the "complex, interlocking carefully woven music that works its way through an enormous dynamic and emotional range", and stating, "I can't think of another album that I'd as unhesitatingly recommend to everyone who's likely to read this". A more reserved review came from Simon Frith in Let It Rock who felt that Tubular Bells was "more than an attractive wall-paper, more than a nature-film score, because of Mike Oldfield's ability to make what happens to the music self-sufficient and satisfying", but questioned why Peel and other critics viewed the album as rock music, and concluded that "Oldfield's concern is the sound of rock, but Tubular Bells lacks rock's other essence — energy. This is no way body music — no sex, no violence, no ecstasy; nothing uncontrolled, nothing uncontrollable."

Paul Gambaccini wrote an enthusiastic review of the album for Rolling Stone, calling it "the most important one-shot project of 1973" and "a debut performance of a kind we have no right to expect from anyone. It took Mike Oldfield half a year to lay down the thousands of overdubs required for his 49 minutes of exhilarating music. I will be playing the result for many times that long." He concluded, "I can say that this is a major work". On the other hand, in an article in the same magazine seven months later which discussed the current top twenty albums on the Billboard chart, Jon Landau dismissed the record as "a clever novelty" and said, "Light, rather showy and cute in places, it probably makes pleasant background music for a dinner or conversation". Writing for Creem, Robert Christgau was also left unmoved, saying, "The best I can come up with here is 'pleasant' and 'catchy'. Oldfield isn't Richard Strauss or even Leonard Cohen — this is a soundtrack because that's the level at which he operates."

In a retrospective review for AllMusic Mike DeGagne called the album "arguably the finest conglomeration of off-centered instruments concerted together to form a single, unique piece" and stated that "the most interesting and overwhelming aspect of this album is the fact that so many sounds are conjured up, yet none go unnoticed, allowing the listener a gradual submergence into each unique portion of the music. Tubular Bells is a divine excursion into the realm of new-age music."

Accolades
Oldfield won the 1974 Grammy Award for Best Instrumental Composition. The album was inducted into the Grammy Hall of Fame in 2018.

In Q magazine's 1998 list of "The 50 Best Albums of the '70s", Tubular Bells was placed at number six. In the Q & Mojo Classic special issue Pink Floyd & The Story of Prog Rock in 2005, the album was listed at number nine in its list of "40 Cosmic Rock Albums". The album was also included in the book 1001 Albums You Must Hear Before You Die.

The Exorcist
The introduction to Part One was chosen to feature in the 1973 horror film The Exorcist. According to the British film critic Mark Kermode, the decision to include the music was the result of chance – the director, William Friedkin, had decided to discard the original score by Lalo Schifrin and was looking for music to replace it. Friedkin was visiting the offices of Ahmet Ertegun, the president of Atlantic Records (which distributed Tubular Bells in the US), and picking up a white label of the album from the selection of records in Ertegun's office, he put it on the record player and instantly decided that the music would be perfect for the film. Although the introduction only features briefly in two scenes, it has become the track most commonly associated with the film. Oldfield said he did not want to see the film because he believed he would find it too frightening.

Sequels and other albums
Tubular Bells remains the album most identified with Oldfield, and he has released three sequels. Tubular Bells II was released in 1992 which, like its predecessor, reached number one in the UK. It was followed by the electronic and dance-oriented Tubular Bells III (1998) and The Millennium Bell (1999). On the thirtieth anniversary of Tubular Bells, Oldfield re-recorded the original Tubular Bells with contemporary technology, making several corrections to what he saw as flaws in the album's original production. Since Stanshall died in 1995, the re-recording features new narration provided by actor John Cleese. Tubular Bells 2003 went to number 51 in the UK.

In 1975, an orchestral arrangement of the original album was released as The Orchestral Tubular Bells.

Compilations:
 * The Best of Tubular Bells (2001)
 * The Complete Tubular Bells (2003)

Oldfield and York's 2013 remix album Tubular Beats contains two remixes of sections of Tubular Bells.

Virgin Group
""I never thought that the word 'tubular bells' was going to play such an important part in our lives ... Virgin going into space most likely wouldn't have existed if we hadn't hired that particular instrument.""

- Richard Branson

Richard Branson recognised the significance of Tubular Bells to the Virgin Group's success, who named one of his first Virgin America aircraft, an Airbus A319-112, N527VA Tubular Belle. Prior to this Virgin Atlantic had named a Boeing 747-4Q8, G-VHOT Tubular Belle, in 1994.

In the United Kingdom Virgin Money signalled its entry into the banking sector in January 2012 with a television advertisement titled '40 Years of Better'. The advertisement opened with an image of a record orbiting the Earth accompanied by the music of the introduction to Tubular Bells, signifying the beginnings of Virgin, and ended with a shot of the same record framed and hanging on the wall of the new bank. Two months later a Virgin Media TV advertisement starring Branson and actor David Tennant also featured the record, where a younger version of Branson is seen holding a copy of Tubular Bells under his arm upon exiting a time machine. However, the advert was withdrawn shortly afterwards following objections from the BBC that it was being used to endorse a rival TV service (in the advert Tennant is shown searching on Virgin's TiVo on-demand service for episodes of Doctor Who, a BBC series in which he formerly played the titular character).

In May 2021, Virgin Orbit, the commercial rocket launch provider subsidiary of the Virgin Group, announced the first operational mission of their LauncherOne air-launched rocket would be named after the first track of the album, Tubular Bells, Part One.

Cultural references
The use of the opening theme in the 1973 film The Exorcist gained the record considerable publicity and introduced the work to a broader audience. Along with a number of other Oldfield pieces the theme was used in the 1979 NASA movie The Space Movie. It has gained cultural significance as a "haunting theme", partly due to the association with The Exorcist, and has been sampled by many other artists.

In television it was used in several episodes of the Dutch children's series Bassie en Adriaan, and an episode ("Ghosts") of the BBC series My Family. It was used in a television advertisement for the Volkswagen Golf Diesel in 2002 and in various films.

Computer tie-ins
With the aid of the software house CRL and distributor Nu Wave, Mike Oldfield released an interactive Commodore 64 version of the album in 1986, which used the computer's SID sound chip to play back a simplified re-arrangement of the album, accompanied by some simple 2D visual effects. The "interactivity" offered by the album/program was limited to controlling the speed and quantity of the visual effects, tuning the sound's volume and filtering, and skipping to any part of the album.

In 2004, Oldfield launched a virtual reality project called Maestro which contains music from Tubular Bells 2003. The original title of the game was The Tube World. This was the second game which was released under the MusicVR banner, the first being Tres Lunas. MusicVR set out to be a real-time virtual reality experience combining imagery and music, as a non-violent and essentially a non-goal driven game.

In 2012 Universal and Indaba Music created a Tubular Bells remix contest, where users could download original stem recordings to create their own pieces and the winner of the $1,000 prize was judged by Oldfield.

2012 Olympic Games
On 27 July 2012 at the 2012 Summer Olympics opening ceremony Mike Oldfield performed during a segment about the NHS and children’s literature. 600 dancers, all of whom were NHS staff, along with 1,200 volunteers recruited from British hospitals, along with people from the Great Ormond Street Hospital, entered along with children on 320 hospital beds, some of which functioned as trampolines. The show's director Danny Boyle stated that he had wanted to make Tubular Bells a "cornerstone" of a 20-minute sequence of the ceremony. A studio version of Oldfield's performance appears on the soundtrack album Isles of Wonder. Although listed as "Tubular Bells"/"In Dulci Jubilo", the track consists of a number of parts, the first being the introduction piece to his Tubular Bells in its normal arrangement, then this is followed by a rearranged version of that same theme that during interviews Oldfield has called "swingular bells". The piece that is used when children's literature villains appear features two arrangements of "Far Above the Clouds" (from Tubular Bells III), and finally as the Mary Poppins characters appear to drive off the villains, there is a rendition of "In Dulci Jubilo" followed by a short coda.

The Olympics version was released as a 500-copy limited edition pink/blue vinyl single on 8 October 2012. This was also released on iTunes as "Tubular Bells/In Dulci Julio (Music from the Opening Ceremony of the London 2012 Olympic Games)" as well as on the official 2012 Olympics album. In 2023, this rendition appeared on the 50th anniversary release of Tubular Bells.

This lists the movements as:
 * 1) "Tubular Bells (Part One Excerpt)"
 * 2) "Tubular Bells (Part One Swing)"
 * 3) "Tubular Bells (Part Two Excerpt)"
 * 4) "Tubular Bells III (Far Above the Clouds)"
 * 5) "Mary Poppins Arrival"
 * 6) "Fanfare for the Isles of Wonder"
 * 7) "In Dulci Jubilo"
 * 8) "Olympic Tubular Bells Coda"

Adaptations
American artist Tori Amos has frequently used the opening Tubular Bells theme in her live shows. It began during the 1996 Dew Drop Inn Tour where she let "Father Lucifer" segue into Tubular Bells on the piano while singing words from Bronski Beat's "Smalltown Boy" as well as playing it on the harpsichord during songs "Love Song" (a Cure cover) and "Bells for Her" (from the album Under the Pink), usually while mixing in lyrics from a third song such as Björk's "Hyperballad" or "Blue Skies". It appeared again in 2005 as part of "Yes, Anastasia", and on the 2007 tour promoting her album American Doll Posse where it was performed with full band as an intro to "Devils and Gods". On the 2011 tour, promoting her album Night of Hunters it is being performed as the intro to and backing melody for "God."

Tubular Bells for Two is a music-theatre production created by two Australian multi-instrumentalists, Aidan Roberts and Daniel Holdsworth, in 2009. The two musicians perform over twenty instruments to recreate the original album 'as faithfully as physically possible'. The show won a Sydney Fringe Award for Best Musical Moment in the 2010 Festival, and has been performed at festivals around Australia and the Pacific. The show made its European debut at the Edinburgh Festival Fringe in 2012, where it won two awards.

In 2013, Oldfield invited Branson to the opening of St. Andrew's International School of The Bahamas, where two of Oldfield's children were pupils. This was the occasion of the debut of Tubular Bells for Schools, a piano solo adaptation of Oldfield's work.

The music from the 1973 film was used in a 1982 Hungarian traffic safety film. (Caution! Railroad crossing!) from 9:20.

Track listing
All music by Mike Oldfield, except on "Tubular Bells, Part Two" which includes "The Sailor's Hornpipe" (traditional).

Side one
 * 1) "Tubular Bells, Part One" – 25:30

Side two
 * 1) "Tubular Bells, Part Two" – 23:22

Personnel
Credits are adapted from album sleeve notes.

 * Mike Oldfield – grand piano, glockenspiel, Farfisa organ, bass guitar, electric guitar (including "speed guitar", "fuzz guitar", "mandolin-like guitar" and "guitars sounding like bagpipes"), taped motor drive amplifier organ chord, assorted percussion, acoustic guitar, flageolet, honky tonk piano, Lowrey organ, tubular bells, concert tympani, Hammond organ, Spanish guitar, vocals ("Piltdown Man" and "Moribund chorus")

Additional musicians
 * Steve Broughton – drums
 * Lindsay Cooper – string basses
 * Jon Field – flutes
 * Mundy Ellis – backing vocals ("Girlie Chorus")
 * Sally Oldfield – backing vocals ("Girlie Chorus")
 * Vivian Stanshall – Master of Ceremonies
 * Nasal Choir (uncredited)
 * Bootleg Chorus – the Manor Choir conducted by Mike Oldfield

Production
 * Mike Oldfield – producer, 2009 stereo and 5.1 surround sound mix
 * Simon Heyworth – producer, engineer, mastering
 * Tom Newman – producer, engineer
 * Trevor Key – artwork

title: Tubular Bells


text: Talk:H. P. Lovecraft

title: Talk:H. P. Lovecraft